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Name ________________________________________ Date __________________ Class __________________ LESSON 1-1 Solving Equations Practice and Problem Solving: A/B Use the guess-and-check method to solve. Show your work. 1. x + 8 = 11 2. 5y − 9 = 16 ________________________________________ ________________________________________ Solve by working backward. Show your work. 3. x − 4 = 9 4. 3y + 4 = 10 ________________________________________ ________________________________________ Solve the equation by using the Properties of Equality. 5. 6c + 3 = 45 6. 11 − a = −23 ________________________________________ 7. ________________________________________ 2 1 +y = 3 4 8. ________________________________________ 7 w = 14 8 ________________________________________ Solve. 9. Houston, Texas has an average annual rainfall about 5.2 times that of El Paso, Texas. If Houston gets about 46 inches of rain, about how many inches does El Paso get? Round to the nearest tenth. _________________________________________________________________________________________ 10. Susan can run 2 city blocks per minute. She wants to run 60 blocks. How long will it take her to finish if she has already run 18 blocks? _________________________________________________________________________________________ 11. Michaela pays her cell phone service provider $49.95 per month for 500 minutes. Any additional minutes used cost $0.15 each. In June, her phone bill is $61.20. How many additional minutes did she use? _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 1 Name ________________________________________ Date __________________ Class __________________ LESSON 1-1 Solving Equations Practice and Problem Solving: C Use the guess-and-check method to solve. Show your work. 1. 26 = t − 19 2. w − 2 = −43 ________________________________________ ________________________________________ Solve by working backward. Show your work. 3. 8n + 6 = 46 4. 15 − 3y = −3 ________________________________________ ________________________________________ Solve the equation by using the Properties of Equality. 5. 2(8 + k) = 22 6. m + 5(m − 1) = 7 ________________________________________ ________________________________________ 7. −13 = 2b − b − 10 8. ________________________________________ 2 5 x − x = 26 3 8 ________________________________________ Solve. 9. Sam is moving into a new apartment. Before he moves in, the landlord asks that he pay the first month’s rent and a security deposit equal to 1.5 times the monthly rent. The total that Sam pays the landlord before he moves in is $3275. What is the monthly rent? _________________________________________________________________________________________ 10. Mr. Rodriguez invests half his money in land, a tenth in stocks, and a twentieth in bonds. He puts the remaining $35,000 in his savings account. What is the total amount of money that Mr. Rodriguez saves and invests? _________________________________________________________________________________________ 11. A work crew has a new pump and an old pump. The new pump can fill a tank in 5 hours, and the old pump can fill the same tank in 7 hours. Write and solve an equation for the time it will take both pumps to fill one tank if the pumps are used together. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 2 Name ________________________________________ Date __________________ Class __________________ LESSON 1-1 Solving Equations Practice and Problem Solving: Modified Use the guess-and-check method to solve, using the suggested numbers as your guesses. The first one is done for you. 1. g + 3 = 9 Guess: 5 2. g + 3 = 9 Guess: 6 Check. Does 5 + 3 = 9? No. Check. Does ____ + 3 = 9? _______ 3. m + 7 = 15 Guess: 6 4. m + 7 = 15 Guess: 8 Check. Does ____ + 7 =15? _______ Check. Does ____ + 7 = 15? _______ Use the steps below to work backward to solve each equation. The first step is done for you. r − 10 = 8 5. If you get 8 after taking away 10, then r is 10 greater than 8. 6. 10______________ than 8 means 10 ____ 8. 7. r = 10 ____ 8 = _______ x−6=3 8. If you get ____ after taking away ____, then x is 6______________ than ____. 9. ____ ______________ than ____ means ______________. 10. x = ____ + ____ = ____ Solve each equation by using the Properties of Equality. The first one is done for you. 11. 4h = 12 12. b + 2 = 38 4h 12 = 4 4 h=3 ________________________________________ ________________________________________ 13. −2d = 6 14. 10 = y − 5 ________________________________________ ________________________________________ Solve. 15. The sales tax rate in Virginia is 4.5%. This is 2.5% less than the sales tax rate in Rhode Island. What is Rhode Island’s sales tax rate? Show your work. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 3 Name ________________________________________ Date __________________ Class __________________ LESSON 1-2 Modeling Quantities Practice and Problem Solving: A/B Use ratios to solve the problems. The diagram below represents a tree and a mailbox and their shadows. The heights of the triangles represent the heights of the objects, and the longer sides represent their shadows. 1. What is the height of the tree? ______________ Use the diagram below for 2–5. 2. If 1 cm represents 10 m, what are the actual measurements of the gym including the closet? ____________________________________________________ 3. What are the actual measurements of the closet? ___________________ 4. If 1 cm represents 12 m, what are the actual measurements of the gym including the closet? ______________ 5. What is the area of the gym? ______________ Solve. Selena rides her bicycle to work. It takes her 15 minutes to go 3 miles. 6. If she continues at the same rate, how long will it take her to go 8 miles? _________________________________________________________________________________________ 7. How many feet will she travel in 3 minutes? _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 6 Name ________________________________________ Date __________________ Class __________________ LESSON 1-2 Modeling Quantities Practice and Problem Solving: C Use a ruler to measure the distance to solve. 1. What is the distance between Bakerstown and Denton? _________________________________________________________________________________________ 2. What is the distance between Bakerstown and Colesville? _________________________________________________________________________________________ 3. If Sarah drives 55 miles per hour, how long will it take her to drive from Amityville to Denton? _________________________________________________________________________________________ 4. Hector drives 60 miles an hour from Amityville to Eaglecroft. If it takes him 5 hours and 45 minutes, what is the distance between the two cities? _________________________________________________________________________________________ 5. If the scale of the map changes, and the new distance between Amityville and Denton is 325 miles, what is the new scale? _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 7 Name ________________________________________ Date __________________ Class __________________ LESSON 1-2 Modeling Quantities Practice and Problem Solving: Modified Use the proportional figures below for problems 1−4. The first one is done for you. 1. Rectangle A’s width is 4 m and its height is 2 m. Write the ratio of width to height of Rectangle A. 4m 2m _________________________________________________________________________________________ 2. Rectangle B’s height is 1 m and its width is unknown. Use a variable to write the ratio of height to width. _________________________________________________________________________________________ 3. Write a proportion. A height B height = A width B width 4. What is the width of Rectangle B? _________________________________________________________________________________________ Solve. 5. You know that 1 hour has 60 minutes. How many minutes are in 3 hours? _________________________________________________________________________________________ 6. If Karen can ride her bike 10 miles in 1 hour, how far can she ride in 2 hours? _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 8 Name ________________________________________ Date __________________ Class __________________ LESSON 1-3 Reporting with Precision and Accuracy Practice and Problem Solving: A/B Identify the more precise measurement. 1. 16 ft; 6 in. ________________________ 4. 9.3 mg; 7.05 mg ________________________ 2. 4.8 L; 2 mL 3. 4 pt; 1 gal _______________________ 5. 74 mm; 2.25 cm ________________________ 6. 12 oz; 11 lb _______________________ ________________________ Find the number of significant digits in each example. 7. 52.9 km ________________________ 10. 0.6 mi ________________________ 8. 800 ft 9. 70.09 in. _______________________ 11. 23.0 g ________________________ 12. 3120.58 m _______________________ ________________________ Order each list of units from most precise to least precise. 13. yard, inch, foot, mile 14. gram, centigram, kilogram, milligram ________________________________________ ________________________________________ Rewrite each number with the number of significant digits indicated in parentheses. 15. 12.32 lb (2) ________________________ 16. 1.8 m (1) 17. 34 mi (4) _______________________ ________________________ Solve. 18. A rectangular garden has length of 24 m and width of 17.2 m. Use the correct number of significant digits to write the perimeter of the garden. _________________________________________________________________________________________ 19. Kelly is making a beaded bracelet with beads that measure 4 mm and 7.5 mm long. If the bracelet is 15 cm long and Kelly uses the same number of each type bead, about how many beads will she use? _________________________________________________________________________________________ 20. When two people each measured a window’s width, their results were 79 cm and 786 mm. Are these results equally precise? Explain. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 11 Name ________________________________________ Date __________________ Class __________________ LESSON 1-3 Reporting with Precision and Accuracy Practice and Problem Solving: C Choose the most precise measurement in each set. 1. 7.0 cm; 700 cm; 7000 cm ________________________ 2. 30 cm; 30 m; 32 mm 3. 9.5 lb; 0.1 oz; 4 oz _______________________ ________________________ For each measurement, find the number of significant digits. 4. 800 kg ________________________ 5. 20.0594 km 6. 0.0009 mm _______________________ ________________________ Rewrite each number with the number of significant digits indicated in parentheses. 7. 0.09 mL (2) ________________________ 8. 5280 ft (1) 9. 9.006 g (3) _______________________ ________________________ Solve. 10. Explain how someone could say the following: “I used to think that 17 and 17.0 were the same. But now I am beginning to wonder.” _________________________________________________________________________________________ _________________________________________________________________________________________ 11. As part of an experiment, a student combined 3.4 g of one chemical with 0.56 g of a second chemical. He then recorded the combined mass as 4 g. Did the student record the combined mass correctly? Explain. _________________________________________________________________________________________ _________________________________________________________________________________________ 12. Building lumber is labeled according to the dimensions (in inches) of its cross section. So, a “two-by-four” measures 2 inches by 4 inches, but not exactly. In fact, the cross section of a two-by-four has the smallest dimensions possible, while still legitimately being called a two-by-four. Find those dimensions. Then find the percent by which the cross-sectional area of a two-by-four is less than that of a “true” two-by-four. _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 12 Name ________________________________________ Date __________________ Class __________________ LESSON 1-3 Reporting with Precision and Accuracy Practice and Problem Solving: Modified Determine which is the more precise measurement in each pair. The first one is done for you. 1. 32 ft; 32 yd 32 ft ________________________ 2. 4 lb; 4.3 lb 3. 23 cm; 23 mm _______________________ ________________________ Find the number of significant digits in each measurement. The first one is done for you. 4. 8.0 g 2 ________________________ 5. 539 mi 6. 2.67 _______________________ ________________________ Order each list of units from most precise to least precise. The first one is done for you. 7. liter, milliliter, kiloliter 8. pound, ton, ounce milliliter, liter, kiloliter ________________________________________ ________________________________________ 9. cup, gallon, quart 10. gram, kilogram, centigram ________________________________________ ________________________________________ Rewrite each number with the number of significant digits indicated in parentheses. The first one is done for you. 11. 583 mi (2) 580 mi ________________________ 12. 24.89 oz (2) 13. 6.22 sec (2) _______________________ ________________________ A rectangle measures 3 m in length and 2.4 m in width. Follow the steps to find the minimum and maximum possible areas. The first step is done for you. 14. Minimum length = 3 − 0.5 = 2.5 m and maximum length = 3 + 0.5 = 3.5 m 15. Minimum width = 2.4 − ____ = ____ m and maximum width = 2.4 + ____ = ____ m 16. Minimum area = _________________________ and maximum area = _________________________ Solve. 17. A carton of milk is labeled 64 oz. If that measurement is correct, what is the greatest amount of milk there could be in the carton? _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 13 Name ________________________________________ Date __________________ Class __________________ LESSON 2-1 Modeling with Expressions Practice and Problem Solving: A/B Identify the terms and coefficients of each expression. 1. 4a + 3c + 8 2. 9b + 6 + 2g 3. 8.1f + 15 + 2.7g terms: ____________ terms: ____________ terms: ____________ coefficients: ________ coefficients: ________ coefficients: ________ 4. 7p − 3r + 6 − 5s 5. 3m − 2 − 5n + p 6. 4.6w − 3 + 6.4x − 1.9y terms: ____________ terms: ____________ terms: ____________ coefficients: ________ coefficients: ________ coefficients: ________ Interpret the meaning of the expression. 7. Frank buys p pounds of oranges for $2.29 per pound and the same number of pounds of apples for $1.69 per pound. What does the expression 2.29p + 1.69p represent? _________________________________________________________________________________________ 8. Kathy buys p pounds of grapes for $2.19 per pound and one pound of kiwi for $3.09 per pound. What does the expression 2.19p − 3.09 represent? _________________________________________________________________________________________ Write an expression to represent each situation. 9. Eliza earns $400 per week plus $15 for each new customer she signs up. Let c represent the number of new customers Eliza signs up. Write an expression that shows how much she earns in a week. _________________________________________________________________________________________ 10. Max’s car holds 18 gallons of gasoline. Driving on the highway, the car uses approximately 2 gallons per hour. Let h represent the number of hours Max has been driving on the highway. Write an expression that shows how many gallons of gasoline Max has left after driving h hours. _________________________________________________________________________________________ 11. A man’s age today is three years less than four times the age of his oldest daughter. Let a represent the daughter’s age. Write an expression to represent the man’s age. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 17 Name ________________________________________ Date __________________ Class __________________ LESSON 2-1 Modeling with Expressions Practice and Problem Solving: C Simplify each expression when you can. Then identify the terms and coefficients of each. 1. 5b + 6d − 5c + 19a 2. 4w − 5 + 6(2x + 7) − 19 terms: ____________________________ terms: ___________________________ coefficients: ________________________ coefficients: _______________________ 3. 12 + 8r − 3(s − 5) + 15t 4. 9g − 2(−h + 3j) + 7 − 8k terms: _____________________________ terms: ___________________________ coefficients: ________________________ coefficients: _______________________ Write a situation that could be represented by the expression. 5. 3a + 6, where a = age in years _________________________________________________________________________________________ 6. 5(p + 2), where p = the number of points scored _________________________________________________________________________________________ Write an expression for each situation. Then solve the problem. 7. A man’s age today is 2 years more than three times the age his son will be 5 years from now. Let a represent the son’s age today. Write an expression to represent the man’s age today. Then find his age if his son is now 8 years old. _________________________________________________________________________________________ 8. Let n represent an even integer. Write an expression for the sum of that number and the next three even integers after it. Simplify your expression fully. _________________________________________________________________________________________ 9. A Fahrenheit temperature, F, can be converted to its corresponding Celsius temperature by subtracting 32° from that temperature and then 5 multiplying the result by . Write an expression that can be used to 9 convert Fahrenheit temperatures to Celsius temperatures. Then find the Celsius temperature corresponding to 95 °F. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 18 Name ________________________________________ Date __________________ Class __________________ LESSON 2-1 Modeling with Expressions Practice and Problem Solving: Modified Identify the terms of each expression. The first one is done for you. 1. 6b + 4 + 3c 2. 7 + 5p + 4r + 6s 6b, 4, 3c ________________________________________ ________________________________________ 3. 7.3w + 2.8v + 1.4 4. 12m + 16n + 5p + 16 ________________________________________ ________________________________________ Identify the terms of each expression. Rewrite the expression if necessary. The first one is done for you. 5. 3(a + 2b) + 5c 6. 7f − 2(g + 3h) + 8 3a, 6b, 5c ________________________________________ ________________________________________ Identify the coefficients in each expression. The first one is done for you. 7. 2f − 6g + 3h − 5 8. 4a + 3b + 6 − 6c 2, −6, 3 ________________________________________ ________________________________________ 9. 4m + 2n − 7p + 5q 10. 3w − 4x − 6y + 9z ________________________________________ ________________________________________ Write an expression for each situation. The first one is done for you. 11. The Blue Team scored two more than five times the number of points, p, scored by the Red Team. 5p + 2 _________________________________________________________________________________________ 12. The Green Team scored seven fewer points, p, than the Orange Team scored. _________________________________________________________________________________________ 13. The Red Team scored three more points, p, than the Brown Team scored. _________________________________________________________________________________________ 14. The Yellow Team scored five times the number of points, p, scored by the Blue Team plus six. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 19 Name ________________________________________ Date __________________ Class __________________ LESSON 2-2 Creating and Solving Equations Practice and Problem Solving: A/B Write an equation for each description. 1. 4 times a number is 16. 2. A number minus 11 is 12. ________________________________________ 3. ________________________________________ 9 times a number plus 6 is 51. 10 4. 3 times the sum of 1 of a number and 3 8 is 11. ________________________________________ ________________________________________ Write and solve an equation to answer each problem. 5. Jan’s age is 3 years less than twice Tritt’s age. The sum of their ages is 30. Find their ages. _________________________________________________________________________________________ 6. Iris charges a fee for her consulting services plus an hourly rate that is 1 1 times her fee. On a 7-hour job, Iris charged $470. What is her fee 5 and her hourly rate? _________________________________________________________________________________________ 7. When angles are complementary, the sum of their measures is 90 degrees. Two complementary angles have measures of 2x − 10 degrees and 3x − 10 degrees. Find the measures of each angle. _________________________________________________________________________________________ 8. Bill wants to rent a car. Rental Company A charges $35 per day plus $0.10 per mile driven. Rental Company B charges $25 per day plus $0.15 per mile driven. After how many miles driven will the price charged by each company be the same? _________________________________________________________________________________________ 9. Katie, Elizabeth, and Siobhan volunteer at the hospital. In a week, Katie volunteers 3 hours more than Elizabeth does and Siobhan volunteers 1 hour less than Elizabeth. Over 3 weeks, the number of hours Katie volunteers is equal to the sum of Elizabeth’s and Siobhan’s volunteer hours in 3 weeks. Complete the table to find out how many hours each person volunteers each week. Volunteer Volunteer Hours per week Volunteer Hours over 3 weeks Katie Elizabeth Siobhan Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 22 Name ________________________________________ Date __________________ Class __________________ LESSON 2-2 Creating and Solving Equations Practice and Problem Solving: C Write an equation for each description. 1. Eight times the difference of a number and 2 is the same as 3 times the sum of the number and 3. _________________________________________________________________________________________ 2. The sum of −7 times a number and 8 times the sum of the number and 1 is the same as the number minus 7. _________________________________________________________________________________________ 3. The quotient of the difference of a number and 24 divided by 8 is the same as the number divided by 6. _________________________________________________________________________________________ Write an equation for each situation. Then use the equation to solve the problem. 4. Sierra has a total of 61 dimes and quarters in her piggybank. She has 3 more quarters than dimes. The coins have a total value of $10.90. How many dimes and how many quarters does she have? [Hint: Use the decimal values of the c coins to write an equation.] _________________________________________________________________________________________ 5. Penn used the formula for the sum of the angles inside a polygon: Sum of the interior angles = (n − 2)180, where n is the number of angles of the polygon. Penn’s answer is 1,980 degrees. How many angles does the polygon have? _________________________________________________________________________________________ 6. Fahrenheit temperature, F, can be found from a Celsius temperature, C, using the formula F = 1.8C + 32. Write an equation to find the temperature at which the Fahrenheit and Celsius readings are equal. Then find that temperature. _________________________________________________________________________________________ 7. Amanda, Bryan, and Colin are in a book club. Amanda reads twice as many books as Bryan per month and Colin reads 4 fewer than 3 times as many books as Bryan in a month. In 4 months, the number of books Amanda reads is 5 equal to the sum of the number of books 8 Bryan and Colin read in 4 months. How many books does each person read each month? Name Books read in 1 month Books read in 4 months Amanda Bryan Colin Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 23 Name ________________________________________ Date __________________ Class __________________ LESSON 2-2 Creating and Solving Equations Practice and Problem Solving: Modified Write an equation for each description. The first one is done for you. 1. A number plus 9 is 15. 2. A number minus 11 is 3. x + 9 = 15 ________________________________________ ________________________________________ 3. 2 times a number is 12. 4. A number divided by 5 is 15. ________________________________________ ________________________________________ Solve each equation. The first one is done for you. 5. d + 23 = 40 d = 17 ________________________ 8. −3z = −21 ________________________ 6. m − 5 = −13 7. 10p = 50 _______________________ 9. w + 9 = 4 ________________________ 10. v + 13 = 19 _______________________ ________________________ Write and solve an equation to solve each problem. The first one is done for you. 11. The perimeter of a square is 44 centimeters. Find the length of each 4s = 44; 11 centimeters side of the square.__________________________________________________________ 12. Pilar wants to save $100. So far, she has saved $63. How much more does Pilar need to save?_____________________________________________________ 13. The price of a bookcase, including 8% sales tax, is $378. What is the price of the bookcase, before sales tax?_________________________________________ 14. Wendi worked 32 hours in all from Monday to Friday. She worked 7 hours each day from Monday through Thursday. How many hours did Wendi work on Friday? ___________________________________________________ 15. Chad bought 8 pounds of strawberries for $25.52. What is the cost, c, of 1 pound of strawberries? __________________________________________________ 16. The area of a rectangle is 117 square centimeters. The width of the rectangle is 9 centimeters. What is the length of the rectangle? ______________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 24 Name ________________________________________ Date __________________ Class __________________ LESSON 2-3 Solving for a Variable Practice and Problem Solving: A/B Solve the equation for the indicated variable. 1. x = 3y for y 2. m + 5n = p for m ________________________ 4. 21 = cd + e for d ________________________ 3. 12r − 6s = t for r _______________________ 5. h = 15 for j j ________________________ 6. _______________________ f −7 = h for f g ________________________ Solve the formula for the indicated variable. 7. Formula for the perimeter of a rectangle: P = 2a + 2b, for b 8. Formula for the circumference of a circle: C = 2π r, for r ________________________________________ ________________________________________ 9. Formula for the sum of angles of a triangle: A + B + C = 180°, for C ________________________________________ 10. Formula for the volume of a cylinder: V = π r 2h, for h ________________________________________ Solve. 11. Jill earns $15 per hour babysitting plus a transportation fee of $5 per job. Write a formula for E, Jill’s earnings per babysitting job, in terms of h, the number of hours for a job. Then solve your formula for h. _________________________________________________________________________________________ 12. A taxi driver charges a fixed rate of r to pick up a passenger. In addition, the taxi driver charges a rate of m for each mile driven. Write a formula to represent T, the total amount this taxi driver will charge for a trip of n miles. _________________________________________________________________________________________ 13. Solve your formula from Problem 12 for m. Then find the taxi driver’s hourly rate if his pickup rate is $2 and he charges $19.50 for a 7-mile trip. _________________________________________________________________________________________ 14. Describe when the formula for simple interest I = prt would be more useful if it were rearranged. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 27 Name ________________________________________ Date __________________ Class __________________ LESSON 2-3 Solving for a Variable Practice and Problem Solving: C Solve the equation for the indicated variable. 1. y = 3 (x + 4) for x 8 2. ab − ac = 2 for a ________________________ 3. h − j = 4(h + j) − 7 for h _______________________ 4. n = m 2 − ( n + 3) 5. ________________________ d −e = e, for d 3d + e _______________________ ________________________ 6. q − 6 = q, for r r ________________________ Solve the formula for the indicated variable. 7. Formula for centripetal force: mv 2 F = , for m r 8. Formula for the volume of a sphere: 4 V = π r 3 , for r 3 ________________________________________ ________________________________________ 9. Formula for half the volume of a right 10. Formula for focal length: 1 1 1 = + , for U V F U circular cylinder: V = π r h , for r 2 2 ________________________________________ ________________________________________ 11. Pythagorean Theorem a 2 + b 2 = c 2 : 12. Formula for the surface area of for a a cone: S = π rs + π r 2 , for s ________________________________________ ________________________________________ Solve. 1 , the mass of an object, m, 2 and the square of its velocity, v. Write a formula for kinetic energy. Then solve your formula for v. 13. Kinetic energy, K, equals the product of _________________________________________________________________________________________ 14. In a circle, area and circumference can be found using the formulas A = π r 2 and C = 2π r , respectively. Write a formula for C in terms of A. (Your answer should not contain π.) _________________________________________________________________________________________ 15. Gina paid $131 for a car stereo on sale for 30% off. There was also 7% sales tax on the purchase. Find the original price of the stereo. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 28 Name ________________________________________ Date __________________ Class __________________ LESSON 2-3 Solving for a Variable Practice and Problem Solving: Modified Solve the equation for the indicated variable. The first one is done for you. 1. y = x − 7 for x 2. K = L + 9 for L x=y+7 ________________________ 4. r = 0.75s for s 3. c = 12d for d _______________________ 5. w = ________________________ 1 for v v 6 ________________________ 6. G = 3j + 4 for j _______________________ ________________________ Solve the formula for the indicated variable. The first one is done for you. 7. Formula for the perimeter of a quadrilateral: P = a + b + c + d, for b b=P−a−c–d ________________________________________ 8. Formula for velocity: d for d v = t ________________________________________ 9. Formula for the area of a triangle: 1 A = bh , for b 2 10. Formula for the volume of a prism: V = lwh, for w ________________________________________ ________________________________________ Solve. The first one is done for you. 11. The area of a parallelogram, A, equals the product of its base, b, and its height, h. Write a formula for the area of a parallelogram. Then find the height of a parallelogram whose area is 24 square centimeters and base is 4 centimeters. A = bh; h = 6 centimeters _________________________________________________________________________________________ 12. A person’s hourly pay rate, r, is found by dividing the total amount paid, p, by the number of hours worked, w. Write a formula for a person’s hourly pay rate. Then solve the formula for w. _________________________________________________________________________________________ 13. An equilateral triangle is a 3-sided polygon in which all sides have the same length, s. Write a formula for the perimeter, P, of an equilateral triangle. Then find the length of a side in an equilateral triangle whose perimeter is 72 inches. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 29 Name ________________________________________ Date __________________ Class __________________ LESSON 2-4 Creating and Solving Inequalities Practice and Problem Solving: A/B Write an inequality for the situation. 1. Cara has $25 to buy dry pet food and treats for the animal shelter. A pound of dog food costs $2 and treats are $1 apiece. If she buys 9 pounds of food, what is the greatest number of treats she can buy? _________________________________________________________________________________________ Solve each inequality for the value of the variable. 2. 2x ≥ 6 3. ________________________________________ a <1 5 ________________________________________ 4. 5x + 7 ≥ 2 5. 5(z + 6) ≤ 40 ________________________________________ ________________________________________ 6. 5x ≥ 7x + 4 7. 3(b − 5) < −2b ________________________________________ ________________________________________ Write and solve an inequality for each problem. 8. By selling old CDs, Sarah has a store credit for $153. A new CD costs $18. What are the possible numbers of new CDs Sarah can buy? _________________________________________________________________________________________ _________________________________________________________________________________________ 9. Ted needs an average of at least 70 on his three history tests. He has already scored 85 and 60 on two tests. What is the minimum grade Ted needs on his third test? _________________________________________________________________________________________ _________________________________________________________________________________________ 10. Jay can buy a stereo either online or at a local store. If he buys online, he gets a 15% discount, but has to pay a $12 shipping fee. At the local store, the stereo is not on sale, but there is no shipping fee. For what regular price is it cheaper for Jay to buy the stereo online? _________________________________________________________________________________________ _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 32 Name ________________________________________ Date __________________ Class __________________ LESSON 2-4 Creating and Solving Inequalities Practice and Problem Solving: C Write an inequality for the situation. 1. Miguel is buying 10 blankets for the animal shelter. If shipping each blanket costs $1.50 and Miguel has $75 to spend, what is the greatest amount he can spend for each blanket? _________________________________________________________________________________________ Solve each inequality. 2. 2( x − 3) + 9 ≥ x 3. ________________________________________ 1 2 a−7< a−9 2 3 ________________________________________ k⎞ ⎛ 5. 8 ⎜ 1 − ⎟ > −5k + 17 2⎠ ⎝ 4. −10(9 − 2 x ) − x ≤ 2 x − 5 ________________________________________ ________________________________________ 6. 100 − 5(7 − 5 y ) > 5(7 + 5 y ) − 100 7. −6(w + 3) − ________________________________________ 3w ≤ −11 − 9w 2 ________________________________________ Solve. 8. One car rental company charges $30 per day plus $0.25 per mile driven. A second company charges $40 per day plus $0.10 per mile driven. How many miles must you drive for a one-day rental at the second company to be less expensive than the same rental at the first company? Write an inequality to solve. _________________________________________________________________________________________ 2x − 1 > 1 , Hal multiplied both sides by x + 8 x+8 and then got the solution x > 9. Is Hal’s work correct? 9. To solve the inequality _________________________________________________________________________________________ 10. To solve 3 ≥ 5 − 2 x, a student typically uses division by −2 and reverses the direction of the inequality. Show how to solve the inequality without using that step. Hint: Use the Addition Property of Equality. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 33 Name ________________________________________ Date __________________ Class __________________ LESSON 2-4 Creating and Solving Inequalities Practice and Problem Solving: Modified Write an inequality for each situation. The first one is done for you. 1. Penny has 7 dollars. Fred has fewer dollars than Penny. Let F = Fred. F <7 _________________________________________________________________________________________ 2. A shelf has room to hold no more than 14 books. Tyrese wants to put poetry books and science books on the shelf. Let p = poetry books and s = science books. _________________________________________________________________________________________ Solve each inequality. The first one is done for you. 3. 2x ≥ 6 4. x≥3 ________________________________________ a <1 5 ________________________________________ 5. 4 ≤ p − 1 6. m + 15 < 6 ________________________________________ ________________________________________ 2 7. − n ≥ −4 3 8. −7x ≤ 0 ________________________________________ ________________________________________ Solve. The first one is done for you. 9. Perdita goes to the fruit market with $9 to buy avocados. Each avocado costs $2. Write and solve an inequality to find the greatest number of avocados Perdita can buy. 2n ≤ 9; n ≤ 4.5; Perdita can buy 4 avocados. _________________________________________________________________________________________ 10. Sam needs an average of 65 on his two science tests. He scored 60 on his first test. What is the minimum grade he needs on his second test? _________________________________________________________________________________________ 11. A car can travel 20 miles on a gallon of gas. Write and solve an inequality to show at least how many gallons of gas are needed to travel 100 miles? Let g = gallons of gas. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 34 Name ________________________________________ Date __________________ Class __________________ LESSON 2-5 Creating and Solving Compound Inequalities Practice and Problem Solving: A/B Solve each compound inequality and graph the solution. 1. x > 2 AND x − 1 ≤ 10 2. 3x + 1 ≥ −8 AND 2x − 3 < 5 ________________________________________ ________________________________________ 3. x > 10 OR x < 0 4. x − 1 > 11 OR 3x ≤ 21 ________________________________________ ________________________________________ 5. 70 < 3x + 10 < 100 6. 2 > 2x − 14 > −14 ________________________________________ ________________________________________ Write the compound inequality shown by each graph. 8. 7. ________________________________________ ________________________________________ Write a compound inequality to model the following situations. Graph the solution. 9. The forecast in Juneau, AK, calls for between 1.2 and 2.0 inches of rain. ____________________________________ 10. Water from industrial plants must be treated before entering the sewer system. Water that is too acidic or too basic will harm the pipes. A semiconductor manufacturer must adjust the pH of any waste water from the process to between 4.0 and 10.0. ____________________________________ 11. A welding shop figures a new welding machine will be cost effective if it runs less than 2 hours or more than 5.5 hours per day. ____________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 37 Name ________________________________________ Date __________________ Class __________________ LESSON 2-5 Creating and Solving Compound Inequalities Practice and Problem Solving: C Write the compound inequality, or inequalities. Draw and label a number line and graph the inequalities. 1. Pilots in the U.S. Air Force must meet certain height requirements. They must be at least 5 feet 4 inches tall, but not taller than 6 feet 2 inches. Convert the heights to inches before completing the problem. _________________________________________________________________________________________ 2. Julie does her homework either between 4:00 and 6:00 p.m. or between 8:00 and 10:00 p.m. _________________________________________________________________________________________ Write a scenario that fits the compound inequality shown. 3. _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ 4. _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 38 Name ________________________________________ Date __________________ Class __________________ LESSON 2-5 Creating and Solving Compound Inequalities Practice and Problem Solving: Modified For the compound inequalities below, determine whether the inequality results in an overlapping region or a combined region. Then determine whether the circles are open are closed. Finally, graph the compound inequality. The first one is done for you. 1. x > 4 x ≤ 13 AND open overlapping closed ________________________________________ 2. x < 4 x ≥ 13 OR ________________________________________ For 3 and 4, first simplify the inequalities. 3. x − 1 ≥ 5 AND 2x < 14 ________________________________________ 4. x − 4 < 0 OR 5x > 30 ________________________________________ Answer the questions below. The first one is done for you. 5. Describe the graphs for compound inequalities formed by using the word AND. The graphs are line segments—they have ends that are either open circles or _________________________________________________________________________________________ closed circles. _________________________________________________________________________________________ 6. Describe the graphs for compound inequalities formed by using the word OR. _________________________________________________________________________________________ _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 39 Name ________________________________________ Date __________________ Class __________________ LESSON 3-1 Graphing Relationships Practice and Problem Solving: A/B Solve. 1. The graph shows the amount of rainfall during one storm. What does segment d represent? ___________________________________________ ___________________________________________ 2. Which segment represents the heaviest rainfall? ___________________________________________ For each situation, tell whether a graph of the situation would be a continuous graph or a discrete graph. 3. the number of cans collected for recycling _______________________ 4. pouring a glass of milk ____________________________ 5. the distance a car travels from a garage _________________________ 6. the number of people in a restaurant ____________________________ Identify which graph represents the situation, the kind of graph, and the domain and range of the graph. 7. Jason takes a shower, but the drain in the shower is not working properly. a. b. c. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 43 Name ________________________________________ Date __________________ Class __________________ LESSON 3-1 Graphing Relationships Practice and Problem Solving: C Sketch a graph for each situation. Be sure to label your graph. 1 of a book, then went to bed. 3 The next day she finished reading the entire book. 1. Sherry read 2. Simon counted the number of red trucks in each section of the parking lot at the mall. 3. On Monday, the furniture truck made three deliveries within 8 miles of the warehouse. 4. Write a situation for which you would use a discrete graph. ___________________________________________________________ ___________________________________________________________ 5. Draw a discrete graph that has a domain of 0 ≤ x ≤ 8 and a range of {2, 4, 6, 8, 10}. Write a situation for the graph. ____________________________________________________________ ____________________________________________________________ 6. Draw a continuous graph that has a domain of 0 ≤ x ≤ 5 and a range of 0 ≤ x ≤ 8. Write a situation for the graph. ____________________________________________________________ ____________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 44 Name ________________________________________ Date __________________ Class __________________ LESSON 3-1 Graphing Relationships Practice and Problem Solving: Modified Identify each part of the graph for the situation. The first one is done for you. Jack took a drive in the country. He drove for a while, then stopped to buy gas. He drove a bit more and stopped at a roadside fruit stand. After more driving, he stopped for lunch. Then he drove straight home. Which part of the graph represents these events? d 1. stopped at a roadside fruit stand _______ 2. drove straight home ___________________ 3. stopped to buy gas ____________________ 4. started his drive _______________________ 5. stopped for lunch ______________________ Complete each sentence. The first one is done for you. input 6. The domain is the set of _________________ numbers, or values of x. 7. The range is the set of _________________ numbers, or values of y. Find the domain and range for each graph. The first one is done for you. 8. 9. 10. domain: domain: domain: 0, 1, 2, 3, 4, 5 ________________________ _______________________ ________________________ range: range: range: 0, 1, 2, 3, 4, 5 ________________________ _______________________ ________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 45 Name ________________________________________ Date __________________ Class __________________ Understanding Relations and Functions LESSON 3-2 Practice and Problem Solving: A/B Express each relation as a table, as a graph, and as a mapping diagram. 1. {(−2, 5), (−1, 1), (3, 1), (−1, −2)} x y 2. {(5, 3), (4, 3), (3, 3), (2, 3), (1, 3)} x y Give the domain and range of each relation. Tell whether the relation is a function. Explain. 3. 4. 5. x y 1 4 2 5 0 6 1 7 2 8 D: _____________________ D: ______________________ D: _____________________ R: _____________________ R: ______________________ R: _____________________ Function? ______________ Function? ______________ Function? ______________ Explain: ________________ Explain: ________________ Explain: ________________ ________________________ _______________________ ________________________ ________________________ _______________________ ________________________ ________________________ _______________________ ________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 48 Name ________________________________________ Date __________________ Class __________________ LESSON 3-2 Understanding Relations and Functions Practice and Problem Solving: C Graph each relation. Then explain whether it is a function or not. 1. {(1, 2), (2, 2), (3, 3), (4, 3)} 2. {(1, 5), (2, 4), (3, 5), (3, 4), (4, 4), (5, 5)} ________________________________________ ________________________________________ ________________________________________ ________________________________________ Solve. 3. Locate 5 points on the first graph so that it shows a function. Then change one number in one of the ordered pairs. Locate the new set of points on the second graph to show a relation that is not a function. Explain your strategy. _________________________________________________________________________________________ 4. Identify whether the graph shows a function or a relation that is not a function. Explain your reasoning. _________________________________________________________________________________________ 5. The function INT(x) is used in spreadsheet programs. INT(x) takes any x and rounds it down to the nearest integer. Find INT(x) for x = 4.6, −2.3, and 2 . Then find the domain and range. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 49 Name ________________________________________ Date __________________ Class __________________ LESSON 3-2 Understanding Relations and Functions Practice and Problem Solving: Modified Identify the domain and range for each set of ordered pairs. The first one is started for you. 1. {(0, 1), (3, −1), (5, 1)} 2. {(2, 2), (3, 4), (−1, 2), (3, −4), (0, 5)} domain: {0, 3, 5} ________________________________________ ________________________________________ ________________________________________ ________________________________________ State whether each mapping diagram shows a function. If not, explain why. The first one is done for you. 3. 4. It is not a function because ________________________________________ ________________________________________ 9 is paired with two outputs. ________________________________________ ________________________________________ 5. 6. ________________________________________ ________________________________________ ________________________________________ ________________________________________ A club’s president kept track of membership over its first 7 years. Use her graph below to solve 7–9. The first one is done for you. 7. What is the range in the graph? {30, 40, 50, 60} ________________________________________ 8. What is the domain in the graph? ________________________________________ 9. Does the graph show a function? Explain your reasoning. ________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 50 Name ________________________________________ Date __________________ Class __________________ LESSON 3-3 Modeling with Functions Practice and Problem Solving: A/B Identify the dependent and independent variables in each situation. 1. The cost of a dozen eggs depends on the size of the eggs. independent: ___________________________ dependent: ___________________________ 2. Ally works in a shop for $18 per hour. dependent: ___________________________ independent: ___________________________ 3. 5 pounds of apples costs $7.45. dependent: ___________________________ independent: ___________________________ For each situation, write a function as a standard equation and in function notation. 4. Keesha will mow grass for $8 per hour. function: ___________________________ standard: ___________________________ 5. Oranges are on sale for $1.59 per pound. standard: ___________________________ function: ___________________________ For each situation, identify the dependent and independent variables. Write a function in function notation, and use the function to solve the problem. 6. A plumber charges $70 per hour plus $40 for the call. What does he charge for 4 hours of work? dependent: _____________________________ Solution: ___________________________ independent: ___________________________ ___________________________ function: ________________________________ ___________________________ 7. A sanitation company charges $4 per bag for garbage pickup plus a $10 weekly fee. A restaurant has 14 bags of g garbage. What will the sanitation company charge the restaurant? dependent: _____________________________ Solution: ___________________________ independent: ___________________________ ___________________________ function: ________________________________ ___________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 53 Name ________________________________________ Date __________________ Class __________________ LESSON 3-3 Modeling with Functions Practice and Problem Solving: C A range for each function is given. Find the domain values from the list: 1, 2, 3, 4, 5, 6, 7, 8. Explain how you arrived at your answer. 1. Function: f(x) = −4x − 8 R: {−16, −28, −36, −40} D: ______________________________________________________________________________________ Explain: ________________________________________________________________________________ _________________________________________________________________________________________ 2. Function: f(x) = 3 x − 17 2 R: {−15.5, −12.5, −9.5, −8} D: ______________________________________________________________________________________ Explain: ________________________________________________________________________________ _________________________________________________________________________________________ 3. Function: f(x) = − 1 x+2 4 R: {1.5, 1, 0.25, 0} D: ______________________________________________________________________________________ Explain: ________________________________________________________________________________ _________________________________________________________________________________________ 4. Function: f(x) = −5x − 13 R: {−28, −38, −43, −48} D: ______________________________________________________________________________________ Explain: ________________________________________________________________________________ _________________________________________________________________________________________ Solve. 5. A bakery has prepared 320 ounces of bread dough. A machine will cut the dough into 5-ounce sections and bake each section into a loaf. The amount of d dough left after m minutes is given by the function d(m) = −5m + 320. How many minutes will it take the machine to use all the dough? Find a reasonable domain and range for this situation. _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 54 Name ________________________________________ Date __________________ Class __________________ LESSON 3-3 Modeling with Functions Practice and Problem Solving: Modified Identify the independent (input) variable and the dependent (output) variable for each situation. The first one is done for you. 1. how much Sam earns when he makes $15 per hour amount earned of hours worked . depends on the number ___________________________ The ___________________________ number of hours worked Independent variable: ___________________________________________________________________ amount earned Dependent variable: ____________________________________________________________________ 2. the cost of a bunch of grapes at $1.19 per pound The ___________________________ depends on the ___________________________. Independent variable: ___________________________________________________________________ Dependent variable: ____________________________________________________________________ Rewrite each equation as a function. The first one is done for you. 3. 2y − 2x = 8 4. y + 5x = 16 5. 4y − 8x = −16 y=x+4 ________________________ _______________________ ________________________ f(x) = x + 4 ________________________ _______________________ ________________________ Write a function for each situation. The first one is done for you. 6. An electrician charges $60 per hour. How much does he charge for 6 hours? total cost of hours worked . depends on the number ___________________________ The ___________________________ the number of hours worked Independent variable: ___________________________________________________________________ the total cost Dependent variable: ____________________________________________________________________ y = 60x Equation: ___________________________ f(x) = 60x Function: ___________________________ 7. A drone costs $300 plus $25 for each set of extra propellers. What is the cost of a drone and 4 extra sets of propellers? The ___________________________ depends on the ___________________________. Independent variable: ___________________________________________________________________ Dependent variable: ____________________________________________________________________ Equation: ___________________________ Function: ___________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 55 Name ________________________________________ Date __________________ Class __________________ Graphing Functions LESSON 3-4 Practice and Problem Solving: A/B Complete the table and graph the function for the given domain. 1. f(x) = 3x − 2 for D = {−3, 1, 5} x y −3 1 5 2. y + 2x + 12 for D = {2, 3, 4} x y 2 3 4 3. 3x − 3y = 9 for D = {0 ≤ x ≤ 8} x y 0 3 8 4. The function f(h) = 2d + 4.3 relates the h height of the water in a fountain in feet to the d diameter of the pipe carrying the water. Graph the function on a calculator and use the graph to find the height of the water when the pipe has a diameter of 1.5 inches. _____________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 58 Name ________________________________________ Date __________________ Class __________________ LESSON 3-4 Graphing Functions Practice and Problem Solving: C Determine the domain for each function. Then graph the function. 1. f ( x ) = 1 x + 4 for R = {5, 6, 7, 8} 2 D = ___________________________ 2. 6x − 3y = 12 for R = {−4 ≤ y ≤ 8} D = ___________________________ 3. 3x = y − 4 for R = {4 ≤ y ≤ 8} D = ___________________________ Solve. 4. A car travels at a speed of 25 miles per hour. The d distance it travels in h hours is given by the equation d = 25h. Write the equation as a function. Use a calculator to graph the function for the domain {0 ≤ h ≤ 5}. What is the meaning of the point (3.5, 87.5) on the graph? Function: ___________________________ Explain: ________________________________________________________________________________ 5. The formula for finding the distance traveled by a free-falling object is D = 16t 2 , where t is the time in seconds. Use a calculator to graph this function for the domain {1, 2, 3, 4, 5, 6}. Find the range. Use the graph to find how much time it takes the object to fall 300 feet. Range: ___________________________ Explain: ________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 59 Name ________________________________________ Date __________________ Class __________________ Graphing Functions LESSON 3-4 Practice and Problem Solving: Modified Use the given domain and range to graph each function. The first one is done for you. 1. f(x) = x + 2 for D = {1, 2, 3, 4} and R = {3, 4, 5, 6} x y 1 3 2 4 3 5 4 6 2. How would the graph be different if the domain were {0 ≤ x ≤ 4)? _____________________________________________________________ Complete the table for the function and then graph the function. The first row is done for you. 3. f(x) = 2x −1 x y 3 5 5 6 8 4. f(x) = 4x − 7 x y 2 3 4 5 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 60 Name ________________________________________ Date __________________ Class __________________ Identifying and Graphing Sequences LESSON 4-1 Practice and Problem Solving: A/B Complete the table and state the domain and range for the sequence. 1. n 1 f(n) 12 2 4 6 36 60 Domain: ___________________________________________________________ Range: ____________________________________________________________ Write the first four terms of each sequence. 2. f (n ) = 3n − 1 3. f (n ) = n 2 + 2n + 5 ________________________________________ ________________________________________ 5. f (n ) = n − 1 4. f (n ) = (n − 1)(n − 2) ________________________________________ ________________________________________ Emma pays $10 to join a gym. For the first 5 months she pays a monthly $15 membership fee. For problems 6–7, use the explicit rule f(n) = 15n + 10. 6. Complete the table. 7. Graph the sequence using the ordered pairs. n f(n) = 15n + 10 f(n) 1 f(1) = 15 (1) + 10 = 25 25 □) = 15 (□) + □ = □ f(□) = 15 (□) + □ = □ f(□) = 15 (□) + □ = □ f(□) = 15 (□) + □ = □ □ □ □ □ 2 3 4 5 f( Use the table to create ordered pairs. The ordered pairs are ________________________________________ ________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 64 Name ________________________________________ Date __________________ Class __________________ LESSON 4-1 Identifying and Graphing Sequences Practice and Problem Solving: C Find the first four terms of each sequence. 2. f (n ) = 1. f (n ) = n 3 − n 2 + 1 ________________________________________ 3. f (n ) = ________________________________________ n(n + 1)(2n + 1) 6 4. f (n ) = ________________________________________ 5. f (n ) = 1 1 − n n +1 n2 − 1 n2 + 1 ________________________________________ n 2 − 12 3 6. f (1) = 9, f (n ) = 13 + f (n − 1) for n ≥ 2 ________________________________________ ________________________________________ Graph the sequence that represents the situation on a coordinate plane. 7. Rebecca had $100 in her savings account in the first week. She adds $45 each week for 5 weeks. The savings account balance can be shown by a sequence. 8. Adam has $300 to donate. For the next five weeks he donates $60 each week to a different charity. His remaining donation money can be shown by a sequence. Solve. 9. In the Fibonacci sequence, f (1) = 1, f (2) = 1, and f (n ) = f (n − 2) + f ( n − 1) for n ≥ 3. Find the first 10 terms of the Fibonacci sequence. _________________________________________________________________________________________ f (n ) . f ( n + 1) Write the first eight terms of this sequence as decimals. If necessary, round a term to three decimal places. Explain any patterns you see. 10. Use f (n ) from Problem 9 to create a new sequence: r (n ) = _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 65 Name ________________________________________ Date __________________ Class __________________ Identifying and Graphing Sequences LESSON 4-1 Practice and Problem Solving: Modified Complete the table and state the domain and range for the sequence. The first one is done for you. 1. n 1 2 3 4 5 f(n) 3 6 9 12 15 1, 2, 3, 4, 5 Domain: ___________________________________________________________ 3, 6, 9, 12,15 Range: ____________________________________________________________ 2. n 1 2 f(n) 10 4 30 50 Domain: ___________________________________________________________ Range: ____________________________________________________________ A taxi charges $4 per ride plus $2 for each mile driven. For 3–4, use the explicit rule f(n) = 2n + 4. The first one in each is done for you. 3. Complete the table. 4. Graph the sequence using the ordered pairs. n f(n) = 2n + 4 f(n) 1 f(1) = 2(1) + 4 = 6 6 □) = 2(□) + 4 = □ f(□) = 2(□) + 4 = □ f(□) = 2(□) + 4 = □ f(□) = 2(□) + 4 = □ □ □ □ □ 2 3 4 5 f( Use the table to create ordered pairs. The ordered pairs are (n, f(n)). □ □ □ □ ), (3, ), (4, ), (5, ) (1, 6), (2, ________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 66 Name ________________________________________ Date __________________ Class __________________ Constructing Arithmetic Sequences LESSON 4-2 Practice and Problem Solving: A/B Write an explicit rule and a recursive rule using the table. 1. n 1 2 3 4 5 f(n) 8 12 16 20 24 2. n 1 2 3 4 5 f(n) 11 7 3 −1 −5 ________________________________________ ________________________________________ ________________________________________ ________________________________________ 3. n 1 2 3 4 5 f(n) −20 −13 −6 1 8 4. n 1 2 3 4 5 f(n) 2.7 4.3 5.9 7.5 9.1 ________________________________________ ________________________________________ ________________________________________ ________________________________________ Write an explicit rule and a recursive rule using the sequence. 5. 45, 50, 55, 60, 65 6. 94, 87, 80, 73, 66 ________________________________________ ________________________________________ ________________________________________ ________________________________________ 8. 83, 43, 3, −37, −77 7. 12, 26, 40, 54, 68 ________________________________________ ________________________________________ ________________________________________ ________________________________________ Solve. 9. The explicit rule for an arithmetic sequence is f(n) = 13 + 6(n − 1). Find the first four terms of the sequence. _________________________________________________________________________________________ 10. Helene paid back $100 in Month 1 of her loan. In each month after that, Helene paid back $50. The graph shows the sequence. Write an explicit rule. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 69 Name ________________________________________ Date __________________ Class __________________ Constructing Arithmetic Sequences LESSON 4-2 Practice and Problem Solving: C Write an explicit rule and a recursive rule for each sequence. 1. n 1 2 3 4 5 f(n) −3.4 −2.1 −0.8 0.5 1.8 2. n 1 2 3 4 5 f(n) 1 6 1 4 1 3 5 12 1 2 ________________________________________ ________________________________________ ________________________________________ ________________________________________ 3. n 1 3 5 6 9 f(n) 82 81 80 79.5 78 4. n 1 4 8 13 19 f(n) −22 2 34 74 122 ________________________________________ ________________________________________ ________________________________________ ________________________________________ Solve. 5. A recursive rule for an arithmetic sequence is f(1) = −8, f(n) = f(n − 1) − 6.5 for n ≥ 2. Write an explicit rule for this sequence. _________________________________________________________________________________________ 6. The third and thirtieth terms of an arithmetic sequence are 4 and 85. Write an explicit rule for this sequence. _________________________________________________________________________________________ 7. f(n) = 900 − 60(n − 1) represents the amount Oscar still needs to repay on a loan at the beginning of month n. Find the amount Oscar pays monthly and the month in which he will make his last payment. _________________________________________________________________________________________ 8. Find the first six terms of the sequence whose explicit formula is f(n) = (−1)n. Explain whether it is an arithmetic sequence. _________________________________________________________________________________________ 9. An arithmetic sequence has common difference of 5.6 and its tenth term is 75. Write a recursive formula for this sequence. _________________________________________________________________________________________ 10. The cost of a college’s annual tuition follows an arithmetic sequence. The cost was $35,000 in 2010 and $40,000 in 2012. According to this sequence, what will tuition be in 2020? _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 70 Name ________________________________________ Date __________________ Class __________________ Constructing Arithmetic Sequences LESSON 4-2 Practice and Problem Solving: Modified Find the common difference for each arithmetic sequence. The first one is done for you. 1. 8, 13, 18, 23, … 2. 9, 23, 37, 51, … 5 ________________________ 3. 28, 22, 16, 10, … _______________________ ________________________ Find the next three terms for each arithmetic sequence. The first one is done for you. 5. 8, 5, 2, −1, … 4. 11, 13, 15, 17, … 19, 21, 23 ________________________ 6. −4, 7, 18, 29, … _______________________ ________________________ Write an explicit rule and a recursive rule for each sequence. The first one is done for you. 7. 9. n 1 2 3 4 5 f(n) 1 3 5 7 9 8. n 1 2 3 4 5 f(n) 15 13 11 9 7 f(n) = 1 + 2(n − 1) ________________________________________ ________________________________________ f(1) = 1, f(n) = f(n − 1) + 2 for n ≥ 2 ________________________________________ ________________________________________ n 1 2 3 4 5 f(n) 16 21 26 31 36 10. n 1 2 3 4 5 f(n) 10 9.5 9 8.5 8 ________________________________________ ________________________________________ ________________________________________ ________________________________________ Solve. 11. The first term of an arithmetic sequence is 20 and the common difference is 15. Find the fifth term of the sequence. _________________________________________________________________________________________ 12. Renata does 30 sit-ups every day from Monday to Friday. The graph shows the sequence. Write an explicit rule for the sequence. _________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 71 Name ________________________________________ Date __________________ Class __________________ LESSON 4-3 Modeling with Arithmetic Sequences Practice and Problem Solving: A/B Complete the table of values to determine the common difference. 1. Mia drives 55 miles per hour. The total miles driven is given by the function C(m) = 55m. 1 Hours 2 3 4 Distance in miles Common difference: ___________________________ 2. Each pound of potatoes costs $1.20. The total cost, in dollars, is given by the function C(p) = 1.2p. 1 Pounds 2 3 4 Cost in dollars Common difference: ___________________________ Solve. Use the following for 3–7. Riley buys a swim pass for the pool in January. The first month costs $30. Each month after that, the cost is $20 per month. Riley wants to swim through December. 3. Complete the table of values. 1 2 3 Cost in dollars 30 50 70 Months 4 5 6 7 8 9 10 11 12 4. What is the common difference? ________________________________________ 5. Write the equation for finding the total cost of a one-year swim pass. ________________________________________ 6. What does f(12) represent? ________________________________________ 7. What is the total amount of money Riley will spend for a one-year swim pass? ________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 74 Name ________________________________________ Date __________________ Class __________________ LESSON 4-3 Modeling with Arithmetic Sequences Practice and Problem Solving: C Use the following diagram for 1–3. 1 2 3 4 5 6 7 8 1. How many sides will Figure 8 have? Is it shaded? _________________________________________________________________________________________ 2. Make a table to show the sequence of figures. Figure Number of Sides 3. How many sides will Figure 21 have? Is it shaded? _________________________________________________________________________________________ 4. Is the sequence of figures an arithmetic sequence? Explain. _________________________________________________________________________________________ Solve. 5. A movie rental club charges $4.95 for the first month’s rentals. The club charges $18.95 for each additional month. What is the total cost for one year? _________________________________________________________________________________________ 6. A photographer charges a sitting fee of $69.95 for one person. Each additional person in the picture is $30. What is the total sitting fee charge for a group of 10 people to be photographed? _________________________________________________________________________________________ 7. Grant is planting one large tree and several smaller trees. He has a budget of $1400. A large tree costs $200. Each smaller tree is $150. How many total trees can Grant purchase on his budget? _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 75 Name ________________________________________ Date __________________ Class __________________ LESSON 4-3 Modeling with Arithmetic Sequences Practice and Problem Solving: Modified Use the table to find the common difference. Then find the value of f(7) for each. The first one is done for you. 1. Membership Fees f(n) 3. 2. 1 2 3 4 12 18 24 30 Number of Weeks n Number of Months n Toys Collected f(n) 1 2 3 4 14 21 28 35 6 Common Difference: _______________ Common Difference: _______________ 48 f(7) = ___________________________________ f(7) = ___________________________________ 4. Number of Kilometers n 1 2 3 4 Number of Pounds n 1 2 3 4 Hours Driving f(n) 12 24 36 48 Boxes of Fruit f(n) 67 70 73 76 Common Difference: ___________________ Common Difference: ____________________ f(7) = ____________________________ f(7) = ____________________________ Each student is training for a race. How many miles did each student run after 6 days? The first one is done for you. 5. 6. f(6) = 30 ________________________________________ ________________________________________ 7. 8. ________________________________________ ________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 76 Name ________________________________________ Date __________________ Class __________________ LESSON 5-1 Understanding Linear Functions Practice and Problem Solving: A/B Tell whether each function is linear or not. 1. y = 3x2 2. 7 − y = 5x + 11 ________________________ 3. −2(x + y) + 9 = 1 _______________________ ________________________ Complete the tables. Is the change constant for equal intervals? If so, what is the change? 5. 4x2 + y = 4 4. 3x + 5y = 4 x −1 y 7 5 0 1 2 6. 6x + 1 = y x −1 0 y 0 4 1 2 x −1 0 1 2 y Constant? _____________ Constant? _____________ Constant? _____________ Change? ______________ Change? ______________ Change? ______________ Graph each line. 7. y = 1 x −3 2 8. 2x + 3y = 8 The solid and dashed lines show how two consultants charge for their services. Use the graph for 9–11. 9. How much does each charge for a 6-hour job? ___________________________________________________________ 10. Does either consultant charge according to a linear function? ___________________________________________________________ 11. For which length of job do A and B charge the same amount? ___________________________________________________________ 12. Are the functions discrete or continuous? Explain. ________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 80 Name ________________________________________ Date __________________ Class __________________ LESSON 5-1 Understanding Linear Functions Practice and Problem Solving: C Tell what constant amount the function changes by over equal intervals. 1. 3x + 4y = 24 2. y = −5x + 10 ________________________________________ ________________________________________ 4. 9 x − 3. x − 7y − 15 = 0 ________________________________________ 2 y = −4 3 ________________________________________ Graph each line. 6. 3( x + y ) − 2( x − y ) = 5(8 + 3 y ) 5. 6x + 5y = 30 Solve. y −8 = 2 and y = 2x + 6 x −1 have identical lines as their graphs. Do you agree? Explain. 7. A student claimed that the two equations _________________________________________________________________________________________ _________________________________________________________________________________________ 8. A line is written in the form Ax + By = 0, where A and B are not both zero. Find the coordinates of the point that must lie on this line, no matter what the choice of A and B. _________________________________________________________________________________________ 9. A line is written in the form Ax + By = C, where A ≠ 0. Find the x-coordinate of the point on the line at which y = 3. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 81 Name ________________________________________ Date __________________ Class __________________ Understanding Linear Functions LESSON 5-1 Practice and Problem Solving: Modified Tell whether each function is linear or not. The first one is done for you. 2. y = 1. x + y = 10 linear ________________________ 2 x 3. y = x 2 − x _______________________ ________________________ Tell the constant amount the function changes by over equal intervals. The first one is done for you. 4. y = 8x + 2 5. y = 3x − 9 8 ________________________ 6. 5x = 11 − y _______________________ ________________________ Complete a table for each linear equation and then graph. The first one is started for you. 7. y = 2x 8. x + y = 5 x 0 1 2 3 x y 0 2 4 6 y Solve. 9. Graph the four lines y = 3, y = −3, x = 4, and x = −4. The points of intersection form the vertices of a geometric figure. State the name of the figure. _________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 82 Name ________________________________________ Date __________________ Class __________________ LESSON 5-2 Using Intercepts Practice and Problem Solving: A/B Find each x- and y-intercept. 1. 2. 3. ________________________ _______________________ ________________________ ________________________ _______________________ ________________________ Use intercepts to graph the line described by each equation. 4. 3x + 2y = −6 5. x − 4y = 4 6. At a fair, hamburgers sell for $3.00 each and hot dogs sell for $1.50 each. The equation 3x + 1.5y = 30 describes the number of hamburgers and hot dogs a family can buy with $30. a. Find the intercepts and graph the function. _____________________________________________ b. What does each intercept represent? _____________________________________________ _____________________________________________ _____________________________________________ _____________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 85 Name ________________________________________ Date __________________ Class __________________ LESSON 5-2 Using Intercepts Practice and Problem Solving: C Find each x- and y-intercept. 1. 4(x − y) + 3 = 2x − 5 2. 5x + 9y = 18 − (x + y) ________________________________________ ________________________________________ Find each x- and y-intercept. Then graph the line described by each equation. 3. x − (y + 2) = 3(x − 2y + 1) 4. 8(4 + x) − 3 = 12(x + y) + 5 Solve. 5. Write the equations of three distinct lines that have the same y-intercept, −1. _________________________________________________________________________________________ 6. A home uses 8 gallons of oil each day for heat. If its oil storage tank is filled to 275 gallons, the function y = 275 − 8x represents the number of gallons remaining in the tank after x days of use. Explain what the x-intercept represents. Then determine when the tank will be half-full. _________________________________________________________________________________________ _________________________________________________________________________________________ 7. The x-intercept of a line is twice as great as its y-intercept. The sum of the two intercepts is 15. Write the equation of the line in standard form. _________________________________________________________________________________________ 8. A linear equation has more than one y-intercept. What can you conclude about the graph of the equation? _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 86 Name ________________________________________ Date __________________ Class __________________ LESSON 5-2 Using Intercepts Practice and Problem Solving: Modified Complete each sentence. The first one is started for you. 1. The y-coordinate of the point where a graph crosses the y-intercept . The x-coordinate y-axis is called the ________________________ of this point is _____________________. 2. The x-coordinate of the point where a graph crosses the x-axis is called the _____________________. The y-coordinate of this point is _____________________. Find each x- and y-intercept. The first one is done for you. 3. 4. 5. (y = 0), 2 x-intercept: ____________ x-intercept: ____________ x-intercept: ____________ (x = 0), 4 y-intercept: ____________ y-intercept: ____________ y-intercept: ____________ 6. x + y = 3 7. 3x + 5y = 30 8. y = 2x − 14 x-intercept: ____________ x-intercept: ____________ x-intercept: ____________ y-intercept: ____________ y-intercept: ____________ y-intercept: ____________ Solve. 9. Jaime bought a jar of 50 vitamins. His two children each take one vitamin each day. The number of vitamins left in the jar after x days is represented by the function f(x) = 50 − 2x. Graph the function and explain what each intercept represents. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 87 Name ________________________________________ Date __________________ Class __________________ LESSON 5-3 Interpreting Rate of Change and Slope Practice and Problem Solving: A/B Find the rise and run between the marked points on each graph. Then find the rate of change or slope of the line. 1. 2. 3. rise = _____ run = _____ rise = _____ run = _____ rise = _____ run = _____ slope = ______________ slope = ______________ slope = _____________ Find the slope of each line. Tell what the slope represents. 4. 5. ________________________________________ ________________________________________ ________________________________________ ________________________________________ Solve. 6. When ordering tickets online, a college theater charges a $5 handling fee no matter how large the order. Tickets to a comedy concert cost $58 each. If you had to graph the line showing the total cost, y, of buying x tickets, what would the slope of your line be? Explain your thinking. _________________________________________________________________________________________ _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 90 Name ________________________________________ Date __________________ Class __________________ LESSON 5-3 Interpreting Rate of Change and Slope Practice and Problem Solving: C Find the rate of change or slope of the line containing each pair of points. 2. (−4, 8) and (3, −9) 1. (4, 5) and (11, 33) ________________________________________ ________________________________________ 1 1⎞ 1 1 4. ⎛⎜ , and ⎛⎜ , ⎞⎟ ⎟ ⎝4 2⎠ ⎝6 3⎠ 3. (0, −8) and (3, 3) ________________________________________ ________________________________________ Find the slope of the line represented by each equation. First find two points that lie on the line. Then find the rate of change or slope. 5. 2x + y = 5 ________________________ 8. y + 5 = 1 ________________________ 6. 3x − 5y = 17 _______________________ 9. −x + 4y = 12 _______________________ 7. y = 4 − 9x ________________________ 10. 6(x − y) = 5(x + y) ________________________ Solve. 11. A line has x-intercept of 6 and y-intercept of −4. Find the slope of the line. _________________________________________________________________________________________ 12. A vertical line contains the points (3, 2) and (3, 6). Use these points and the formula for slope to explain why a vertical line’s slope is undefined. _________________________________________________________________________________________ _________________________________________________________________________________________ 13. The steepness of a road is called its grade. The higher the grade, the steeper the road. For example, an interstate highway is considered out of standard if its grade exceeds 7%. Interpret a grade of 7% in terms of slope. Use feet to explain the meaning for a driver. _________________________________________________________________________________________ _________________________________________________________________________________________ 14. Ariel was told the x-intercept and the y-intercept of a line with a positive slope. Yet, it was impossible for Ariel to find the slope of the line. What can you conclude about this line? Explain your thinking. _________________________________________________________________________________________ _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 91 Name ________________________________________ Date __________________ Class __________________ LESSON 5-3 Interpreting Rate of Change and Slope Practice and Problem Solving: Modified Find the rise and run between the two points indicated on each line. Then find the rate of change or slope of the line. The first one is done for you. 1. 2. 1 3 slope = _________________ 3. slope = _________________ slope = ______________ Tell whether the slope of each line is positive, negative, zero, or undefined. The first one is done for you. 4. 5. zero ________________________ 6. _______________________ ________________________ Solve. 7. The table shows a truck driver’s distance from home during one day’s deliveries. Find the rate of change for each time interval. Times (h) 0 1 4 5 8 10 Distance (mi) 0 35 71 82 199 200 Hour 0 to Hour 1: ________ Hour 1 to Hour 4: ________ Hour 5 to Hour 8: _________ Hour 8 to Hour 10: _________ Hour 4 to Hour 5: _______ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 92 Name ________________________________________ Date __________________ Class __________________ LESSON 6-1 Slope-Intercept Form Practice and Problem Solving: A/B Write the equation for each line in slope-intercept form. Then identify the slope and the y-intercept. 1. 4x + y = 7 2. 2x − 3y = 9 Equation: ____________________________ Equation: ____________________________ Slope: _______________________________ Slope: _______________________________ y-intercept: __________________________ y-intercept: __________________________ 3. 5x + 1 = 4y + 7 4. 3x + 2y = 2x + 8 Equation: ____________________________ Equation: ____________________________ Slope: _______________________________ Slope: _______________________________ y-intercept: __________________________ y-intercept: __________________________ Graph the line described by each equation. 5. y = −3 x + 4 6. y = 5 x −1 6 Solve. 7. What are the slope and y-intercept of y = 3 x − 5 ? _________________________________________________________________________________________ 8. A line has a y-intercept of −11 and slope of 0.25. Write its equation in slope-intercept form. _________________________________________________________________________________________ 9. A tank can hold 30,000 gallons of water. If 500 gallons of water are used each day, write the equation that represents the amount of water in the tank x days after it is full. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 96 Name ________________________________________ Date __________________ Class __________________ LESSON 6-1 Slope-Intercept Form Practice and Problem Solving: C Write the equation for each line in slope-intercept form. Then identify the slope and the y-intercept. y x 1. 3(x − 2y) = 5(x − 3y) + 9 2. − =1 5 7 Equation: ____________________________ Equation: ____________________________ Slope: _______________________________ Slope: _______________________________ y-intercept: __________________________ y-intercept: __________________________ Write an equation for each line. Then graph the line. 3. A line whose slope and y-intercept are equal and the sum of the two is −4 4. A line that has a slope half as great as its y-intercept and the sum of the two is 1 ________________________________________ ________________________________________ Let f(x) = mx + b be a function with real numbers for m and b. Use this for Problems 5 and 6. 5. Show that the domain of this function is the set of all real numbers. _________________________________________________________________________________________ _________________________________________________________________________________________ 6. Show that the range of the function may or may not be the set of all real numbers. _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 97 Name ________________________________________ Date __________________ Class __________________ LESSON 6-1 Slope-Intercept Form Practice and Problem Solving: Modified Write each equation in slope-intercept form, y = mx + b. The first one is done for you. 1. 4x + 2y = 8 2. 8x + y = 17 y = −2x + 4 ________________________ 4. 5x + 4y = 4 3. −3x + y = −11 _______________________ 5. 2y + 6 = x ________________________ ________________________ 6. 5x − 20y = 2 _______________________ ________________________ Find the slope of each line. The first one is done for you. 7. 10x + y = 1 8. y = −x + 7 −10 ________________________ 9. 2 + y = 8 + 4x _______________________ ________________________ Find the y-intercept of each line. The first one is done for you. 10. x + y + 8 = 0 11. 4y = −6x + 8 −8 ________________________ 12. y + 5 = 3x − 9 _______________________ ________________________ Name the slope and y-intercept for each line. Then graph the line. The first one is started for you. 13. y = 3x − 5 14. 2x + 5y = 15 3 slope ___________________________ slope ___________________________ −5 y-intercept ___________________________ y-intercept ___________________________ Solve. 15. The price of a bus ride rose from $2 to $2.50. If you graphed the function f(x) = the cost of x bus rides, how would the graph change after the fare rose? _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 98 Name ________________________________________ Date __________________ Class __________________ LESSON 6-2 Point-Slope Form Practice and Problem Solving: A/B Write each in point-slope form. 2. Line with a slope of −3 and passes through point (−1, 7). 1. Line with a slope of 2 and passes through point (3, 5). ________________________________________ ________________________________________ 3. (−6, 3) and (4, 3) are on the line. 4. (0, 0 ) and (5, 2) are on the line. ________________________________________ 5. ________________________________________ 6. x y 0 −2 18 2 9 1 9 4 18 4 0 x y 0 ________________________________________ ________________________________________ 7. 8. ________________________________________ ________________________________________ Solve. 9. For 4 hours of work, a consultant charges $400. For 5 hours of work, she charges $450. Write a point-slope equation to show this, then find the amount she will charge for 10 hours of work. _________________________________________________ _________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 101 Name ________________________________________ Date __________________ Class __________________ LESSON 6-2 Point-Slope Form Practice and Problem Solving: C Write each in point-slope form. 1. The graph of the function has slope of 3 8⎞ ⎛ − and contains ⎜ −2, − ⎟ . 4 5⎠ ⎝ 2. The graph of the function has slope of 2 and contains (−35, 39). 5 ________________________________________ 3. ________________________________________ 4. x x y 1 6 −5 1 1 2 −10 2 3 ________________________________________ y − 2 3 2 ________________________________________ 5. 6. ________________________________________ ________________________________________ Solve. 7. Ben claims that the points (2, 4), (4, 8), and (8, 12) lie on a line. Show that Ben is incorrect. _________________________________________________________________________________________ _________________________________________________________________________________________ 8. Prove the following statement: If the x- and y-intercepts of a line are identical nonzero numbers, the line must have a slope of −1. _________________________________________________________________________________________ _________________________________________________________________________________________ 9. A consignment store charges a flat rate plus a percent of the sale price for any items it sells. An item priced at $500 carries a total fee of $120 while an item priced at $800 carries a total fee of $180. Use the pointslope equation to find the total fee for an item priced at $300. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 102 Name ________________________________________ Date __________________ Class __________________ Point-Slope Form LESSON 6-2 Practice and Problem Solving: Modified Find each slope of the line containing the two points. The first one is done for you. 1. (1, 9) and (3, 3) 2. (2, 5) and (7, 25) −3 ________________________ 3. (0, 4) and (8, 8) _______________________ ________________________ Write the point-slope form for each. The first one is done for you. 5. The slope is −2 and (1, 6) is on the line. 4. The slope is 3 and (0, 5) is on the line. y − 5 = 3( x − 0) ________________________________________ 6. ________________________________________ 7. x y 6 −4 −2 1 10 1 −7 2 14 4 −10 x y 0 ________________________________________ ________________________________________ 8. 9. ________________________________________ ________________________________________ On Day 1 of the state fair, Austin sold 15 handmade chairs. On each of the next 9 days, he sold 5 chairs. Solve the problems below. The first one is done for you. 10. What is the slope of the line representing Austin’s chair sales? 5 _________________________________________________________________________________________ 11. On which day did Austin reach a total of 30 chairs sold? _________________________________________________________________________________________ 12. Write the point-slope equation. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 103 Name ________________________________________ Date __________________ Class __________________ LESSON 6-3 Standard Form Practice and Problem Solving: A/B Tell whether each function is written in standard form. If not, rewrite it in standard form. 1. y = 3x 2. 7 − y = 5x + 11 3. −2(x + y) + 9 = 1 ________________________ _______________________ ________________________ ________________________ _______________________ ________________________ Given a slope and a point, write an equation in standard form for each line. 4. slope = 6, (3, 7) ________________________ 5. slope = −1, (2, 5) 6. slope = 9, (−5, 2) _______________________ ________________________ Graph the line of each equation. 7. x − 2y = 6 8. 2x + 3y = 8 Solve. 9. A swimming pool was filling with water at a constant rate of 200 gallons per hour. The pool had 50 gallons before the timer started. Write an equation in standard form to model the situation. ___________________________________________________________ 10. A grocery bag containing 4 potatoes weighs 2 pounds. An identical bag that contains 12 potatoes weighs 4 pounds. Write an equation in standard form that shows the relationship of the weight (y) and the number of potatoes (x). ___________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 106 Name ________________________________________ Date __________________ Class __________________ LESSON 6-3 Standard Form Practice and Problem Solving: C Write each equation in slope-intercept form. Then identify its intercepts. 1. 3x + 4y = 24 2. y = −5x + 10 ________________________________________ ________________________________________ 4. 9 x − 3. x − 7y − 15 = 0 ________________________________________ 2 y = −4 3 ________________________________________ Graph each line. Rewrite the equation in standard form if necessary. 5. 6x + 5y = 30 6. 3( x + y ) − 2( x − y ) = 5(8 + 3 y ) Solve. y −8 = 2 and y = 2 x + 6 have x −1 identical lines as their graphs. Do you agree? Explain. 7. A student claims that the two equations _________________________________________________________________________________________ _________________________________________________________________________________________ 8. A line is written in standard form Ax + By = 0, where A and B are not both zero. Find the coordinates of the point that must lie on this line, no matter what the choice of A and B. _________________________________________________________________________________________ 9. A line is written in standard form Ax + By = C, where A ≠ 0. Find the x-coordinate of the point on the line at which y = 3. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 107 Name ________________________________________ Date __________________ Class __________________ Standard Form LESSON 6-3 Practice and Problem Solving: Modified Tell whether each function is in standard form or not. If not, rewrite it in standard form. The first one is done for you. 1 1. x + y = 10 2. y = x − 8 3. y − 8 = 2(x − 3) 2 standard form ________________________ _______________________ ________________________ Identify the form of each equation. The first one is done for you. 4. y − 1 = 5(x + 9) 5. y = 3x − 9 point-slope ________________________ 6. 6x + 4y = 12 _______________________ ________________________ Complete a table for each standard equation and then graph it. The first one is started for you. 7. 2x − y = 0 8. x + y = 5 x 0 1 2 3 x y 0 2 4 6 y Match each equation on the left with its equivalent in standard form. 9. A y − 1 = 3( x + 4) 5 x − 2y = −1 _________ B 5 x + 1 = 2y x − y = 6 _________ C 6+y = x 1 ( x − 2) = y D 2 x − 2y = 2 _________ 3 x − y = −13 _________ Solve. 10. Is 3x + 2y = 6 the standard form of y = 3 x + 3? Explain why or why not. 2 _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 108 Name ________________________________________ Date __________________ Class __________________ LESSON 6-4 Transforming Linear Functions Practice and Problem Solving: A/B Identify the steeper line. 1. y = 3x + 4 or y = 6x + 11 2. y = −5x − 1 or y = −2x − 7 ________________________________________ ________________________________________ Each transformation is performed on the line with the equation y = 2x − 1. Write the equation of the new line. 3. vertical translation down 3 units 4. slope increased by 4 ________________________________________ ________________________________________ 5. slope divided in half 6. shifted up 1 unit ________________________________________ ________________________________________ 7. slope increased by 50% 8. shifted up 3 units and slope doubled ________________________________________ ________________________________________ A salesperson earns a base salary of $4000 per month plus 15% commission on sales. Her monthly income, f(s), is given by the function f(s) = 4000 + 0.15s, where s is monthly sales, in dollars. Use this information for Problems 9–12. 9. Find g(s) if the salesperson’s commission is lowered to 5%. _________________________________________________________________________________________ 10. Find h(s) if the salesperson’s base salary is doubled. _________________________________________________________________________________________ 11. Find k(s) if the salesperson’s base salary is cut in half and her commission is doubled. _________________________________________________________________________________________ 12. Graph f(s) and k(s) on the coordinate grid below. Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 111 Name ________________________________________ Date __________________ Class __________________ LESSON 6-4 Transforming Linear Functions Practice and Problem Solving: C Identify the steeper line. 1. y = 2x − 3 or x − 5y = 20 2. x + 10y = 1 or 3x + 20y = 1 ________________________________________ ________________________________________ Each transformation is performed on the line with the equation y = 4x − 20. Write the equation of the new line. 3. slope cut in half 4. vertical translation 25 units upward ________________________________________ ________________________________________ 5. shifted up 8 units and slope tripled 6. reflection across the y-axis ________________________________________ ________________________________________ Solve. 7. Compare the steepness of the lines whose equations are 8x + y = 1 and −8x + y = 2. Explain your reasoning. _________________________________________________________________________________________ _________________________________________________________________________________________ 8. f(x) is an increasing linear function that passes through the point (4, 0). Show that if written in the form f ( x ) = mx + b, m > 0 and b < 0. _________________________________________________________________________________________ _________________________________________________________________________________________ 9. A salesperson earns a base salary of $400 per week plus 20% commission on sales. He is offered double his base salary if he’ll accept half his original commission. Graph and label the original deal and the new deal below. Next to the graph, find when the original deal is a better choice. Explain your thinking. ______________________________________________________ ______________________________________________________ ______________________________________________________ ______________________________________________________ ______________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 112 Name ________________________________________ Date __________________ Class __________________ LESSON 6-4 Transforming Linear Functions Practice and Problem Solving: Modified A line’s y-intercept is changed from b to b*. State whether the line shifts up or down. The first one is done for you. 1. b = 8 and b* = 2 2. b = −4 and b* = −6 down ________________________ 3. b = −1 and b* = 0 _______________________ ________________________ A line’s slope is changed from m to m*. State whether the line becomes more steep or less steep. The first one is done for you. 4. m = 2 and m* = 3 5. m = 5 and m* = more steep ________________________ 1 5 6. m = −4 and m* = −9 _______________________ ________________________ A taxi charges an initial fee of $3 plus $2 for each mile driven. This is shown in each graph below. For each situation described, draw the new graph. The first one is done for you. 7. The initial fee is decreased to $1. 8. The fee per mile is decreased to $1. 9. The initial fee is eliminated. 10. Both fees are changed to $4. Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 113 Name ________________________________________ Date __________________ Class __________________ LESSON 6-5 Comparing Properties of Linear Functions Practice and Problem Solving: A/B The linear functions f(x) and g(x) are defined by the graph and table below. Assume that the domain of g(x) includes all real numbers between the least and greatest values of x shown in the table. x 1 2 3 4 5 6 7 8 1. Find the domain of f(x). g(x) 35 30 25 20 15 10 5 0 2. Find the domain of g(x). ________________________________________ ________________________________________ 3. Find the range of f(x). 4. Find the range of g(x). ________________________________________ ________________________________________ 5. Find the initial value of f(x). 6. Find the initial value of g(x). ________________________________________ ________________________________________ 7. Find the slope of the line 8. Find the slope of the line represented by f(x). represented by g(x). ________________________________________ ________________________________________ 9. How are f(x) and g(x) alike? How are they different? _________________________________________________________________________________________ 10. Describe a situation that could be represented by f(x). _________________________________________________________________________________________ 11. Describe a situation that could be represented by g(x). _________________________________________________________________________________________ 12. If the domains of f(x) and g(x) were extended to include all real numbers greater than or equal to 0, what would their y-intercepts be? _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 116 Name ________________________________________ Date __________________ Class __________________ LESSON 6-5 Comparing Properties of Linear Functions Practice and Problem Solving: C The linear functions f(x), g(x), h(x), and k(x) are defined by the graphs and table below. Assume that the domains of h(x) and k(x) include all real numbers between the least and greatest values of x shown in the table. x 1 2 3 4 5 6 7 8 9 10 h(x) 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 k(x) 12 14.5 17 19.5 22 24.5 27 29.5 32 34.5 1. Find the domains of the functions. _________________________________________________________________________________________ 2. Find the ranges of the functions. _________________________________________________________________________________________ 3. Find the initial values of the functions. _________________________________________________________________________________________ 4. Find the slopes of the functions. _________________________________________________________________________________________ 5. Describe a situation that could be represented by two of the functions. _________________________________________________________________________________________ _________________________________________________________________________________________ 6. If the domains of the functions were extended to include all real numbers greater than or equal to 0, what would their y-intercepts be? _________________________________________________________________________________________ 7. If the domain of f(x) and g(x) were extended to include all real numbers, at what point would their graphs intersect? _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 117 Name ________________________________________ Date __________________ Class __________________ LESSON 6-5 Comparing Properties of Linear Functions Practice and Problem Solving: Modified The linear functions f(x) and g(x) are defined by the graph and table below. Assume that the domain of g(x) includes all real numbers between the least and greatest values of x shown in the table. The first one is done for you. 1. Find the domain of f(x). x g(x) 2 3 4 5 6 7 8 9 3 4 5 6 7 8 9 10 2. Find the domain of g(x). 2≤x≤9 ________________________________________ ________________________________________ 3. Find the range of f(x). 4. Find the range of g(x). ________________________________________ ________________________________________ 5. Find the initial value of f(x). 6. Find the initial value of g(x). ________________________________________ ________________________________________ 7. Find the slope of the line represented by f(x). 8. Find the slope of the line represented by g(x). ________________________________________ ________________________________________ 9. How are f(x) and g(x) alike? How are they different? _________________________________________________________________________________________ _________________________________________________________________________________________ 10. If f(x) and g(x) were drawn as lines with all real numbers in their domains, how many times would the lines intersect? At what point would the lines intersect? _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 118 Name ________________________________________ Date __________________ Class __________________ LESSON 7-1 Modeling Linear Relationships Practice and Problem Solving: A/B Solve. 1. A recycling center pays $0.10 per aluminum can and $0.05 per plastic bottle. The cheerleading squad wants to raise $500. a. Write a linear equation that describes the problem. _________________ b. Graph the linear equation. c. If the cheerleading squad collects 6000 plastic bottles, how many cans will it need to collect to reach the goal? _________________ 2. A bowling alley charges $2.00 per game and will rent a pair of shoes for $1.00 for any number of games. The bowling alley has an earnings goal of $300 for the day. a. Write a linear equation that describes the problem. _________________ b. Graph the linear equation. c. If the bowling alley rents 40 pairs of shoes, how many games will need to be played to reach its goal? _________________ 3. The members of a wheelchair basketball league are playing a benefit game to meet their fundraising goal of $900. Tickets cost $15 and snacks cost $6. a. Write a linear equation that describes the problem. _________________ b. Graph the linear equation. c. If the team sells 50 tickets, how many snacks does it need to sell to reach the goal? _________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 122 Name ________________________________________ Date __________________ Class __________________ LESSON 7-1 Modeling Linear Relationships Practice and Problem Solving: C Solve. 1. Mr. Malone can heat his house in the winter by burning three cords of wood, by using natural gas, or by a combination of the two. His heating budget for the winter is $600. a. Write a linear equation that describes the problem. _______________________ b. Graph the linear equation and label both axes. c. If Mr. Malone spends $275 on natural gas, about how many cords of wood will he need? _______________________ 2. Timber Hill Tennis Club sells monthly memberships for $72 and tennis rackets for $150 each. The tennis club has a sales goal of $5400 per month. a. Write a linear equation that describes the problem. _______________________ b. Graph the linear equation and label both axes. c. If the club sells 50 memberships, how many rackets must be sold to meet the goal? _______________________ 3. Brian’s Bakery sells loaves of Italian bread for $3.50 and loaves of rye bread for $2.80. Brian’s goal is to bring in $420 per day from sales of these two items. a. Write a linear equation that describes the problem. _______________________ b. Graph the linear equation and label both axes. c. If Brian sells 100 Italian loaves, how many rye loaves must he sell to meet his goal? _______________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 123 Name ________________________________________ Date __________________ Class __________________ LESSON 7-1 Modeling Linear Relationships Practice and Problem Solving: Modified Solve. The first one is started for you. 1. Van’s Deli sells hot dogs for $2 and hamburgers for $5. His daily sales goal is $200. a. Complete the chart. Hamburgers Hot Dogs 0 100 40 0 14 65 2d + 5b = 200 b. Write a linear equation that describes the problem. _______________________ c. Graph the linear equation. d. If Van sells 14 hamburgers, how many hot dogs must he sell to reach his goal? _______________________ 2. The Good Fruit stand sells baskets of cherries for $4 and baskets of blueberries for $3. Its daily sales goal is $720. a. Complete the chart. Cherries Blueberries 0 0 30 b. Write a linear equation that describes the problem. _______________________ c. Graph the linear equation. d. If the fruit stand sells 60 baskets of cherries, how many baskets of blueberries must it sell to meet the goal? _______________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 124 Name ________________________________________ Date __________________ Class __________________ LESSON 7-2 Using Functions to Solve One-Variable Equations Practice and Problem Solving: A/B Use the following for 1–5. Locksmith Larry charges $90 for a house call plus $20 per hour. Locksmith Barry charges $50 for a house call plus $30 per hour. 1. Write a one-variable equation for the charges of Locksmith Larry. f(x) = ___________________________ 2. Write a one-variable equation for the charges of Locksmith Barry. g(x) = ___________________________ 3. Complete the table for f(x) and g(x). Hours f(x) 4. Plot f(x) and g(x) on the graph below. Find the intersection. g(x) 0 1 2 3 4 5 5. After how many hours will the two locksmiths charge the same amount? ____________ 6. Jill has $600 in savings. She has a recurring monthly bill of $75 but no income. a. Write an equation, f(x), representing her savings each month. ______________________ b. Let g(x) = 0 represent the point when Jill has no money left. In how many months, x, will her savings account reach zero? _____________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 127 Name ________________________________________ Date __________________ Class __________________ Using Functions to Solve One-Variable Equations LESSON 7-2 Practice and Problem Solving: C Use the following for 1–5. DJ A charges $75.30 plus $12.50 per hour. DJ B charges $52.90 plus $18.10 per hour. When will their charges be equal? 1. Write a one-variable equation for the charges of DJ A. f(x) = ___________________________ 2. Write a one-variable equation for the charges of DJ B. g(x) = ___________________________ 3. Complete the table for f(x) and g(x). Hours f(x) g(x) 0 1 2 3 4 5 4. Use a graph to solve for x. Plot f(x) and g(x) on the graph below. Find the intersection. 5. After how many hours will the two DJs charge the same amount? ____________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 128 Name ________________________________________ Date __________________ Class __________________ Using Functions to Solve One-Variable Equations LESSON 7-2 Practice and Problem Solving: Modified Use the following for 1–5. The first is done for you. Dog Walker A charges $15 per day for each dog plus $4 per hour. Dog Walker B charges $25 per day and $2 per hour. 1. Write a one-variable equation for the charges of Dog Walker A. 15 + 4x f(x) = ___________________________ 2. Write a one-variable equation for the charges of Dog Walker B. g(x) = ___________________________ 3. Complete the table for f(x) and g(x). The first line has been filled in for you. Hours f(x) g(x) 0 15 25 1 2 3 4 5 4. Use a graph to solve for x. Plot f(x) and g(x) on the graph below. Find the intersection. f(x) is done for you. 5. After how many hours will the two dog walkers charge the same amount? Set f(x) equal to g(x) and solve for x the number of hours. x = ________________ hours Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 129 Name ________________________________________ Date __________________ Class __________________ LESSON 7-3 Linear Inequalities in Two Variables Practice and Problem Solving: A/B Use substitution to tell whether each ordered pair is a solution of the given inequality. 1. (3, 4); y > x + 2 3. (2, −1); y < −x 2. (4, 2); y ≤ 2x − 3 ________________________ _______________________ ________________________ Rewrite each linear inequality in slope-intercept form. Then graph the solutions in the coordinate plane. 5. 6x + 2y > −2 4. y − x ≤ 3 ________________________________________ ________________________________________ 6. Trey is buying peach and blueberry yogurt cups. He will buy at most 8 cups of yogurt. Let x be the number of peach yogurt cups and y be the number of blueberry yogurt cups he buys. a. Write an inequality to describe the situation. ____________________________________ b. Graph the solutions. c. Give two possible combinations of peach and blueberry yogurt that Trey can choose. ____________________________________ ____________________________________ Write an inequality to represent each graph. 7. 8. ________________________ 9. _______________________ ________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 132 Name ________________________________________ Date __________________ Class __________________ LESSON 7-3 Linear Inequalities in Two Variables Practice and Problem Solving: C Graph the solution set for each inequality. 1. 2x − 3y ≤ 15 2. 1 1 1 x+ y< 4 3 2 Write and graph an inequality for each situation. 3. Hats (x) cost $5 and scarves (y) cost $8. Joel can spend at most $40. 4. Juana wants to sell more than 1 million dollars worth of $1000 laptops (x) and $2000 desktop computers (y) this year. ________________________________________ ________________________________________ Solve. 5. To graph y ≤ 2 x + 8, you first draw the line y = 2 x + 8. Explain how you can then tell, without doing any arithmetic, which region to shade. _________________________________________________________________________________________ 6. Why does the graph of y ≥ x contain a solid line while the graph of y > x contains a dotted line? _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 133 Name ________________________________________ Date __________________ Class __________________ LESSON 7-3 Linear Inequalities in Two Variables Practice and Problem Solving: Modified Use substitution to tell whether the ordered pair is a solution of the given inequality. The first one is done for you. 1. (4, 3); y ≥ 2x no ________________________ 4. (1, 5); y ≥ 3x + 2 ________________________ 2. (1, 1); y > 4x − 3 _______________________ 5. (3, −2); 4x + 3y ≤ 10 _______________________ 3. (8, 0); 2x + 4y < 18 ________________________ 6. (−1, 6); 4y − x > 27 ________________________ Graph each linear inequality. The first one is done for you. 7. y ≤ x − 3 9. y ≥ − 8. y > 2x − 4 1 x+2 2 10. y < −3 x − 5 Solve. 11. Carla has $45. Pineapples cost $5 each and mangos cost $2 each. Let p stand for pineapples and let m stand for mangos. Write an inequality to show how many of each Carla can buy. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 134 Name ________________________________________ Date __________________ Class __________________ LESSON 8-1 Two-Way Frequency Tables Practice and Problem Solving: A/B Solve. 1. Nancy’s school conducted a recycling drive. Students collected 20pound bags of plastic, glass, and metal containers. The first chart shows the data: the bags of each type of container that were collected. Complete the frequency table. Bags containing 20-pound Bags Collected glass glass plastic metal plastic plastic metal glass glass plastic metal metal metal glass glass plastic plastic plastic plastic metal Frequency plastic glass 2. A school administrator conducted a survey in her school. Students were asked to choose the science or the natural history museum for an upcoming field trip. Complete the two-way frequency table. Gender Boys Girls Total Science Field Trip Preferences History 56 Total 102 54 200 3. Teresa surveyed 100 students about whether they wanted to join the math club or the science club. Thirty-eight students wanted to join the math club only, 34 wanted to join the science club only, 21 wanted to join both math and science clubs, and 7 did not want to join either. Complete the two-way frequency table. Math Yes No Total Yes Science No 38 Total 34 4. A pet-shop owner surveyed 200 customers about whether they own a cat or a dog. Partial results of the survey are recorded below. Complete the two-way frequency table. One-half of the respondents own a dog but not a cat. The number of customers who own neither a dog nor a cat is 38. There are no customers who own both a dog and a cat. Dog Yes No Total Cat No Yes 0 Total 38 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 138 Name ________________________________________ Date __________________ Class __________________ LESSON 8-1 Two-Way Frequency Tables Practice and Problem Solving: C 1. A surveyor asked students whether they favored or did not favor a change in the Friday lunch menu at their school. i The survey involved 200 students. i The number of boys surveyed equaled the number of girls surveyed. i Fifty percent of the girls favored the change. i The number of boys who did not favor the change was two-thirds of the number of boys who favored the change. Complete the two-way frequency table. Explain your reasoning. Favor or Disfavor the Change Gender Yes No Total Girls Boys Total _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ 2. A pet-shop owner surveyed 150 customers about whether they owned birds, cats, or dogs. Partial results of the survey are recorded below. In the table, B represents bird, C represents cat, and D represents dog. Ownership B and C B and D C and D Only Only Only Reply B C D Yes 80 77 84 32 30 29 12 6 150 No 70 73 66 128 120 121 138 144 150 150 150 150 150 150 150 150 150 150 Total B, C, D Not B, C, or D Total Write the correct number in each of the eight regions in the Venn diagram below. Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 139 Name ________________________________________ Date __________________ Class __________________ LESSON 8-1 Two-Way Frequency Tables Practice and Problem Solving: Modified Solve. The first one is started for you. 1. The table below shows the results of a survey about student vegetable preferences. Complete each sentence to help complete the table. How many students in grade 9 prefer carrots? Fill in the blank. ______ + 18 + 13 = 45 How many students in grade 10 prefer cucumbers? Fill in the blank. 20 + 22 + _____ = 55 How do you find the total number of students in the survey that prefer carrots? What is that total? ____________________________________________________________ How do you find the total number of students in the survey? What is that total? ____________________________________________________________ Complete the two-way frequency table using your responses. Grade 9 10 Total Carrots 20 Preferred Vegetable Celery Cucumber 18 13 22 Total 45 55 2. Frank and Lisa surveyed classmates on whether they prefer apples, oranges, or berries packed with their lunches. Each respondent made exactly one choice. Complete the two-way frequency table. Grade 9 10 Total Apple 21 24 Preferred Fruit Orange Berries 18 13 19 19 Total 3. Sixty students were asked to identify a language they plan to study next year. The partial results are shown below. Complete the two-way frequency table. Foreign Language Gender Italian Spanish Boys 6 10 Girls 6 French Total 28 7 32 Total Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 140 Name ________________________________________ Date __________________ Class __________________ LESSON 8-2 Relative Frequency and Probability Practice and Problem Solving: A/B The two-way frequency table below represents the results of a survey about favorite forms of entertainment. In Exercises 1–3, write fractions that are not simplified as responses. Like Board Games Like Reading Yes No Total Yes 48 25 73 No 43 9 52 Total 91 34 125 1. Find the joint relative frequency of people surveyed who like to read but dislike playing board games. _______________ 2. What is the marginal relative frequency of people surveyed who like to read? _______________ 3. Given someone interested in reading, is that person more or less likely to take an interest in playing board games? Explain. _________________________________________________________________________________________ _________________________________________________________________________________________ 4. Given someone interested in board games, is that person more or less likely to take an interest in reading? Explain your response. _________________________________________________________________________________________ _________________________________________________________________________________________ The two-way frequency table below represents the results of a survey about ways students get to school. Type of Transportation Grade On Foot By Car By Bus Total 9 15 28 64 107 10 20 30 43 93 Total 35 58 107 200 5. Find the joint relative frequency of students surveyed who walk to school and are in grade 9. _______________ 6. Given grade level, is that person more or less likely to travel to school by bus? Explain your response. _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 143 Name ________________________________________ Date __________________ Class __________________ LESSON 8-2 Relative Frequency and Probability Practice and Problem Solving: C The two-way frequency table below represents the results of a survey about men, women, and televised sports. 1. Complete the table. Like Televised Sports Yes No 48 43 9 34 Gender Men Women Total Total 73 125 In Exercise 2–5, write your answers as percents. 2. Find the joint relative frequency of men surveyed who like televised sports. ___________ 3. Find the marginal relative frequency of people surveyed who like televised sports. ______ 4. Given someone is male, is that person more or less likely to like televised sports? Explain. _________________________________________________________________________________________ _________________________________________________________________________________________ 5. Given someone likes televised sports, is that person more or less likely to be male? Explain your response. _________________________________________________________________________________________ _________________________________________________________________________________________ The two-way frequency table below represents the results of a survey about ways people get to the movies. 6. Complete the table. Age Adult Child Total On Foot 15 35 Type of Transportation By Car By Bus 28 64 30 43 107 7. Find the joint relative frequency of people surveyed who walk to the movies and are adults. Total 93 ________________ 8. Given that a person is an adult, is that person more or less likely to travel to the movies by bus? What about if the person is a child? Explain your response. _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 144 Name ________________________________________ Date __________________ Class __________________ LESSON 8-2 Relative Frequency and Probability Practice and Problem Solving: Modified The two-way frequency table below represents results of a survey about favorite amusement park rides on a list provided by grade 9 and grade 10 students. Complete each item. Some items are started for you. Amusement Park Ride Grade Roller Coaster Ferris Wheel Whip Total 9 36 10 28 74 10 30 8 28 66 Total 66 18 56 140 1. Determine the joint relative frequency of a grade 9 student preferring the Whip. What number is in the space where the grade 9 row intersects the Whip column? ____________________ What is the grand total of all student responses? ____________________ Write the ratio of the number of grade 9 students preferring the Whip to the grand total as a fraction. ____________________ 2. Determine the marginal relative frequency of choosing the Whip. What number is in the space where the Total row intersects the Whip column? ____________________ What is the grand total of all student responses? ____________________ Write the ratio of the number of students who chose the Whip to the Total number of students surveyed. ____________________ 3. Determine the joint relative frequency of a grade 10 student preferring the roller coaster. ____________________ 4. Determine the conditional relative frequency of a student preferring the Whip given the student is in grade 9. What number is in the space where the grade 9 row intersects the Whip column? ____________________ What is the total of all students in grade 9? ____________________ Write the ratio of the number of grade 9 students preferring the Whip to the total number of grade 9 students as a fraction. ____________________ 5. Determine the conditional relative frequency of a student preferring the Ferris wheel given that the student is in grade 10. ____________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 145 Name ________________________________________ Date __________________ Class __________________ LESSON 9-1 Measures of Center and Spread Practice and Problem Solving: A/B Find the mean, median, and range for each data set. 1. 18, 24, 26, 30 2. 5, 5, 9, 11, 13 Mean: _________________________________ Mean: _________________________________ Median: _______________________________ Median: ________________________________ Range: ________________________________ Range: ________________________________ 3. 72, 91, 93, 89, 77, 82 4. 1.2, 0.4, 1.2, 2.4, 1.7, 1.6, 0.9, 1.0 Mean: _________________________________ Mean: _________________________________ Median: _______________________________ Median: ________________________________ Range: ________________________________ Range: ________________________________ The data sets below show the ages of the members of two clubs. Use the data for 5–9. Club A: 42, 38, 40, 34, 35, 48, 38, 45 Club B: 22, 44, 43, 63, 22, 27, 58, 65 5. Find the mean, median, range, and interquartile range for Club A. _________________________________________________________________________________________ 6. Find the mean, median, range, and interquartile range for Club B. _________________________________________________________________________________________ 7. Find the standard deviation for each club. Round to the nearest tenth. _________________________________________________________________________________________ 8. Use your statistics to compare the ages and the spread of ages on the two clubs. _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ 9. Members of Club A claim that they have the “younger” club. Members of Club B make the same claim. Explain how that could happen. _________________________________________________________________________________________ _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 149 Name ________________________________________ Date __________________ Class __________________ LESSON 9-1 Measures of Center and Spread Practice and Problem Solving: C The data sets below show the price that a homeowner paid, per therm, for natural gas during each of the first ten months of 2011 and 2012. Use the data for 1–4. 2011: $1.59, $1.72, $1.71, $1.86, $2.32, $2.54, $2.45, $2.80, $2.38, $2.25 2012: $1.57, $1.61, $1.96, $1.71, $1.98, $2.17, $2.51, $2.44, $2.52, $2.10 1. Find the mean, median, range, and interquartile range for 2011. _________________________________________________________________________________________ 2. Find the mean, median, range, and interquartile range for 2012. _________________________________________________________________________________________ 3. Find the standard deviation for each year. Round to the nearest hundredth. _________________________________________________________________________________________ 4. Use your statistics to compare the overall trend in prices for the two years. _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ Solve. 5. To earn an exemption from the final exam, Aaron needs his mean test score to be 92 or greater. If Aaron scored 90, 96, 87, and 90 on the first four tests and he has one test still to take, what is the lowest he can score and still earn an exemption? _________________________________________________________________________________________ 6. A, B, and C are positive integers with A < B < C. The mean of A, B, and C is 25, and their median is 10. Find all possible values for C. _________________________________________________________________________________________ 7. A teacher gave a test to 24 students and recorded the scores as a data set. Afterward, the teacher realized that the total number of points on the test added up to 96 instead of 100. To correct this, she added four points to each student’s score. How did the mean, median, range, interquartile range, and standard deviation change from the original data set of scores when she added four points to each score? _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 150 Name ________________________________________ Date __________________ Class __________________ LESSON 9-1 Measures of Center and Spread Practice and Problem Solving: Modified Two students, Brad and Jin, had the test scores shown below. Use their data for 1–10. The first one is done for you. Brad: Jin: 70, 76, 78, 80, 90, 94, 94, 98 80, 82, 84, 84, 86, 86, 88, 90 1. Find Brad’s mean test score. 2. Find Jin’s mean test score. 85 ________________________________________ ________________________________________ 3. Find Brad’s median test score. 4. Find Jin’s median test score. ________________________________________ ________________________________________ 5. Find Brad’s range. 6. Find Jin’s range. ________________________________________ ________________________________________ 7. Find Brad’s first and third quartiles. 8. Find Jin’s first and third quartiles. ________________________________________ ________________________________________ 9. Find Brad’s interquartile range. 10. Find Jin’s interquartile range. ________________________________________ ________________________________________ Use your statistics from 1–10 to solve. The first one is done for you. 11. In what ways are Brad’s and Jin’s test scores similar? Possible answer: Their means are equal and their medians are equal. _________________________________________________________________________________________ 12. In what ways are Brad’s and Jin’s test scores different? _________________________________________________________________________________________ 13. Which of the two students would you consider a more consistent test taker? Explain your thinking. _________________________________________________________________________________________ _________________________________________________________________________________________ 14. One of the students has test scores with a standard deviation of 3 and the other has test scores with a standard deviation of 9.6. Without calculating, how can you tell which student has each standard deviation? _________________________________________________________________________________________ _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 151 Name ________________________________________ Date __________________ Class __________________ LESSON 9-2 Data Distributions and Outliers Practice and Problem Solving: A/B For each data set, determine if 100 is an outlier. Explain why or why not. 1. 60, 68, 100, 70, 78, 80, 82, 88 2. 70, 75, 77, 78, 100, 80, 82, 88 ________________________________________ ________________________________________ ________________________________________ ________________________________________ The table below shows a major league baseball player’s season home run totals for the first 14 years of his career. Use the data for Problems 3–8. Season 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Home Runs 18 22 21 28 30 29 32 40 33 34 28 29 22 20 3. Find the mean and median. 4. Find the range and interquartile range. ________________________________________ ________________________________________ 5. Make a dot plot for the data. 6. Examine the dot plot. Do you think any of the season home run totals are outliers? Then test for any possible outliers. _________________________________________________________________________________________ _________________________________________________________________________________________ 7. The player wants to predict how many home runs he will hit in his 15th season. Could he use the table or the dot plot to help him predict? Explain your reasoning. _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ 8. Suppose the player hits 10 home runs in his 15th season. Which of the statistics from Problems 3 and 4 would change? _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 154 Name ________________________________________ Date __________________ Class __________________ LESSON 9-2 Data Distributions and Outliers Practice and Problem Solving: C For each data set, determine if 100 is an outlier. Explain why or why not. 1. 90, 56, 78, 82, 75, 68, 88, 100, 75 2. 123, 111, 122, 100, 109, 117, 125, 121, 130 ________________________________________ ________________________________________ ________________________________________ ________________________________________ The table below shows the age of 20 presidents of the United States upon first taking office. Use the data for Problems 3–8. 54 42 51 56 55 51 54 51 60 62 43 55 56 61 52 69 64 46 54 47 3. Find the mean and median. 4. Find the range and interquartile range. ________________________________________ ________________________________________ 5. Make a dot plot for the data. 6. Examine the dot plot. Describe any patterns you see in the data. Could these patterns be seen in the original data set? _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ 7. Examine the dot plot. Test for any possible outliers. _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ 8. The most recent president of the United States not included in the data set above was Grover Cleveland, who took office on March 4, 1893. Based on your work so far, make an educated guess as to his age that day. Explain your reasoning. Then find his age. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 155 Name ________________________________________ Date __________________ Class __________________ LESSON 9-2 Data Distributions and Outliers Practice and Problem Solving: Modified Identify each dot plot as symmetric, skewed to the left, or skewed to the right. The first one is done for you. 1. 2. skewed to the left ________________________________________ ________________________________________ 3. 4. ________________________________________ ________________________________________ The table below shows the scores of ten golfers in a tournament. Use the data for Problems 5–10. The first one is done for you. 68 69 70 73 74 74 74 75 75 76 5. Find the mean and median. 6. Find the range and interquartile range. mean: 72.8; median: 74 ________________________________________ ________________________________________ 7. Make a dot plot for the data. 8. Identify the dot plot as symmetric, skewed to the left, or skewed to the right. _________________________________________________________________________________________ 9. Suppose an 11th golfer with a score of 95 is added to the tournament scores. Which of the statistics from Problems 5 and 6 would change? _________________________________________________________________________________________ 10. If a score of 95 were added, would it be an outlier? Explain. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 156 Name ________________________________________ Date __________________ Class __________________ LESSON 9-3 Histograms and Box Plots Practice and Problem Solving: A/B Solve each problem. Fire and Rescue Service 1. The number of calls per day to a fire and rescue service for three weeks is given below. Use the data to complete the frequency table. Number of Calls 0−3 4−7 Calls for Service 5 17 2 12 0 19 16 8 2 6 3 8 15 1 11 13 18 3 10 6 Frequency 8−11 4 12−15 16−19 2. Use the frequency table in Exercise 1 to make a histogram with a title and axis labels. 3. Which intervals have the same frequency? ________________________________________ 4. Use the histogram to estimate the mean. Then compare your answer with the actual mean, found by using the original data. _________________________________________________________________________________________ _________________________________________________________________________________________ Use the box plot for Problems 5–7. 5. Find the median temperature. 6. Find the range. ________________________________________ ________________________________________ 7. Determine whether the temperature of 50 °F is an outlier. _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 159 Name ________________________________________ Date __________________ Class __________________ LESSON 9-3 Histograms and Box Plots Practice and Problem Solving: C The histogram below shows the population distribution, by age, for the city of Somerville. Use the histogram to solve the problems that follow. 1. What is the approximate total population of the city? _________________________________________________ 2. Which age intervals have approximately the same total population? _________________________________________________ 3. Use the histogram to estimate the mean age. Show your work. _________________________________________________ _________________________________________________ _________________________________________________ _________________________________________________ 4. A student claims that the distribution is roughly symmetric. Do you agree? Why or why not? _________________________________________________________________________________________ _________________________________________________________________________________________ Use the data for Problems 5–7. Harmon Killebrew and Willie Mays were two of baseball’s all-time greatest home run hitters. Their season home run totals are shown below. Harmon Killebrew: 0, 4, 5, 2, 0, 42, 31, 46, 48, 45, 49, 25, 39, 44, 17, 49, 41, 28, 26, 5, 13, 14 Willie Mays: 20, 4, 41, 51, 36, 35, 29, 34, 29, 40, 49, 38, 47, 52, 37, 22, 23, 13, 28, 18, 8, 6 5. Make a double box plot for Killebrew and Mays. 6. Find mean and median season home run totals for Killebrew and Mays. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 160 Name ________________________________________ Date __________________ Class __________________ LESSON 9-3 Histograms and Box Plots Practice and Problem Solving: Modified The heights of players at a basketball game are given in the table below. Use the data for 1–7. Players’ Heights (in.) 75 78 80 87 72 80 81 83 85 78 76 81 77 78 83 83 78 82 79 80 75 84 82 90 1. Use the data to make a frequency table. The first one is started for you. 2. Use your frequency table to make a histogram for the data. Players’ Heights Heights (in.) Frequency 72–76 4 77–81 82–86 87–91 3. Which interval has the most players? 4. Describe the shape of the distribution. ________________________________________ ________________________________________ 5. Use the midpoints of each interval to estimate the mean height of a player. _________________________________________________________________________________________ Use the box plot for 6–9. The first one is done for you. 6. Find the median age. 7. Find the range. 28 ________________________________________ ________________________________________ 8. Find the interquartile range. 9. Find the age of the oldest player. ________________________________________ ________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 161 Name ________________________________________ Date __________________ Class __________________ LESSON 9-4 Normal Distributions Practice and Problem Solving: A/B A collection of data follows a normal distribution. Find the percent of the data that falls within the indicated range of the mean. 1. one standard deviation of the mean 2. three standard deviations of the mean ________________________________________ ________________________________________ 3. two standard deviations above the mean 4. one standard deviation below the mean ________________________________________ ________________________________________ The amount of cereal in a carton is listed as 18 ounces. The cartons are filled by a machine, and the amount filled follows a normal distribution with mean of 18 ounces and standard deviation of 0.2 ounce. Use this information for 5–7. 5. Find the probability that a carton of cereal contains less than its listed amount. _________________________________________________________________________________________ 6. Find the probability that a carton of cereal contains between 18 ounces and 18.4 ounces. _________________________________________________________________________________________ 7. Find the probability that a carton of cereal contains between 17.6 ounces and 18.2 ounces. _________________________________________________________________________________________ Suppose the manufacturer of the cereal above is concerned about your answer to Problem 5. A decision is made to leave the amount listed on the carton as 18 ounces while increasing the mean amount filled by the machine to 18.4 ounces. The standard deviation remains the same. Use this information for 8–11. 8. Find the probability that a carton contains less than its listed amount. _________________________________________________________________________________________ 9. Find the probability that a carton contains more than its listed amount. _________________________________________________________________________________________ 10. Find the probability that a carton now contains more than 18.2 ounces. _________________________________________________________________________________________ 11. Find the probability that a carton is more than 0.2 ounce under the weight listed on the carton. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 164 Name ________________________________________ Date __________________ Class __________________ LESSON 9-4 Normal Distributions Practice and Problem Solving: C When a fair coin is tossed, it has a probability p of 0.5 that it will land showing Heads. If the coin is tossed n times, it can land showing Heads anywhere from 0 to n times. 1. Find the probability that a fair coin tossed n times will never land showing Heads. Evaluate for n = 5 and write as a percent. _________________________________________________________________________________________ 2. Suppose a fair coin is tossed 1000 times. If you had to predict the number of times it will land showing Heads, what would your prediction be? Justify your answer. _________________________________________________________________________________________ The number of Heads obtained when a coin is tossed n times obeys a probability rule called the Binomial Distribution. For large n, this rule can be approximated using a normal distribution. In the case of a fair coin, the mean is 0.5n and the standard deviation is 0.5 n . Use the normal distribution to estimate the following probabilities. 3. The probability that a fair coin tossed 100 times lands showing Heads between 45 and 55 times _________________________________________________________________________________________ 4. The probability that a fair coin tossed 100 times lands showing Heads fewer than 45 times _________________________________________________________________________________________ 5. The probability that a fair coin tossed 100 times lands showing Heads more than 65 times _________________________________________________________________________________________ 6. The probability that a fair coin tossed 2500 times lands showing Heads between 1200 and 1300 times _________________________________________________________________________________________ Solve. 7. A coin is tossed 400 times as part of an experiment and lands showing Heads 221 times. A student concludes that this is not a fair coin. What do you think? Justify your reasoning. _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 165 Name ________________________________________ Date __________________ Class __________________ LESSON 9-4 Normal Distributions Practice and Problem Solving: Modified Solve. The first one is started for you. 1. The graph below shows a normal curve that has been divided into eight sections. Each section represents one standard deviation above or below the mean. Fill in the percentage of the area under the curve that each section contains. 2. Find the sum of the percents written above the curve in Problem 1. Explain why that sum makes sense. _________________________________________________________________________________________ 3. Use your curve above to complete each sentence with the correct percent for normally distributed data. (a) _________________ % lie within 1 standard deviation of the mean. (b) _________________% lie within 2 standard deviations of the mean. (c) _________________% lie within 3 standard deviations of the mean. A company selling light bulbs claims in its advertisements that its light bulbs’ average life is 1000 hours. In fact, the life span of these light bulbs is normally distributed with a mean of 1000 hours and a standard deviation of 100 hours. Use this information for Problems 4–7. The first one is done for you. 4. Find the probability that a randomly chosen light bulb will last between 34% 1000 and 1100 hours. _______________________________________________________ 5. Find the probability that a randomly chosen light bulb will last less than 900 hours. _______________________________________________________________ 6. Find the probability that a randomly chosen light bulb will last more than 1200 hours. ___________________________________________________________ 7. Find the probability that a randomly chosen light bulb will last between 800 and 1100 hours. ________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 166 Name ________________________________________ Date __________________ Class __________________ LESSON 10-1 Scatter Plots and Trend Lines Practice and Problem Solving: A/B Graph a scatter plot and find the correlation. 1. The table shows the number of juice drinks sold at a small restaurant from 11:00 am to 1:00 pm. Graph a scatter plot using the given data. Time 11:00 11:30 12:00 12:30 1:00 Number of Drinks 20 29 34 49 44 2. Name the two variables. ____________________________ 3. Write positive, negative, or none to describe the correlation illustrated by the scatter plot you drew in problem 1. Estimate the value of the correlation coefficient, r. Indicate whether r is closer to −1, −0.5, 0, 0.5, or 1. _________________________________________________________________________________________ A city collected data on the amount of ice cream sold in the city each day and the amount of suntan lotion sold at a nearby beach each day. 4. Do you think there is causation between the city’s two variables? If so, how? If not, is there a third variable involved? Explain. _________________________________________________________________________________________ Solve. 5. The number of snowboarders and skiers at a resort per day and the amount of new snow the resort reported that morning are shown in the table. Amount of New Snow (in inches) 2 Number of Snowsliders 1146 4 6 8 10 1556 1976 2395 2490 a. Make a scatterplot of the data. b. Draw a line of fit on the graph above and find the equation for the linear model. __________ c. If the resort reports 15 inches of new snow, how many skiers and snowboarders would you expect to be at the resort that day? _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 170 Name ________________________________________ Date __________________ Class __________________ LESSON 10-1 Scatter Plots and Trend Lines Practice and Problem Solving: C Graph a scatter plot and find the correlation. 1. A biologist in a laboratory comes up with the following data points. Make a scatter plot using the data in the table. x 2 6 9 14 16 21 25 28 y 3 7 15 33 38 35 40 41 2. Draw a line of fit on the graph and find the equation for the liner model. Estimate the correlation coefficient, r (choose 1, 0.5, 0, −0.5, or −1). ________________________________________ ________________________________________ 3. Use a graphing calculator to find the equation for the line of best fit for the data presented in the table above. Use a graphing calculator to find the correlation coefficient, r. ________________________________________ ________________________________________ 4. Compare the results you found in step 3, using a graphing calculator, to those you found in step 2, estimating. The calculator provides a line of BEST fit, while the line you drew by hand is called a line of fit. Explain the difference. _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 171 Name ________________________________________ Date __________________ Class __________________ LESSON 10-1 Scatter Plots and Trend Lines Practice and Problem Solving: Modified Write positive, negative, or none to describe the correlation in each scatter plot. The first one is done for you. 1. 2. positive ________________________________________ ________________________________________ 3. 4. ________________________________________ ________________________________________ State whether you would expect positive, negative, or no correlation between the two data sets. The first one is done for you. 5. temperature and ice cream sales positive ________________________________________ 6. a child’s age and the time it takes him or her to run a mile ________________________________________ 7. the month of a person's birth and the time it takes to run a mile ________________________________________ Solve. 8. Look at your answer for Exercise 6. Explain your thinking. Then discuss whether you think there is causation. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 172 Name ________________________________________ Date __________________ Class __________________ LESSON 10-2 Fitting a Linear Model to Data Practice and Problem Solving: A/B The table below lists the ages and heights of 10 children. Use the data for 1−5. 2 3 3 4 4 4 5 5 5 6 H, height in inches 30 33 34 37 35 38 40 42 43 42 A, age in years 1. Draw a scatter plot and line of fit for the data. 2. A student fit the line H = 3.5A + 23 to the data. Graph the student’s line above. Then calculate the student’s predicted values and residuals. A, age in years 2 3 3 4 4 4 5 5 5 6 H, height in inches 30 33 34 37 35 38 40 42 43 42 Predicted Values Residuals 3. Use the graph below to make a residual plot. 4. Use your residual plot to discuss how well the student’s line fits the data. _________________________________________________________________________________________ 5. Use the student’s line to predict the height of a 20-year-old man. Discuss the reasonableness of the result. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 175 Name ________________________________________ Date __________________ Class __________________ LESSON 10-2 Fitting a Linear Model to Data Practice and Problem Solving: C Use the scatter plot, fitted line, and residual plot for 1−5. 1. Find the equation of the line of fit shown above. _________________________________________________________________________________________ 2. Use the line of fit to predict the height of a 20-year old man. Discuss the suitability of the linear model for extrapolation in this case. _________________________________________________________________________________________ _________________________________________________________________________________________ 3. Examine the residual plot. Does the distribution seem suitable? Discuss any issues you see. _________________________________________________________________________________________ _________________________________________________________________________________________ 4. The data for the scatter plot is shown in the first two rows of the table below. Complete the next two rows of the table. A 2 3 3 4 4 4 5 5 5 6 H 30 33 34 37 35 38 40 42 43 42 AH A2 5. The row sums in the table above can be used to find a line of fit. This line is called the least-squares line of best fit. Use these formulas to find the slope and y-intercept of that line: m = 10 i sum( AH ) − sum( A) i sum(H ) 10 i sum( A2 ) − (sum( A))2 b = sum(H ) i sum( A2 ) − sum( A) i sum( AH ) 10 i sum( A2 ) − (sum( A))2 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 176 Name ________________________________________ Date __________________ Class __________________ LESSON 10-2 Fitting a Linear Model to Data Practice and Problem Solving: Modified Use the table below to complete 1−6. You will complete the table at the end. x 1 2 3 4 5 6 8 8 y 7 5 4 6 3 5 3 4 Predicted Values Residuals Each scatter plot shows the data given in the table. On each plot, graph the line of the equation given. The first one is done for you. 1. y = 2x + 3 2. y = − 1 x+6 3 3. y = − 2 x + 6 4. Examine the lines you drew. Which of the lines do you think best fits the data? Explain. _________________________________________________________________________________________ _________________________________________________________________________________________ 5. Does the scatter plot show positive, negative, or no correlation? Explain. _________________________________________________________________________________________ _________________________________________________________________________________________ 6. One student used the equation y = 6 − 0.5x as his line of fit for the data. Use this equation to complete the table at the top of the page. Explain whether this line is a good fit or not and why. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 177 Name ________________________________________ Date __________________ Class __________________ LESSON 11-1 Solving Linear Systems by Graphing Practice and Problem Solving: A/B Tell the number of solutions for each system of two linear equations and state if the system is consistent or inconsistent and dependent or independent. 1. 2. 3. ________________________ _______________________ ________________________ ________________________ _______________________ ________________________ ________________________ _______________________ ________________________ Solve each system of linear equations by graphing. ⎧ 6x + 3y = 12 5. ⎨ ⎩8 x + 4y = 24 ⎧ x +y =3 4. ⎨ ⎩− x + y = 1 solution: __________________________ solution: __________________________ 6. Jill babysits and earns y dollars at a rate of $8 per hour plus a $5 transportation fee. Samantha babysits and earns 2y dollars at $16 per hour plus a $10 transportation fee. Write a system of equations and graph to determine the number of hours each needs to babysit to earn the same amount of money. _______________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 181 Name ________________________________________ Date __________________ Class __________________ LESSON 11-1 Solving Linear Systems by Graphing Practice and Problem Solving: C Draw a graph of a system of linear equations that is: 1. consistent and independent 2. consistent and dependent 3. inconsistent Solve each linear system by graphing. State if there is no solution or an infinite number of solutions. ⎧4x + 3y = 9 4. ⎨ ⎩2x + y = 4 ________________________ ⎧ x − 3y = −6 7. ⎨ ⎩ x − 3y = 21 ________________________ ⎧4x − 5y = 20 5. ⎨ ⎩8 x − 12 = 10y _______________________ ⎧3x + y = 4 8. ⎨ ⎩2x − 2y = 8 _______________________ ⎧ x + 3y = 6 6. ⎨ ⎩3x + 9y = 18 ________________________ ⎧6x + 12 = 2y 9. ⎨ ⎩18 − 3y = −9 x ________________________ Write a linear system and tell how to solve by graphing. 10. The sum of two integers is 12 and the difference of the two integers is 6. What are the two integers? _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 182 Name ________________________________________ Date __________________ Class __________________ LESSON 11-1 Solving Linear Systems by Graphing Practice and Problem Solving: Modified Tell the number of solutions for each system of two linear equations and if the system is consistent or inconsistent. The first one is done for you. 1. 2. One solution, consistent __________________________ 3. _______________________ ________________________ Solve each system of linear equations by graphing. The first one is done for you. 4. x + y = 9, x -int = 9 , y -int = 9 x − y = 1, x -int = 1, y -int = −1 5. 2x + y = 8, x -int = x − y = 7, x -int = (5, 4) ________________________________________ 6. 6 x − 2y = 12, x -int = 3 x − y = 6, x -int = y -int = y -int = ________________________________________ 7. x − 2y = −4, x -int = x − 2y = 6, x -int = y -int = y -int = ________________________________________ y -int = y -int = ________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 183 Name ________________________________________ Date __________________ Class __________________ LESSON 11-2 Solving Linear Systems by Substitution Practice and Problem Solving: A/B Solve each system by substitution. Check your answer. 1. ⎧y = x − 2 ⎨ ⎩y = 4 x + 1 ________________________ 4. ⎧2x − y = 6 ⎨ ⎩ x + y = −3 ________________________ 7. ⎧3x − 2y = 7 ⎨ ⎩ x + 3y = −5 ________________________ 2. ⎧y = x − 4 ⎨ ⎩y = − x + 2 _______________________ 5. ⎧2x + y = 8 ⎨ ⎩y = x − 7 _______________________ 8. ⎧−2x + y = 0 ⎨ ⎩5 x + 3y = −11 _______________________ 3. ⎧y = 3x + 1 ⎨ ⎩y = 5 x − 3 ________________________ 6. ⎧2x + 3y = 0 ⎨ ⎩ x + 2y = −1 ________________________ 9. ⎧ 1 1 ⎪⎪ 2 x + 3 y = 5 ⎨ ⎪ 1 x + y = 10 ⎪⎩ 4 ________________________ Write a system of equations to solve. 10. A woman’s age is three years more than twice her son’s age. The sum of their ages is 84. How old is the son? _________________________________________________________________________________________ 11. The length of a rectangle is three times its width. The perimeter of the rectangle is 100 inches. What are the dimensions of the rectangle? _________________________________________________________________________________________ 12. Benecio worked 40 hours at his two jobs last week. He earned $20 per hour at his weekday job and $18 per hour at his weekend job. He earned $770 in all. How many hours did he work at each job? ________________________________________ 13. Choose one of Exercises 1–9 and graph its solution. Does the answer you found by substitution agree with the answer you got by graphing? ________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 186 Name ________________________________________ Date __________________ Class __________________ LESSON 11-2 Solving Linear Systems by Substitution Practice and Problem Solving: C Solve each system by substitution. Check your answer. 1. ⎧4 x − 9 y = 1 ⎨ ⎩ 2 x + y = −5 ________________________ 2. ⎧ 1 ⎪⎪ 2 x + y = 2 ⎨ ⎪ 2 x − 1 y = 28 ⎪⎩ 3 4 3. ⎧2 x + 4 y = 1 ⎨ ⎩ x + 6y = 1 _______________________ ________________________ Write a system of equations to solve. 4. Aaron is three times as old as his son. In ten years, Aaron will be twice as old as his son. How old is Aaron now? _________________________________________________________________________________________ 5. Kitara has 100 quarters and dimes. Their total value is $19. How many of each coin does Kitara have? _________________________________________________________________________________________ 6. A cleaning company charges a fixed amount for a house call and a second amount for each room it cleans. The total cost to clean six rooms is $250 and the total cost to clean eight rooms is $320. How much would this company charge to clean two rooms? _________________________________________________________________________________________ 7. Willie Mays and Mickey Mantle hit 88 home runs one season to lead their leagues. Mays hit 14 more home runs than Mantle that year. How many home runs did Willie Mays hit? _________________________________________________________________________________________ 8. Coco has a jar containing pennies and nickels. There is $9.20 worth of coins in the jar. If she could switch the number of pennies with the number of nickels, there would be $26.80 worth of coins in the jar. How many pennies and nickels are in the jar? _________________________________________________________________________________________ 9. Fabio paid $15.50 for five slices of pizza and two sodas. Liam paid $19.50 for six slices of pizza and three sodas. How much does a slice of pizza cost? _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 187 Name ________________________________________ Date __________________ Class __________________ LESSON 11-2 Solving Linear Systems by Substitution Practice and Problem Solving: Modified For each linear system, tell whether it is more efficient to solve for x and then substitute for x or to solve for y and then substitute for y. The first one is done for you. 1. ⎧2x + 3y = 8 ⎨ ⎩ x + 4y = 9 x ________________________ 2. ⎧−4x + y = −6 ⎨ ⎩ 3x − 2y = 2 3. ⎧7 x + 4y = 3 ⎨ ⎩ 5x − y = 6 _______________________ ________________________ For each linear system, write the expression you could substitute for x from the first equation to solve the second equation. The first one is done for you. 4. ⎧ x + 2y = 17 ⎨ ⎩3x + 5y = 94 −2y + 17 ________________________ 5. ⎧ x − 5y = 5 ⎨ ⎩2x − y = 10 6. ⎧ − x + 6y = 16 ⎨ ⎩3x + 10y = 8 _______________________ ________________________ Solve each system by substitution and check your answer. The first one is done for you. 7. ⎧ y =x +6 ⎨ ⎩3x + y = 18 (3, 9) ________________________ 10. ⎧ x − 4y = 1 ⎨ ⎩2x + y = 11 ________________________ 8. ⎧ x = 2y − 3 ⎨ ⎩2x + 5y = 30 9. ⎧ y = −x − 7 ⎨ ⎩3x + 2y = 3 _______________________ 11. ⎧ 6x + y = 17 ⎨ ⎩5 x + 2y = 6 ________________________ 12. ⎧ 4x − 3y = 2 ⎨ ⎩−7 x + y = 5 _______________________ ________________________ Write a system of equations to solve. The first one is done for you. 13. Jan is five years older than her brother Dan. The sum of their ages is 27. How old are Jan and Dan? Jan is 16 years old and Dan is 11 years old. _________________________________________________________________________________________ 14. Mariko has 30 nickels and dimes. She has 12 more nickels than dimes. How many dimes does Mariko have? _________________________________________________________________________________________ 15. It costs $35 for one adult and two children to attend a show. It costs $60 for two adults and three children to attend the same show. How much does it cost one adult to attend the show? _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 188 Name ________________________________________ Date __________________ Class __________________ LESSON 11-3 Solving Linear Systems by Adding or Subtracting Practice and Problem Solving: A/B Solve each system of linear equations by adding or subtracting. Check your answer. 1. x − 5 y = 10 2. x + y = −10 2x + 5y = 5 5 x + y = −2 ________________________________________ ________________________________________ 3. 4 x + 10 y = 2 4. −3 x − 7 y = 8 3 x − 2y = −44 −4 x + 8 y = 16 ________________________________________ ________________________________________ 5. − x + 4 y = 15 6. −4 x + 11y = 5 4 x − 11y = −5 3x + 4y = 3 ________________________________________ ________________________________________ 7. − x − y = 1 8. 3 x − 5 y = 60 − x + y = −1 4 x + 5 y = −4 ________________________________________ ________________________________________ Write a system of equations to solve. 9. A plumber charges an initial amount to make a house call plus an hourly rate for the time he is working. A 1-hour job costs $90 and a 3-hour job costs $210. What is the initial amount and the hourly rate that the plumber charges? _________________________________________________________________________________________ 10. A man and his three children spent $40 to attend a show. A second family of three children and their two parents spent $53 for the same show. How much does a child’s ticket cost? _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 191 Name ________________________________________ Date __________________ Class __________________ LESSON 11-3 Solving Linear Systems by Adding or Subtracting Practice and Problem Solving: C Solve each system of linear equations by adding or subtracting. Check your answer. 1.5 x + 3 y = 9 1 y =6 2 1 2x + y = 8 4 ________________________________________ ________________________________________ 1. 0.5 x − 3 y = 1 2. 2 x + 3. −4 x + 7 y = 11 4. 4 x − 9 y = −13 ________________________________________ 1 x+y =0 3 2 x+y =5 5 ________________________________________ 5. A theater charges $25 for adults and $15 for children. When the theater increases its prices next year, the price of a child’s ticket will increase to $18 and the cost for the members of a dance club to attend the theater will increase from $450 to $480. Write and solve a system of equations to find how many adults are in the dance club. _________________________________________________________________________________________ 6. Pearl solved a system of two linear equations. In the final step, she found herself writing “0 = 6.” Pearl thought she had done something wrong, but she had not. Explain what occurred here and how the graphs of the two equations are related. _________________________________________________________________________________________ _________________________________________________________________________________________ 7. The equations ax + by = c and dx − by = e form a system of equations where a, b, c, d, and e are real numbers with a ≠ −d . Solve the system for x. _________________________________________________________________________________________ _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 192 Name ________________________________________ Date __________________ Class __________________ LESSON 11-3 Solving Linear Systems by Adding or Subtracting Practice and Problem Solving: Modified Which method is easier to use to solve the system of equations: substitution or addition/subtraction? The first one is done for you. 1. y = 4y − 9 2x + 5y = 11 substitution ________________________ 2. 4x − 3y = 6 3. 5x + 2y = 11 2x + 3y = 18 −3x + y = 0 _______________________ ________________________ Solve each system of linear equations by adding or subtracting. Check your answer. The first one is done for you. 4. 2x − 5y = 4 5. −2x + 8y = 8 3x − 4y = −4 (12, 4) ________________________________________ ________________________________________ 6. 6x + y = 13 7. 10x − 4y = 2 3x + y = 4 9x + 4y = 17 ________________________________________ 8. x + 4y = 4 ________________________________________ 7x − y = 9 9. x − 2y = 8 7x + 2y = 24 −x + 2y = 13 ________________________________________ ________________________________________ Write a system of equations to solve. The first one is started for you. 10. The sum of two numbers is 70. When the smaller number is subtracted from the bigger number, the result is 24. Find the numbers. x + y = 70; x − y = 24 _________________________________________________________________________________________ 11. Two pairs of socks and a pair of slippers cost $30. Five pairs of socks and a pair of slippers cost $42. How much does a pair of socks cost? _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 193 Name ________________________________________ Date __________________ Class __________________ LESSON 11-4 Solving Linear Systems by Multiplying First Practice and Problem Solving: A/B Solve each system of equations. Check your answer. ⎧−3 x − 4 y = −2 1. ⎨ ⎩6 x + 4 y = 3 ⎧2 x − 2y = 14 2. ⎨ ⎩ x + 4 y = −13 ________________________________________ ________________________________________ ⎧ y − x = 17 3. ⎨ ⎩2y + 3 x = −11 ⎧ x + 6y = 1 4. ⎨ ⎩2 x − 3 y = 32 ________________________________________ ________________________________________ ⎧3 x + y = −15 5. ⎨ ⎩2 x − 3 y = 23 ⎧5 x − 2y = −48 6. ⎨ ⎩2 x + 3 y = −23 ________________________________________ ________________________________________ Solve each system of equations. Check your answer by graphing. ⎧ 4 x − 3 y = −9 7. ⎨ ⎩5 x − y = 8 ⎧3 x − 3 y = −1 8. ⎨ ⎩12 x − 2y = 16 Solve. 9. Ten bagels and four muffins cost $13. Five bagels and eight muffins cost $14. What are the prices of a bagel and a muffin? _________________________________________________________________________________________ 10. John can service a television and a cable box in one hour. It took him four hours yesterday to service two televisions and ten cable boxes. How many minutes does John need to service a cable box? _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 196 Name ________________________________________ Date __________________ Class __________________ LESSON 11-4 Solving Linear Systems by Multiplying First Practice and Problem Solving: C Solve each system of equations. Check your answer. 1 y =8 2 3 x − 4 y = −39 1. − x + 2. ________________________________________ 1 1 x+ y =5 3 4 1 2 x − y = 31 6 3 ________________________________________ 3. 5 x = 3 y + 18 4. 3x + 5y = 4 ________________________________________ 0.25 x − 6 y = 17 0.07 x + 0.4 y = −2 ________________________________________ Write a system of equations to solve. 5. Travis has $60 in dimes and quarters. If he could switch the numbers of dimes with the number of quarters, he would have $87. How many of each coin does Travis have? _________________________________________________________________________________________ 6. The total cost of a bus ride and a ferry ride is $8.00. In one month, bus fare will increase by 10% and ferry fare will increase by 25%. The total cost will then be $9.25. How much is the current bus fare? _________________________________________________________________________________________ 7. A truckload of 10-pound and 50-pound bags of fertilizer weighs 9000 pounds. A second truck carries twice as many 10-pound bags and half as many 50-pound bags as the first truck. That load also weighs 9000 pounds. How many of each bag are on the first truck? _________________________________________________________________________________________ 8. The hundreds digit and the ones digit of a three-digit number are the same. The sum of its three digits is 16. If the tens digit and the ones digit are exchanged, the number increases by 45. What is the number? _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 197 Name ________________________________________ Date __________________ Class __________________ LESSON 11-4 Solving Linear Systems by Multiplying First Practice and Problem Solving: Modified For each linear system, tell whether you would multiply the terms in the first or second equation in order to eliminate one of the variables. Then write the number by which you could multiply. The first one is done for you. 1. 3x + 2y = 12 2. x + 5y = 17 −2x + y = 16 −3x + 2y = 22 2nd Multiply the ____________________ equation Multiply the ____________________ equation 3 or −3 by ___________________________. by ___________________________. 3. 4x − 3y = 19 4. 4x + 7y = 8 5x + 12y = 32 −x + 2y = −2 Multiply the ____________________ equation Multiply the ____________________ equation by ___________________________. by ___________________________. Solve each system of equations from 1–4 and check your answer. The first one is done for you. 5. 3 x + 2y = 12 x + 5 y = 17 6. −3 x + 2y = 22 (2, 3) ________________________________________ 7. −2 x + y = 16 ________________________________________ 4 x − 3 y = 19 8. 4 x + 7 y = 8 − x + 2 y = −2 5 x − 12y = 32 ________________________________________ ________________________________________ Write a system of equations to solve. The first one is started for you. 9. A newspaper and three hot chocolates cost $7. Two newspapers and two hot chocolates cost $6. How much does one hot chocolate cost? n + 3h = 7; 2n + 2h = 6 _________________________________________________________________________________________ 10. Ariel scored on 12 two-point and three-point shots. She scored 27 points in all. How many of each shot did Ariel make? _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 198 Name ________________________________________ Date __________________ Class __________________ LESSON 12-1 Creating Systems of Linear Equations Practice and Problem Solving: A/B Write and solve a system of equations for each situation. 1. One week Beth bought 3 apples and 8 pears for $14.50. The next week she bought 6 apples and 4 pears and paid $14. Find the cost of 1 apple and the cost of 1 pear. _________________________________________________________________________________________ 2. Brian bought beverages for his coworkers. One day he bought 3 lemonades and 4 iced teas for $12.00. The next day he bought 5 lemonades and 2 iced teas for $11.50. Find the cost of 1 lemonade and 1 iced tea, to the nearest cent. _________________________________________________________________________________________ Two campgrounds rent a campsite for one night according to the following table. Use the table for 3–5. Number of campers Sunnyside Campground Green Mountain Campground 1 2 3 4 $58 $66 $74 $82 $40 $50 $60 $70 3. Write the equation for the rate charged by Sunnyside Campground. _________________________________________________________________________________________ 4. Write the equation for the rate charged by Green Mountain. _________________________________________________________________________________________ 5. Solve the system of the equations you found in Problems 3 and 4. For how many campers do the campgrounds charge the same rate? What is the rate charged for that number of campers? _________________________________________________________________________________________ Use the graph for 6–8. 6. Write the functions represented by the graph. 7. What does each function represent? What does the variable represent? _____________________________________________________ 8. Solve the system of equations. Is the intersection point shown on the graph correct? ____________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 202 Name ________________________________________ Date __________________ Class __________________ LESSON 12-1 Creating Systems of Linear Equations Practice and Problem Solving: C Use the graph for 1–3. 1. Write the equation for the line of the graph. 2. Develop a real-world scenario that could be solved by this equation. Examples may be “the number of bales of hay needed to feed 4 elephants,” or “the cost of 6 sandwiches and 4 iced teas.” Record your idea: _________________________________________________ _________________________________________________ 3. Select one point on the line. Write two more equations that also have this point as a solution. Graph the two new equations. Let x = _____________________ equations: _____________________ Let y = ______________________ __________________________________ 4. Make a chart of the information another student could use to write the equations and find the solution for all three equations. What information will you need to show? Label the columns and rows according to the scenario you chose. 5. Write your own problem, asking students to find the equations from the chart above. Write a complete solution for your problem on another sheet of paper. ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 203 Name ________________________________________ Date __________________ Class __________________ LESSON 12-1 Creating Systems of Linear Equations Practice and Problem Solving: Modified Use the situation below to complete 1–2. Gym Rats Health Club has a starting membership fee of $25 and charges $12 per month. Greens and Soy Health Club has a starting membership fee of $35 and charges $10 per month. After how many months would the cost for the two health clubs be the same? What is that cost? Write an equation for the cost of each health club, using the slope and the y-intercept. The first one is done for you. 12 1. Gym Rats: slope: ______________ 25 y-intercept: ____________ y = 12x + 25 equation: ______________________________________ 2. Greens and Soy: slope: _____________ y-intercept: _____________ equation: ______________________________________ Solve the system of equations by filling in the blanks. The first one is done for you. 10x + 35 3. 12x + 25= ______________________________ 4. 12x + 25 − 25= ______________________________ 5. 12x= ______________________________ 6. 12x − 10x= ______________________________ 7. 2x= ______________________________ 8. 2x = ______________________________ 2 9. x = ______________________________ 10. The cost for Gym Rats and Greens and Soy are the same after ________ months. 11. Using your answer from Exercise 10, what is the cost for the number of months that each health club charges the same price for? ____________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 204 Name ________________________________________ Date __________________ Class __________________ LESSON 12-2 Graphing Systems of Linear Inequalities Practice and Problem Solving: A/B Tell whether the ordered pair is a solution of the given system. ⎧y < x − 3 1. (2, −2); ⎨ ⎩y > − x + 1 ________________________ ⎧y > 2x 2. (2, 5); ⎨ ⎩y ≥ x + 2 _______________________ ⎧y ≤ x + 2 3. (1, 3); ⎨ ⎩y > 4x − 1 ________________________ Graph the system of linear inequalities. a. Give two ordered pairs that are solutions. b. Give two ordered pairs that are not solutions. ⎧y ≤ x + 4 4. ⎨ ⎩ y ≥ −2 x 1 ⎧ ⎪y ≤ x + 1 5. ⎨ 2 ⎪⎩ x + y < 3 ⎧y > x − 4 6. ⎨ ⎩y < x + 2 a. _____________________ a. _____________________ a. _____________________ b. _____________________ b. _____________________ b. _____________________ 7. Charlene makes $10 per hour babysitting and $5 per hour gardening. She wants to make at least $80 a week, but can work no more than 12 hours a week. a. Write a system of linear equations. __________________________________________________ b. Graph the solutions of the system. c. Describe all the possible combinations of hours that Charlene could work at each job. _____________________________________________________________________________________ _____________________________________________________________________________________ d. List two possible combinations. ______________________________________________________ _____________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 207 Name ________________________________________ Date __________________ Class __________________ LESSON 12-2 Graphing Systems of Linear Inequalities Practice and Problem Solving: C The coordinate grid below shows a system of two linear equations. For each problem, state the system of inequalities that generates the region indicated as its solution. Write the inequalities in terms of y. 1. Region I 2. Region II 3. Region III 4. Region IV ________________ ________________ _______________ ________________ ________________ ________________ _______________ ________________ The inequalities x ≥ −5, y ≥ −5, x + y ≤ 1, and 2 x − y ≤ 5 form a system. Use this system for Problems 5–7. 5. Graph the system. 6. Describe geometrically the shaded region that represents the system’s solution. Identify the vertices of that region. 7. Each square on the coordinate grid has an area of 1 square unit. Find the area of the shaded region in your graph above. Show your method fully. Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 208 Name ________________________________________ Date __________________ Class __________________ LESSON 12-2 Graphing Systems of Linear Inequalities Practice and Problem Solving: Modified For each inequality, write the equation of the corresponding line in slope-intercept form. Then state whether you shade above or below the line to graph the inequality. The first one is done for you. 1. 2x + y < 4 y = −2x + 4; below ________________________ 2. y ≥ 3x − 6 3. 4x − y ≤ 7 _______________________ ________________________ Tell whether the ordered pair (3, 2) is a solution of the given system. The first one is done for you. ⎧ y < 2x − 5 4. ⎨ ⎩ y > −x + 2 no ________________________ ⎧x + y ≤ 5 5. ⎨ ⎩ 3 x + 2y > 10 _______________________ ⎧ x < 3y − 2 6. ⎨ ⎩ y > 3x − 7 ________________________ Graph the system of linear inequalities. a. Give two ordered pairs that are solutions. b. Give two ordered pairs that are not solutions. The first one is started for you. ⎧y ≥ x + 1 7. ⎨ ⎩ y ≤ −2 x ⎧ y < 2x + 4 8. ⎨ ⎩ y > x −1 (−1, 0) and (−3, 2) a. _________________ a. ____________________ (0, −3) and (4, 0) b. _________________ b. ____________________ Solve. 9. Coach Jules bought more than five bats. Some were wood and some were composite. The wood bats cost $49 each and the composite bats cost $100 each. Coach Jules spent less than $400. Write the system of equations that could be used to represent this situation. Let w stand for wood bats and c stand for composite bats. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 209 Name ________________________________________ Date __________________ Class __________________ LESSON 12-3 Modeling with Linear Systems Practice and Problem Solving: A/B Write a system of equations to solve each problem. 1. For a small party of 12 people, the caterer offered a choice of a steak dinner for $12.00 per meal or a chicken dinner for $10.50 per meal. The final cost for the meals was $138.00. How many of each meal was ordered? Equations: __________________________________________ Solution: ____________________________________________ 2. A clubhouse was furnished with a total of 9 couches and love seats so that all 23 members of the club could be seated at once. Each couch seats 3 people and each love seat seats 2 people. How many couches and how many love seats are in the clubhouse? Equations: __________________________________________ Solution: ____________________________________________ 3. A small art museum charges $5 for an adult ticket and $3 for a student ticket. At the end of the day, the museum had sold 89 tickets and made $371. How many student tickets and how many adult tickets were sold? Equations: __________________________________________ Solution: ____________________________________________ 4. Cassie has a total of 110 coins in her piggy bank. All the coins are quarters and dimes. The coins have a total value of $20.30. How many quarters and how many dimes are in the piggy bank? Equations: ___________________________________________ Solution: _____________________________________________ Write a system of inequalities and graph them to solve the problem. 5. Jack is buying tables and chairs for his deck party. Tables cost $25 and chairs cost $15. He plans to spend no more than $285 and buy at least 16 items. Show and describe the solution set, and suggest a reasonable solution to the problem. Equations: ___________________________________________ Solution: _____________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 212 Name ________________________________________ Date __________________ Class __________________ LESSON 12-3 Modeling with Linear Systems Practice and Problem Solving: C Write and solve a system of linear equations for each problem. Solve each problem using two different methods. 1. A flower shop displays 41 vases for sale throughout the shop. Large vases cost $22 each and small vases cost $14 each. The vases on display have a combined value of $710. How many of each size of vase are on display? Equations: _______________________________ _______________________________ Solution: _________________________________ 2. Some members of the ski club and some faculty chaperones are on an overnight ski trip. They reserved one $120 hotel room for every 4 students and one $90 hotel room for every 2 faculty chaperones, or 27 rooms in all for $2880. How many students and how many faculty chaperones are on the trip? Equations: _______________________________ _______________________________ Solution: _________________________________ Write a system of inequalities and graph them to solve the problem. 3. Lane is buying fish for his aquarium. Tetras cost $5 each and cichlids cost $19 each. Lane would like to have at least 8 fish in all, but he can spend no more than $100. Describe the solution set and give a reasonable solution. Equations: _______________________________ _______________________________ Solution set: ______________________________ Solution: __________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 213 Name ________________________________________ Date __________________ Class __________________ LESSON 12-3 Modeling with Linear Systems Practice and Problem Solving: Modified Solve each problem. The first one is started for you. 1. A student bought some markers and pads of paper. The markers cost $2 each and the pads of paper cost $4 each. He bought 8 items in all and spent $26. How many markers and how many pads of paper did he buy? markers pads of paper Let m = _________________ and let p = _____________________ 2m + 4p = 26, m + p = 8 Equations: ____________________________________________ Multiply m + p = 8 by −2 to make opposite coefficients: −2(m + p = 8) ___________________________________ Add: 2m + 4p = 26 −2m −2p = −16 ______ = 10 p = ______ m+5=8 He bought _____ pads of paper. He bought ______ markers. m = ____________ 2. A student bought some music CDs and some movie DVDs. The CDs cost $9 each and the DVDs cost $17 each. He bought 7 items in all for $87. How many CDs and how many DVDs did he buy? Equations: _________________________________ Solution: ___________________________________ Write a system of inequalities and graph them to solve the problem. The work is started for you. 3. Alie needs to buy at least 12 candles. Plain candles sell for $4 each and scented candles sell for $7 each. She can spend no more than $57. Give one possible solution. 4p + 7s ≤ 57 Inequalities: _____________________________ Possible solution: ________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 214 Name ________________________________________ Date __________________ Class __________________ LESSON 13-1 Understanding Piecewise-Defined Functions Practice and Problem Solving: A/B Graph each piecewise-defined function. ⎧ ⎪⎪0.5 x − 1.5 1. f ( x ) = ⎨ x + 1 ⎪ 4 ⎪⎩ ⎧ ⎪⎪ −4 x − 16 2. f ( x ) = ⎨0.5 x − 4.5 ⎪ −2 ⎪⎩ x < −1 −1 ≤ x ≤ 3 x >3 x < −3 −3 ≤ x < 3 x≥3 Write equations to complete the definition of each function. 3. 4. ________________________________________ ________________________________________ 5. The graph at the right shows shipping cost as a function of purchase amount. Find the shipping cost for each purchase amount. purchase amount: $8.49 _______________ purchase amount: $20.00 ______________ purchase amount: $89.50 ______________ purchase amount: $40.01 ______________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 218 Name ________________________________________ Date __________________ Class __________________ LESSON 13-1 Understanding Piecewise-Defined Functions Practice and Problem Solving: C 1. The incomplete piecewise-defined function at the right is represented by this graph. Find real numbers a and c to complete the definition of f. Show your work. ⎧ 2 ⎪ 2 f (x) = ⎨ ax + c ⎪ −1 ⎩ x < −2 −2 ≤ x ≤ 1 x >1 _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ 2. The graph at the left below represents a piecewise-defined function f. It is defined for all real numbers x. The pattern shown continues as suggested both to the left and to the right indefinitely. Which is greater, f(48) or f(30)? Explain. ___________________________________________________________ ___________________________________________________________ ___________________________________________________________ ___________________________________________________________ 3. The diagram at the right shows the left half of the letter W. The right half of the letter is formed by reflection in the dotted line. Represent the four parts of the letter as a function f defined piecewise. Show your work. _______________________________________________________ _______________________________________________________ _______________________________________________________ _______________________________________________________ _______________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 219 Name ________________________________________ Date __________________ Class __________________ LESSON 13-1 Understanding Piecewise-Defined Functions Practice and Problem Solving: Modified ⎧ To graph f ( x ) = ⎪⎨ 2 x + 3 ⎪⎩ −2 x + 4 done for you. x ≤ 1 , fill in each blank. The first one is x >1 1. Evaluate 2x + 3 for x = 1 and x = 0. Complete the ordered pairs. 5 ) and (0, ____ 3 ) (1, ____ 2. On the grid, draw the ray that describes the first part of the function. Use the appropriate type of dot. 3. Evaluate −2x + 4 for x = 1 and x = 2. Complete the ordered pairs. (1, ____) and (2, ____) 4. On the grid, draw the ray that describes the second part of the function. Use the appropriate type of dot. To write a function f for this graph, fill in each blank. The first one is done for you. 5. For the left part of the graph, complete the ordered pairs. 1 ) and (−3, ____ 2 ) (−1, ____ 6. Find the slope of the left part. ______________ 7. Write an equation for the left part of the graph. ______________________________________________ 8. For the right part of the graph, complete the ordered pairs. (−1, ____) and (1, ____) 9. Write an equation for the right part of the graph. ______________________________________________ 4 x<2 ⎧ 10. Graph f (x) = ⎨ . ⎩− x + 4 x ≥ 2 11. Write a function for this graph. Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 220 Name ________________________________________ Date __________________ Class __________________ LESSON 13-2 Absolute Value Functions and Transformations Practice and Problem Solving: A/B Create a table of values for f(x), graph the function, and tell the domain and range. 1. f ( x ) = x − 3 + 2 x 2. f ( x ) = 2 x + 1 − 2 x f(x) ________________________________________ f(x) ________________________________________ Write an equation for each absolute value function whose graph is shown. 3. 4. ________________________________________ ________________________________________ Solve. 5. A machine is used to fill bags with sand. The average weight of a bag filled with sand is 22.3 pounds. Write an absolute value function describing the difference between the weight of an average bag of sand and a bag of sand with an unknown weight. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 223 Name ________________________________________ Date __________________ Class __________________ LESSON 13-2 Absolute Value Functions and Transformations Practice and Problem Solving: C Create a table of values for f(x), graph the function, and tell the domain and range. 1. f ( x ) = −2 x − 1 + 2 x 2. f ( x ) = − x f(x) ________________________________________ 1 x +1 + 3 2 f(x) ________________________________________ Write an equation for each absolute value function whose graph is shown. 3. 4. ________________________________________ ________________________________________ Solve. 5. Suppose you plan to ride your bicycle from Portland, Oregon, to Seattle, Washington, and back to Portland. The distance between Portland and Seattle is 175 miles. You plan to ride 25 miles each day. Write an absolute value function d(x), where x is the number of days into the ride, that describes your distance from Portland and use your function to determine the number of days it will take to complete your ride. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 224 Name ________________________________________ Date __________________ Class __________________ LESSON 13-2 Absolute Value Functions and Transformations Practice and Problem Solving: Modified Create a table of values for f(x), graph the function, and tell the domain and range. The first one is done for you. 1. f ( x ) = x + 1 x f(x) −2 3 −1 2 0 1 1 2 2 3 3 4 2. f ( x ) = x + 1 + 1 x Domain = {Real Numbers}, Range = { y ≥ 1} ____________________________________________ 3. f ( x ) = 2 x + 1 − 1 x f(x) ________________________________________ 4. f ( x ) = − x + 1 + 2 x f(x) ________________________________________ f(x) ________________________________________ Write an equation for each absolute value function whose graph is shown. The first one is done for you. 5. 6. f(x) = |x + 1| + 1 ________________________________________ ________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 225 Name ________________________________________ Date __________________ Class __________________ LESSON 13-3 Solving Absolute Value Equations Practice and Problem Solving: A/B Solve. 1. How many solutions does the equation x + 7 = 1 have? _________________________ 2. How many solutions does the equation x + 7 = 0 have? _________________________ 3. How many solutions does the equation x + 7 = −1 have? _________________________ Solve each equation algebraically. 4. x = 12 5. x = ________________________ 7. 5 + x = 14 1 2 6. x − 6 = 4 _______________________ 9. x + 3 = 10 8. 3 x = 24 ________________________ ________________________ _______________________ ________________________ Solve each equation graphically. 10. x − 1 = 2 11. 4 x − 5 = 12 ________________________________________ ________________________________________ Leticia sets the thermostat in her apartment to 68 degrees. The actual temperature in her apartment can vary from this by as much as 3.5 degrees. 12. Write an absolute-value equation that you can use to find the minimum and maximum temperature. _______________________________ 13. Solve the equation to find the minimum and maximum temperature. ______________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 228 Name ________________________________________ Date __________________ Class __________________ LESSON 13-3 Solving Absolute Value Equations Practice and Problem Solving: C Solve each equation algebraically. 1. x + 6 = −4 2. −9 x = −63 ________________________ 4. x − 1 =2 2 3. x + 11 = 0 _______________________ 5. 3 x − 1 = −15 ________________________ ________________________ 6. x − 1 − 1.4 = 6.2 _______________________ ________________________ Solve each equation graphically. 7. 4x − 1 =1 2 8. −3 5 x − 2 = −12 ________________________________________ ________________________________________ Solve. 9. A carpenter cuts boards for a construction project. Each board must be 3 meters long, but the length is allowed to differ from this value by at most 0.5 centimeters. Write and solve an absolute-value equation to find the minimum and maximum acceptable lengths for a board. _________________________________________________________________________________________ _________________________________________________________________________________________ 10. The owner of a butcher shop keeps the shop’s freezer at −5 °C. It is acceptable for the temperature to differ from this value by 1.5 °C. Write and solve an absolute-value equation to find the minimum and maximum acceptable temperatures. _________________________________________________________________________________________ _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 229 Name ________________________________________ Date __________________ Class __________________ Solving Absolute Value Equations LESSON 13-3 Practice and Problem Solving: Modified Fill in the blanks to solve each equation. The first one is done for you. 2. x + 4 = 7 1. x + 3 = 5 ____ −3 ____ −3 x = ____ 2 Case 1 −2 x = ____ Case 2 2 x = ____ 3. 5 x − 1 = 30 x − 1 = ____ Case 1 Case 2 x + 4 = ____ x + 4 = ____ − ____ − ____ − ____ − ____ x = ____ Case 1 Case 2 x − 1 = ____ x − 1 = ____ x = ____ x = ____ x = ____ Solve each equation algebraically. The first one is done for you. 4. x = 8 5. x = 14 x = −8 or x = 8 ________________________________________ ________________________________________ 6. x − 7 = 10 7. 4 x + 2 = 20 ________________________________________ ________________________________________ Solve each equation graphically. The first one is done for you. 8. 3 x = 6 9. x + 2 = 4 x = −2 or x = 2 ________________________________________ ________________________________________ Troy’s car can go 24 miles on one gallon of gas. However, his gas mileage can vary by 2 miles per gallon depending on where he drives. The first one is done for you. 10. Write an absolute-value equation that you can use to find the minimum and maximum gas mileage. x − 24 = 2 _________________________________________________________________________________________ 11. Solve the equation to find the minimum and maximum gas mileage. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 230 Name ________________________________________ Date __________________ Class __________________ LESSON 13-4 Solving Absolute Value Inequalities Practice and Problem Solving: A/B Solve each inequality and graph the solutions. 1. x − 2 ≤ 3 2. x + 1 + 5 < 7 ________________________________________ ________________________________________ 3. 3 x − 6 ≤ 9 4. x + 3 − 1.5 < 2.5 ________________________________________ ________________________________________ 5. x + 17 > 20 6. x − 6 − 7 > − 3 ________________________________________ 7. ________________________________________ 1 x+5 ≥ 2 2 8. 2 x − 2 ≥ 3 ________________________________________ ________________________________________ Solve. 9. The organizers of a drama club wanted to sell 350 tickets to their show. The actual sales were no more than 35 tickets from this goal. Write and solve an absolute-value inequality to find the range of the number of tickets that could have been sold. _________________________________________________________________________________________ 10. The temperature at noon in Los Angeles on a summer day was 88 °F. During the day, the temperature varied from this by as much as 7.5 °F. Write and solve an absolute-value inequality to find the range of possible temperatures for that day. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 233 Name ________________________________________ Date __________________ Class __________________ LESSON 13-4 Solving Absolute Value Inequalities Practice and Problem Solving: C Solve each inequality and graph the solutions. 1. x − 7 < − 4 2. x − 3 + 0.7 < 2.7 ________________________________________ 3. ________________________________________ 1 x+2 ≤1 3 4. x − 5 − 3 > 1 ________________________________________ ________________________________________ 6. x + 5. 5 x ≥ 15 ________________________________________ 1 −2≥2 2 ________________________________________ 7. x − 2 + 7 ≥ 3 8. 4 x − 6 ≥ − 8 ________________________________________ ________________________________________ Solve. 9. The ideal temperature for a refrigerator is 36.5 °F. It is acceptable for the temperature to differ from this value by at most 1.5 °F. Write and solve an absolute-value inequality to find the range of acceptable temperatures. _________________________________________________________________________________________ 10. At a trout farm, most of the trout have a length of 23.5 cm. The length of some of the trout differs from this by as much as 2.1 cm. Write and solve an absolute-value inequality to find the range of lengths of the trout. _________________________________________________________________________________________ 11. Ben says that there is no solution for this absolute-value inequality. Is he correct? If not, solve the inequality. Explain how you know you are correct. x −7 32 + <7 13 _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 234 Name ________________________________________ Date __________________ Class __________________ LESSON 13-4 Solving Absolute Value Inequalities Practice and Problem Solving: Modified Fill in the blanks to solve each inequality. The first one is done for you. 1. x + 7 ≤ 9 2. x − 1 > 3 7 − ____ 7 − ____ x − 1 < ____ OR x − 1 > ____ 2 x ≤ ____ 2 AND x ≤ ____ 2 x ≥ − ____ + ____ + ____ + ____ + ____ x < ____ OR x > ____ Solve each inequality and graph the solutions. The first one is done for you. 3. x + 1 < 5 4. x + 2 ≤ 2 x > −4 AND x < 4 ________________________________________ ________________________________________ 5. 5 x ≤ 25 6. x − 4 > − 2 ________________________________________ ________________________________________ 7. x − 1 ≥ 3 8. x + 3 − 3 > − 1 ________________________________________ ________________________________________ Solve. The first one is done for you. 9. In Mr. Garcia's class, a student receives a B if the average of the student's test scores is 85 or if the average of the scores differs from this value by at most 4 points. Write an absolute-value inequality that represents the range of scores that results in a B. x − 85 ≤ 4 _________________________________________________________________________________________ 10. Solve the inequality. What range of scores results in a B? _________________________________________________________________________________________ _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 235 Name ________________________________________ Date __________________ Class __________________ Understanding Rational Exponents and Radicals LESSON 14-1 Practice and Problem Solving: A/B Write the name of the property that is demonstrated by each equation. 1. (2a )4 = 16a 4 2. (36 )3 = 318 ________________________________________ ________________________________________ Simplify each expression. 2 3 1 3. 8 3 4. 15 5. 9 2 ________________________ _______________________ ________________________ 3 5 1 6. 25 2 7. 16 4 8. 27 3 ________________________ 1 _______________________ 2 1 9. 814 + 4 2 2 10. 343 3 • 32 5 ________________________ ________________________ 11. 100 _______________________ − 1 2 ________________________ Find the value of the expression for the value indicated. 3 1 1 12. 6a 4 for a = 16 13. c 2 + c 3 for c = 64 ________________________________________ ________________________________________ 3 5 m5 for m = 32 14. 8 15. 0.5d 7 for d = 128 ________________________________________ ________________________________________ Solve. 1 16. The equation t = 0.25d 2 can be used to find the number of seconds, t, that it takes an object to fall a distance of d feet. How long does it take an object to fall 64 feet? _________________________________________________________________________________________ 3 ⎛ 1⎞ 17. Show that ⎜ 16 4 ⎟ and 163 ⎝ ⎠ ( ) 1 4 are equivalent. _________________________________________________________________________________________ 18. The surface area, S, of a cube with volume V can be found using the 2 formula S = 6V 3 . Find the surface area of a cube whose volume is 125 cubic inches. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 239 Name ________________________________________ Date __________________ Class __________________ Understanding Rational Exponents and Radicals LESSON 14-1 Practice and Problem Solving: C Simplify each expression. Assume all variables represent positive numbers. 2 1 1. 27 3 − 125 3 2. ________________________ ( 4. 16b 7. 3 4 4 ) (a ) 4 3 3. 25 _______________________ 5. 25 42 − _______________________ 2 512 3 k4 • ________________________ (w ) −2 9. 1 k2 3 ⎞2 ⎟⎟ ⎠ ________________________ 3 8. 3 2 ________________________ ⎛ 2 6. ⎜⎜ n 3 ⎝ 25 83 ________________________ 2 100,000 5 − 3 w −8 _______________________ ________________________ Find the value of the expression for the value indicated. 10. 100m −2 for m = 5 11. ________________________________________ 12. ( 81 ) a a 1 2 for a = ________________________________________ 27 x 2 for x = −x 3 27 13. ________________________________________ ( 441 k + 784k ) k for k = 1 2 ________________________________________ Solve. 14. Use the Quotient of Powers Property to explain why a0 must equal a2 1 for all positive values of a. Hint: Examine 2 . a _________________________________________________________________________________________ _________________________________________________________________________________________ 15. Use your knowledge of fractional exponents to show that the following statement is true: The square root of the cube root of a number equals the sixth root of that number. _________________________________________________________________________________________ _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 240 Name ________________________________________ Date __________________ Class __________________ Understanding Rational Exponents and Radicals LESSON 14-1 Practice and Problem Solving: Modified Simplify each expression. The first one is done for you. 1. 3 2. 16 4 ________________ 27 3. ________________ 4. 10,000 _______________ 5 32 ________________ Rewrite each expression using a fractional exponent. The first one is done for you. 5. 3 125 4 6. 1 3 125 ________________ 53 7. ________________ 6 645 8. _______________ ________________ Use a property of rational exponents to simplify each expression. The first one is done for you. m8 9. a 4 • a5 10. 11. c 6 2 m ( ) 9 a ________________________ _______________________ 10 7 ________________________ Simplify each expression. The first one is done for you. 1 1 1 12. 25 2 13. 16 4 14. 27 3 5 ________________________ _______________________ ________________________ 3 3 4 15. 16 4 16. 25 2 17. 8 3 ________________________ _______________________ ________________________ Solve. The first one is done for you. 1 18. The equation t = 0.25d 2 can be used to find the number of seconds, t, that it takes an object to fall a distance of d feet. How long does it take an object to fall 100 feet? 2.5 seconds _________________________________________________________________________________________ 19. The side length, s, of a cube with volume V can be found using the 1 formula s = V 3 . Find the side length of a cube whose volume is 216 cubic inches. _________________________________________________________________________________________ 20. Use a fractional exponent to write the expression the fourth root of 81 raised to the ninth power. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 241 Name ________________________________________ Date __________________ Class __________________ LESSON 14-2 Simplifying Expressions with Rational Exponents and Radicals Practice and Problem Solving: A/B Simplify each expression. 1. 5 y5 ________________________ 25y 4 4. x 4 y 12 2. ________________________ 3. _______________________ 5. 3 3 a 6b 3 ________________________ x 6y 9 (9y 2 )2 6. _______________________ (9y 2 )2 ________________________ 1 7. 5 (32y 5 )3 ________________________ 10. 4 8 ( xy ) ________________________ 8. ( x 3 y )3 x 2 y 2 9. _______________________ 1 2 4 11. ( x ) 3 (27y 3 )4 6 (27y 3 )4 ________________________ 1 x 6 12. _______________________ ( x 4 )8 3 x3 ________________________ Solve. 1 13. Given a cube with volume V, you can use the formula P = 4V 3 to find the perimeter of one of the cube’s square faces. Find the perimeter of a face of a cube that has volume 125 m3. _________________________________________________________________________________________ 14. The Beaufort Scale measures the intensity of tornadoes. For a tornado 3 2 with Beaufort number B, the formula v = 1.9B may be used to estimate the tornado’s wind speed in miles per hour. Estimate the wind speed of a tornado with Beaufort number 9. _________________________________________________________________________________________ 1 ⎛ V ⎞2 15. At a factory that makes cylindrical cans, the formula r = ⎜ ⎟ is used ⎝ 12 ⎠ to find the radius of a can with volume V. What is the radius of a can with a volume of 192 cm3? _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 244 Name ________________________________________ Date __________________ Class __________________ Simplifying Expressions with Rational Exponents and Radicals LESSON 14-2 Practice and Problem Solving: C Find a path from start to finish in the maze below. Each box that you pass through must have a value that is greater than or equal to the value in the previous box. You may only move horizontally or vertically to go from one box to the next. START 1 1 1 36 2 − 216 3 1 1 1 4 2 − 27 3 05 1 16 4 512 2 9 1 3 1 15 64 6 3 2 1000 3 9 +3 1 4 4 32 3 1 100 2 3 2 5 1 16 4 1 1 1 1 92 − 83 814 49 2 + 0 2 3 2 216 3 2 1 2 16 4 + 32 5 2 125 3 3 12 − 9 2 2 1 25 2 − 32 5 1 13 − 8 3 4 27 1 102410 1 4 625 128 7 2 3 1 9 − 30 4 1 814 + 49 2 1 243 5 1 1 144 2 − 812 3 2 64 3 1 1 125 3 − 20 814 − 32 5 1 625 4 2 2 2 3 1 64 6 − 125 3 243 5 125 3 128 7 243 5 5 16 4 + 4 2 3 32 5 + 0 4 1 2 1 2 121 + 1 16 4 − 12 1 1 0 2 1 1 1 3 3 100 2 + 27 3 16 2 − 16 4 1 3 1 83 + 92 256 2 64 3 32 5 + 100 2 FINISH Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 245 Name ________________________________________ Date __________________ Class __________________ Simplifying Expressions with Rational Exponents and Radicals LESSON 14-2 Practice and Problem Solving: Modified Match each expression with a fractional exponent to an equivalent radical expression. The first one is done for you. 1 1. x 2 B A. ( x )3 1 2. x 3 ____________ B. ____________ C. ( 3 x )2 ____________ D. x 2 3. x 3 3 4. x 2 3 x Rewrite each expression using a fractional exponent. The first one is done for you. 5. 5 x 6. 4 x5 3 7. 182 8. 2 106 1 x5 ________________ ________________ _______________ ________________ Simplify each expression. The first one is done for you. 1 1 1 9. 49 2 10. 814 11. 13 7 ________________________ 1 _______________________ 5 1 12. 8 3 + 100 2 ________________________ 13. 8 3 ________________________ x 16 14. _______________________ ________________________ Solve. The first one is started for you. 1 2 15. Given a square with area x, you can use the formula d = 1.4x to estimate the length of the diagonal of the square. Use the formula to estimate the length of the diagonal of a square with area 100 cm2. 1 d = 1.4(100 2 ) = _________________________________________________________________________________________ 16. For a pendulum with a length of L meters, the time in seconds that it 1 takes the pendulum to swing back and forth is 2L2 . How long does it take a pendulum that is 9 meters long to swing back and forth? _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 246 Name ________________________________________ Date __________________ Class __________________ Understanding Geometric Sequences LESSON 15-1 Practice and Problem Solving: A/B Find the common ratio r for each geometric sequence and use r to find the next three terms. 1. 3, 9, 27, 81, … r = ______ 2. 972, 324, 108, 36, … Next three terms: _______________________ r = ______ Next three terms: _______________________ Complete. 3. The 11th term in a geometric sequence is 48 and the common ratio is 4. The 12th term is _________ and the 10th term is ________. 4. 7 and 105 are successive terms in a geometric sequence. The term following 105 is _________________________ . Find the common difference d of the arithmetic sequence and write the next three terms. 5. 6, 11, 16, 21, … d = _____ 6. 7, 4, 1, −2, … d = _____ Next three terms: _______________________ Next three terms: _______________________ Use the table to answer Exercise 7. Bounce Height 1 24 2 12 3 6 7. A ball is dropped from the top of a building. The table shows its height in feet above ground at the top of each bounce. What is the height of the ball at the top of bounce 5? _______ 8. Tom’s bank balances at the end of months 1, 2, and 3 are $1600, $1664, and $1730.56. What will Tom’s balance be at the end of month 5? ____________ 9. Consider the geometric sequence 6, −18, 54…. Select all that apply. • A. The common ratio is 3. • B. The 6th term is −1458. • C. The 4th term is −3 times 54. • D. 6(−3)11 is smaller than 6(−3)10. Find the indicated term by using the common ratio. 10. 108, −72, 48, …; 5th term ________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 250 Name ________________________________________ Date __________________ Class __________________ Understanding Geometric Sequences LESSON 15-1 Practice and Problem Solving: C Find the common ratio r for each geometric sequence and use r to find the next three terms. 1. 4, 5, 6.25, … r = ______ 2. 864, −288, 96, … Next three terms: _______________________ r = ______ Next three terms: _______________________ Complete. 3. The 11th term in a geometric sequence is 48 and the common ratio is −0.8. The 12th term is _________ and the 10th term is ________. 4. 8.5 and 11.9 are successive terms in a geometric sequence. The term following 11.9 is ______________________________ . Find the common difference d of the arithmetic sequence and write the next three terms. 5. 8, 17.6, 27.2, … d = _____ 6. 4, −2.5, −9, … d = ______ Next three terms: _______________________ Next three terms: _______________________ Use the table to answer Exercise 7. Bounce Height 1 36 2 27 3 20.25 7. A ball is dropped from the top of a building. The table shows its height in feet above ground at the top of each bounce. To the nearest hundredth, what is the height of the ball at the top of bounce 5? 8. Lee’s bank balances at the end of months 1, 2, and 3 are $1600, $1640, and $1681. What will Lee’s balance be at the end of month 5? ____________ 9. Consider the geometric sequence −12, 19.2, −30.72…. Select all that apply. • A. The common ratio is −1.6. • B. The 5th term is 78.6432. • • D. −12(−1.6)9 is smaller than −12(−1.6)8. C. The 4th term is 1.6 times −30.72. Find the indicated term by using the common ratio. 10. 108, −27, 6.75, …; 5th term ________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 251 Name ________________________________________ Date __________________ Class __________________ Understanding Geometric Sequences LESSON 15-1 Practice and Problem Solving: Modified Find the common ratio, r, for each geometric sequence and use r to find the next three terms. The first one is done for you. 5 1. 2, 10, 50, 250, … r = ______ 2. 4, 24, 144, 864, … 1250, 6250, 31,250 Next three terms: _______________________ r = ______ Next three terms: _______________________ Complete. 3. The 4th term in a geometric sequence is 24 and the common ratio is 2. The 5th term is _________ and the 3rd term is ________. 4. 6 and 24 are successive terms in a geometric sequence. The term following 24 is ___________________________ . Find the common difference, d, of the arithmetic sequence and write the next three terms. The first one is started for you. 3 5. 6, 9, 12, 15, … d = ______ 6. 5, 2, −1, −4, … d = ______ Next three terms: _______________________ Next three terms: _______________________ Complete the tables. The first one is started for you. 7. 8. Arithmetic Arithmetic Common Term Number Term Difference Geometric Term Number Geometric Common Term Ratio 1 6 5 ____ 1 4 _____ 2 11 _____ 2 24 _____ 3 16 _____ 3 144 _____ 4 21 _____ 4 864 _____ 9. A population of animals declines in a manner that closely resembles a geometric sequence. Given this table of values, how large is the population: Year Number of Animals 1 36 In year 4? ________ animals 2 27 In year 5? ________ animals 3 20.25 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 252 Name ________________________________________ Date __________________ Class __________________ Constructing Geometric Sequences LESSON 15-2 Practice and Problem Solving: A/B Complete. 1. Below are the first five terms of a geometric series. Fill in the bottom row by writing each term as the product of the first term and a power of the common ratio. N 1 2 3 4 5 f (n) 3 12 48 192 768 f (n) The general rule is f(n) = __________. Each rule represents a geometric sequence. If the given rule is recursive, write it as an explicit rule. If the rule is explicit, write it as a recursive rule. Assume that f(1) is the first term of the sequence. 2. f(n) = 11(2)n − 1 3. f(1) = 2.5; f(n) = f(n − 1) • 3.5 for n ≥ 2 ________________________________________ 4. f(1) = 27; f(n) = f(n − 1) • ________________________________________ 1 for n ≥ 2 3 5. f ( n) = −4(0.5)n − 1 ________________________________________ ________________________________________ Write an explicit rule for each geometric sequence based on the given terms from the sequence. Assume that the common ratio r is positive. 6. a1 = 90 and a2 = 360 7. a1 = 16 and a3 = 4 ________________________________________ ________________________________________ 8. a1 = 2 and a5 = 162 9. a2 = 30 and a3 = 10 ________________________________________ ________________________________________ A bank account earns a constant rate of interest each month. The account was opened on March 1 with $18,000 in it. On April 1, the balance in the account was $18,045. Use this information for 10–12. 10. Write an explicit rule and a recursive rule that can be used to find A(n), the balance after n months. _________________________________________________________________________________________ 11. Find the balance after 5 months. __________ 12. Find the balance after 5 years. __________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 255 Name ________________________________________ Date __________________ Class __________________ LESSON 15-2 Constructing Geometric Sequences Practice and Problem Solving: C Each rule represents a geometric sequence. If the given rule is recursive, write it as an explicit rule. If the rule is explicit, write it as a recursive rule. Assume that f(1) is the first term of the sequence. 1. f(1) = 2 ; f(n) = f(n − 1) • 8 for n ≥ 2 3 2. f(n) = −10(0.4)n − 1 ________________________________________ ________________________________________ Write an explicit rule for each geometric sequence based on the given terms from the sequence. Assume that the common ratio r is positive. 3. a1 = 6 and a4 = 162 4. a2 = 9 and a4 = 2.25 ________________________________________ ________________________________________ 5. a4 = 0.01 and a5 = 0.0001 6. a3 = ________________________________________ 7. a3 = 32 and a6 = 1 1 and a4 = 48 192 ________________________________________ 256 125 8. a2 = −4 and a4 = −9 ________________________________________ ________________________________________ Solve. 9. A geometric sequence contains the terms a3 = 40 and a5 = 640. Write the explicit rules for r > 0 and for r < 0. _________________________________________________________________________________________ 10. The sum of the first n terms of the geometric sequence f(n) = ar n − 1 a(r n − 1) . Use this formula to find the can be found using the formula r −1 sum 1 + 3 + 32 + 33 + ... + 310. Check your answer the long way. _________________________________________________________________________________________ 11. An account earning interest compounded annually was worth $44,100 after 2 years and $48,620.25 after 4 years. What is the interest rate? _________________________________________________________________________________________ 12. There are 64 teams in a basketball tournament. All teams play in the first round but only winning teams move on to subsequent rounds. Write an explicit rule for T(n), the number of games in the nth round of the tournament. State the domain: _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 256 Name ________________________________________ Date __________________ Class __________________ Constructing Geometric Sequences LESSON 15-2 Practice and Problem Solving: Modified Complete: The first one is done for you. 1. Below are the first five terms of a geometric series. Fill in the bottom row by writing each term as the product of the first term and a power of the common ratio. N 1 f (n) 3 f (n) 3(4) 2 3 12 0 3(4) 4 48 1 3(4) 5 192 2 3(4) 3 3(4)n−1 The general rule is f(n) = __________. 768 3(4)4 2. Below are the first five terms of a geometric series. Fill in the bottom row by writing each term as the product of the first term and a power of the common ratio. N 1 2 3 4 5 f (n) 6 12 24 48 96 The general rule is f(n) = __________. f (n) Evaluate each geometric sequence written as an explicit rule for n = 4. The first one is done for you. 3. f(n) = 10(3)n − 1 4. f(n) = 2(5)n − 1 3 f(4) = 10(3) = 270 ________________________________________ ________________________________________ Evaluate each geometric sequence written as a recursive rule for n = 4. Assume that f(1) is the first term of the sequence. The first one is done for you. 5. f(1) = 7; f(n) = f(n − 1) • 3 for n ≥ 2 6. f(1) = 4; f(n) = f(n − 1) • 2 for n ≥ 2 3 f(4) = 7(3) = 189 ________________________________________ ________________________________________ Write an explicit rule for each geometric sequence based on the given terms from the sequence. Assume that the common ratio r is positive. The first one is done for you. 7. a1 = 9 and a2 = 18 8. a1 = 2 and a2 = 20 n−1 f(n) = 9(2) ________________________________________ ________________________________________ The population of a town is 20,000. It is expected to grow at 4% per year. Use this information for 9–10. The first one is started for you. 9. Write a recursive rule and an explicit rule to predict the population p(n) n years from today. p(1) = 20,000; p(n) = p(n − 1) • 1.04 for n ≥ 2 _________________________________________________________________________________________ 10. Use a rule to predict the population in 5 years and in 10 years. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 257 Name ________________________________________ Date __________________ Class __________________ LESSON 15-3 Constructing Exponential Functions Practice and Problem Solving: A/B Use two points to write an equation for each function shown. 1. x 0 1 2 3 f (x) 6 18 54 162 2. ________________________________________ x −2 0 2 4 f (x) 84 21 5.25 1.3125 ________________________________________ Complete the table using domain of {−2, −1, 0, 1, 2} for each function shown. Graph each. 3. f(x) = 3(2)x x −2 −1 4. f(x) = 4(0.5)x 0 1 2 x f (x) −2 −1 0 1 2 f (x) Graph each function. 5. y = 5(2)x 6. y = −2(3)x ⎛ 1⎞ 7. y = 3 ⎜ ⎟ ⎝2⎠ x Solve. 8. If a basketball is bounced from a height of 15 feet, the function f ( x ) = 15(0.75)x gives the height of the ball in feet at each bounce, where x is the bounce number. What will be the height of the fifth bounce? Round to the nearest tenth of a foot. _______________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 260 Name ________________________________________ Date __________________ Class __________________ LESSON 15-3 Constructing Exponential Functions Practice and Problem Solving: C Graph each function. On your graph, include points to indicate the ordered pairs for x = −1, 0, 1, and 2. 2. f ( x ) = 5(4)− x 1. f ( x ) = 0.75(2)x Solve. 3. An exponential function, f(x), passes through the points (2, 360) and (3, 216). Write an equation for f(x). _________________________________________________________________________________________ 4. The half-life of a radioactive substance is the average amount of time it takes for half of its atoms to disintegrate. Suppose you started with 200 grams of a substance with a half-life of 3 minutes. How many minutes have passed if 25 grams now remain? Explain your reasoning. _________________________________________________________________________________________ _________________________________________________________________________________________ 5. If A is deposited in a bank account at r% annual interest, compounded annually, its value at the end of n years, V(n), can be found using the n r ⎞ ⎛ formula V (n ) = A ⎜ 1 + ⎟ . Suppose that $5000 is invested in an ⎝ 100 ⎠ account paying 4% interest. Find its value after 10 years. _________________________________________________________________________________________ 6. The graph of f ( x ) = 5(4)− x from Problem 2 moves closer and closer to the x-axis as x increases. Does the graph ever reach the x-axis? Explain your reasoning and what your conclusion implies about the range of the function. _________________________________________________________________________________________ _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 261 Name ________________________________________ Date __________________ Class __________________ Constructing Exponential Functions LESSON 15-3 Practice and Problem Solving: Modified Find the value of each exponential expression. The first one is done for you. 3. 2−1 2. 50 1. 2(3)2 18 ________________ ________________ ⎛ 1⎞ 6. 8 ⎜ ⎟ ⎝2⎠ 5. 100(0.6)3 ________________ 4. 4 −2 _______________ ________________ 3 ⎛ 1⎞ 8. 18 ⎜ ⎟ ⎝7⎠ 7. 12(4−3 ) ________________ _______________ 0 ________________ Use two points to write an equation for the function shown. The first one is done for you. 9. x 0 1 2 3 f(x) 1 5 25 125 10. x f(x) = 5 ________________________________________ x 0 1 2 3 f(x) 81 27 9 3 ________________________________________ Solve. The first problem is started for you. x ⎛ 1⎞ 11. Make a table of values and a graph for the function f ( x ) = 6 ⎜ ⎟ . ⎝2⎠ x −2 f(x) 24 −1 0 1 2 12. A blood sample has 50,000 bacteria present. A drug fights the bacteria such that every hour the number of bacteria remaining, r(n), decreases by half. If r(n) is an exponential function of the number, n, of hours since the drug was taken, find the bacteria present four hours after administering the drug. ________________________________________________________________________________________ . Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 262 Name ________________________________________ Date __________________ Class __________________ LESSON 15-4 Graphing Exponential Functions Practice and Problem Solving: A/B Graph each exponential function. Identify a, b, the y-intercept, and the end behavior of the graph. 1 x 1. f(x) = 4(2)x 2. f ( x ) = ( 3 ) 3 x –2 –1 0 1 2 x f(x) −2 –1 0 1 2 f(x) a = ____ b = ____ y-intercept = ____ a = ____ b = ____ y-intercept = ____ end behavior: x → −∞ = ____ , x → +∞ = ____ end behavior: x → −∞ = ____ , x → +∞ = ____ ⎛ 1⎞ 4. f ( x ) = 3 ⎜ ⎟ ⎝2⎠ 3. f(x) = −3(2)x x −2 −1 0 1 2 x −2 x −1 0 1 2 f(x) f(x) a = ____ b = ____ y-intercept = ____ a = ____ b = ____ y-intercept = ____ end behavior: x → −∞ = ____ , x → +∞ = ____ end behavior: x→ −∞ = ____ , x → +∞ = ____ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 265 Name ________________________________________ Date __________________ Class __________________ LESSON 15-4 Graphing Exponential Functions Practice and Problem Solving: C Graph each exponential function. Identify a, b, the y-intercept, and the end behavior of the graph. 1. f(x) = 3.5(2)x x −2 −1 2. f ( x ) = 0 1 x 2 1 x (3) 2 −2 −1 0 1 2 f(x) f(x) a = ____ b = ____ y-intercept = ____ a = ____ b = ____ y-intercept = ____ end behavior: x → −∞ = ____ , x → +∞ = ____ end behavior: x → −∞ = ____ , x → +∞ = ____ Graph each function. On your graph, include points to indicate the ordered pairs for x = −1, 0, 1, and 2. 4. f ( x ) = 5(4)− x 3. f ( x ) = −3(2)x Solve. 5. The half-life of a radioactive substance is the average amount of time it takes for half of its atoms to disintegrate. Suppose you started with 200 grams of a substance with a half-life of 3 minutes. How many minutes have passed if 25 grams now remain? Explain your reasoning. _________________________________________________________________________________________ _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 266 Name ________________________________________ Date __________________ Class __________________ Graphing Exponential Functions LESSON 15-4 Practice and Problem Solving: Modified Solve. The first problem is started for you. x 1. Make a table of values and a graph for the function f ( x ) = 3 ( 2 ) . x −2 f(x) 3 4 −1 0 1 2 x ⎛ 1⎞ 2. Make a table of values and a graph for the function f ( x ) = 6 ⎜ ⎟ . ⎝2⎠ x −2 f(x) 24 −1 0 1 2 Graph each exponential function. Identify a, b, the y-intercept, and the end behavior of the graph. 1 x 3. f(x) = 4(2)x 4. f ( x ) = ( 3 ) 3 x −2 −1 0 1 x 2 −2 −1 0 1 2 f(x) f(x) a = ____ b = ____ y-intercept = ____ a = ____ b = ____ y-intercept = ____ end behavior: x → −∞ = ____, x → +∞ = ____ end behavior: x → −∞ = ____, x → +∞ = ____ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 267 Name ________________________________________ Date __________________ Class __________________ LESSON 15-5 Transforming Exponential Functions Practice and Problem Solving: A/B A parent function has equation Y1 = ( 0.25) . For 1–4, find the x equation for each Y2 . 1. Y2 is a vertical stretch of Y1 . The values of Y2 are 6 times those of Y1 . _________________________________________________________________________________________ 2. Y2 is a vertical compression of Y1 . The values of Y2 are half those of Y1 . _________________________________________________________________________________________ 3. Y2 is a translation of Y1 4 units down. _________________________________________________________________________________________ 4. Y2 is a translation of Y1 11 units up. _________________________________________________________________________________________ Values for f(x), a parent function, and g(x), a function in the same family, are shown below. Use the table for 5–8. x −2 −1 0 1 2 f (x ) 0.04 0.2 1 5 25 g (x ) 0.016 0.08 0.4 2 10 5. Write equations for the two functions. _________________________________________________________________________________________ 6. Is g(x) a vertical stretch or a vertical compression of f(x)? Explain how you can tell. _________________________________________________________________________________________ 7. Do the graphs of f(x) and g(x) meet at any points? If so, find where. If not, explain why not. _________________________________________________________________________________________ _________________________________________________________________________________________ 8. Let h(x) be the function defined by h(x) = −f(x). Describe how the graph of h(x) is related to the graph of f(x). _________________________________________________________________________________________ _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 270 Name ________________________________________ Date __________________ Class __________________ LESSON 15-5 Transforming Exponential Functions Practice and Problem Solving: C A parent function has equation Y1 = ( 0.8 ) . Find the equation x for each Y2, a function created by transforming Y1. 1. To form Y2 , there is first a vertical stretch of Y1 such that the values of Y2 are twice those of Y1 . Then the resulting graph is shifted 8 units up. _________________________________________________________________________________________ 2. To form Y2 , there is first a vertical compression of Y1 such that the values of Y2 are one-third those of Y1 . Then the resulting graph is shifted 12 units down. _________________________________________________________________________________________ 3. To form Y2 , the graph of Y1 is reflected across the x-axis. _________________________________________________________________________________________ 4. To form Y2 , the graph of Y1 is reflected across the y-axis. _________________________________________________________________________________________ 5. To form Y2 , the graph of Y1 is shifted 3 units down and then reflected across the x-axis. _________________________________________________________________________________________ 6. To form Y2 , the graph of Y1 is reflected across the x-axis and then shifted 3 units up. _________________________________________________________________________________________ 7. To form Y2 , the graph of Y1 is shifted 10 units down and then reflected across the y-axis. _________________________________________________________________________________________ 8. To form Y2 , the graph of Y1 is reflected across the y-axis and then shifted 10 units down. _________________________________________________________________________________________ 9. To form Y2 , the graph of Y1 is reflected first across the x-axis and then across the y-axis. _________________________________________________________________________________________ 10. To form Y2 , the graph of Y1 is reflected across the x-axis, then across the y-axis, then across the x-axis again, and finally across the y-axis. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 271 Name ________________________________________ Date __________________ Class __________________ LESSON 15-5 Transforming Exponential Functions Practice and Problem Solving: Modified The graphs of the parent function Y1 = ( 0.4 ) and the function x Y2 = 3 ( 0.4 ) are shown to the right. x Use the graphs for 1–4. The first one is done for you. 1. What is the value of a for Y1 and Y2 ? Y 1 : 1, Y2 : 3 ___________________________ 2. Explain how you can tell that Y2 is a vertical stretch of Y1 . ________________________________________________________________ 3. Write an equation for a function that is a vertical compression of Y1 . _________________________________________________________________________________________ 4. Write an equation for a function that translates Y1 5 units up. _________________________________________________________________________________________ Values for f(x), a parent function, and g(x), a function in the same family, are shown below. Use the table for Problems 5–8. The first one is done for you. x −2 −1 0 1 2 f (x ) 1 4 1 2 1 2 4 g (x ) 1 2 4 8 16 5. Write an equation for the parent function. x f (x ) = 2 _________________________________________________________________________________________ 6. How does the value for g(x) compare with the value for f(x) in each column? _________________________________________________________________________________________ 7. Write an equation for g(x). _________________________________________________________________________________________ 8. Is g(x) a vertical stretch or a vertical compression of f(x)? Explain how you can tell. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 272 Name ________________________________________ Date __________________ Class __________________ LESSON 16-1 Using Graphs and Properties to Solve Equations with Exponents Practice and Problem Solving: A/B Solve each equation without graphing. 1. 5 x = 625 ________________________ 1 (6)x = 108 4. 12 ________________________ 7. 2 (10)x = 40 5 ________________________ 2. 4(2)x = 128 3. _______________________ x 6x = 81 16 ________________________ x 64 ⎛4⎞ 5. ⎜ ⎟ = 125 ⎝5⎠ 2⎛ 1⎞ 1 6. ⎜ ⎟ = 3⎝2⎠ 6 _______________________ ________________________ x 8. (0.1)x = 0.00001 9. _______________________ 2⎛3⎞ 9 = ⎜ ⎟ 3⎝8⎠ 256 ________________________ Solve each equation by graphing. Round your answer to the nearest tenth. Write the equations of the functions you graphed first. 10. 9 x = 11 11. 12 x = 120 Equation: ___________________________ Equation: ___________________________ Equation: ___________________________ Equation: ___________________________ Solution: ___________________________ Solution: ____________________________ Solve using a graphing calculator. Round your answers to the nearest tenth. 12. A town with a population of 600 is expected to grow at an annual rate of 5%. Write an equation and find the number of years it is expected to take the town to reach a population of 900. _________________________________________________________________________________________ 13. How long will it take $20,000 earning 3.5% annual interest to double in value? _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 276 Name ________________________________________ Date __________________ Class __________________ LESSON 16-1 Using Graphs and Properties to Solve Equations with Exponents Practice and Problem Solving: C Solve each equation without graphing. 1. 1 (3)x = 9 27 2. ________________________ x 1⎛ 1⎞ 1 4. ⎜ ⎟ = 2⎝2⎠ 2 5 (2)x = 160 16 _______________________ x 11 ⎛7⎞ 5. ⎜ ⎟ = 7 ⎝ 11 ⎠ ________________________ _______________________ x 3. 25 ⎛ 3 ⎞ 3 = ⎜ ⎟ 27 ⎝ 5 ⎠ 25 ________________________ x ⎛ 1⎞ 6. ⎜ ⎟ = 64 ⎝8⎠ ________________________ Solve each equation by graphing. Round your answer to the nearest tenth. 7. (2.72) = 3.14 x 8. 16 ( 3 ) = 40 ________________________ x _______________________ x 1⎛7⎞ 3 9. ⎜ ⎟ = 7⎝8⎠ 50 ________________________ Solve using a graphing calculator. 10. Does $10,000 invested at 6% interest double its value in half the time as $10,000 invested at 3% interest? Show your work. _________________________________________________________________________________________ _________________________________________________________________________________________ 11. Suppose you were a Revolutionary War veteran and had the foresight to put one penny in a bank account when George Washington became President in 1789. If the bank promised you 5% interest on your account, how much would it be worth in 2014? _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 277 Name ________________________________________ Date __________________ Class __________________ Using Graphs and Properties to Solve Equations with Exponents Practice and Problem Solving: Modified LESSON 16-1 Solve each equation without graphing. The first one is done for you. 1. 2x = 32 x=5 ________________________ x 1 ⎛ 1⎞ 4. ⎜ ⎟ = 27 ⎝3⎠ ________________________ 2. 5 x = 125 3. 3 x = 81 _______________________ ________________________ x 1 ⎛ 1⎞ 5. ⎜ ⎟ = 16 ⎝4⎠ 6. 5(2)x = 80 _______________________ ________________________ Solve each equation by graphing. Round your answer to the nearest tenth. For each, write the equations of the functions you graphed first. The first one is done for you. 7. 2x = 22 8. 3 x = 32 f(x) = 22 Equation: ___________________________ Equation: ___________________________ g(x) = 2x Equation: ___________________________ Equation: ___________________________ x ≈ 4.5 Solution: ____________________________ Solution: ___________________________ Solve using a graphing calculator. The first one is done for you. 9. A wolf population is 400. The population is growing at an annual rate of 8%. Write an equation in one variable to represent, t, the number of years it will take the population to reach 700. t 400(1 + 0.08) = 700, about 6.5 years _________________________________________________________________________________________ 10. Write an equation in one variable and find the number of years it will take an investment of $10,000 earning 4% annual interest to double in value. Round your answer to the nearest tenth. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 278 Name ________________________________________ Date __________________ Class __________________ LESSON 16-2 Modeling Exponential Growth and Decay Practice and Problem Solving: A/B Write an exponential growth function to model each situation. Determine the domain and range of each function. Then find the value of the function after the given amount of time. 1. Annual sales for a fast food restaurant are $650,000 and are increasing at a rate of 4% per year; 5 years ___________________________ _________________________________________________________________________________________ 2. The population of a school is 800 students and is increasing at a rate of 2% per year; 6 years ___________________________ _________________________________________________________________________________________ Write an exponential decay function to model each situation. Determine the domain and range of each function. Then find the value of the function after the given amount of time. 3. The population of a town is 2500 and is decreasing at a rate of 3% per year; 5 years ___________________________ _________________________________________________________________________________________ 4. The value of a company’s equipment is $25,000 and decreases at a rate of 15% per year; 8 years ___________________________ _________________________________________________________________________________________ Write an exponential growth or decay function to model each situation. Then graph each function. 5. The population is 20,000 now and expected to grow at an annual rate of 5%. ________________________________________ 6. A boat that cost $45,000 is depreciating at a rate of 20% per year. ________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 281 Name ________________________________________ Date __________________ Class __________________ LESSON 16-2 Modeling Exponential Growth and Decay Practice and Problem Solving: C Use this information for Problems 1–4. Odette has two investments that she purchased at the same time. Investment 1 cost $10,000 and earns 4% interest each year. Investment 2 cost $8000 and earns 6% interest each year. 1. Write exponential growth functions that could be used to find v1(t) and v2(t), the values of the investments after t years. _________________________________________________________________________________________ 2. Find the value of each investment after 5 years. Explain why the difference between their values, which was initially $2000, is now significantly less. _________________________________________________________________________________________ _________________________________________________________________________________________ 3. Will the value of Investment 2 ever exceed the value of Investment 1? If not, why not? If so, when? _________________________________________________________________________________________ _________________________________________________________________________________________ 4. Instead of calculating 4% interest for one year, suppose the interest for Investment 1 was calculated every day at a rate of (4/365)%. This is called daily compounding. Would Odette earn more, the same, or less using this daily method for one year? Provide an example to show your thinking. _________________________________________________________________________________________ _________________________________________________________________________________________ Solve. 5. A car depreciates in value by 20% each year. Graham argued that the value of the car after 5 years must be $0, since 20% × 5 = 100%. Do you agree or disagree? Explain fully. _________________________________________________________________________________________ _________________________________________________________________________________________ 6. Workers at a plant suffered pay cuts of 10% during a recession. When the economy returned to normal, their salaries were raised 10%. Should the workers be satisfied? Explain your thinking. _________________________________________________________________________________________ _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 282 Name ________________________________________ Date __________________ Class __________________ LESSON 16-2 Modeling Exponential Growth and Decay Practice and Problem Solving: Modified Write an exponential growth function to model each situation. Then find the value of the function after the given amount of time. The first one is done for you. 1. Annual sales for a clothing store are $270,000 and are increasing at a rate of 7% per year; 3 years 3 y = 270,000(1 + 0.07) = $330,761.61 _________________________________________________________________________________________ 2. The population of a school is 2200 and is increasing at a rate of 2%; 6 years _________________________________________________________________________________________ 3. The value of a vase is $200 and is increasing at a rate of 8%; 12 years _________________________________________________________________________________________ Write an exponential decay function to model each situation. Then find the value of the function after the given amount of time. The first one is done for you. 4. The population of a school is 800 and is decreasing at a rate of 2% per year; 4 years y = 800(1 − 0.02)4 ≈ 738 _________________________________________________________________________________________ 5. The bird population in a forest is about 2300 and is decreasing at a rate of 4% per year; 10 years _________________________________________________________________________________________ Write an exponential decay function to model the situation. Then graph the function. The table is started for you. 6. A car that cost $30,000 when new depreciates at a rate of 18% per year. Function: ___________________________ x 0 2 4 y 30,000 20,172 13,564 6 8 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 283 Name ________________________________________ Date __________________ Class __________________ LESSON 16-3 Using Exponential Regression Models Practice and Problem Solving: A/B The table below shows the total attendance at major league baseball games, at 10-year intervals since 1930. Use the table for the problems that follow. Major League Baseball Total Attendance (yd), in millions, vs. Years Since 1930 (x) x 0 10 20 30 40 50 60 70 80 yd 10.1 9.8 17.5 19.9 28.7 43.0 54.8 72.6 73.1 ym residual 1. Use a graphing calculator to find the exponential regression equation for this data. Round a and b to the nearest thousandth. _________________________________________________________________________________________ 2. According to the regression equation, by what percent is attendance growing each year? _________________________________________________________________________________________ 3. Complete the row labeled ym above. This row contains the predicted y-values for each x-value. Round your answers to the nearest tenth. 4. Calculate the row of residuals above. 5. Analyze the residuals from your table. Does it seem like the equation is a good fit for the data? _________________________________________________________________________________________ _________________________________________________________________________________________ 6. Use your graphing calculator to find the correlation coefficient for the equation and write it below. Does the correlation coefficient make it seem like the equation is a good fit for the data? _________________________________________________________________________________________ 7. Use the exponential regression equation to predict major league baseball attendance in 2020. Based on your earlier work on this page, do you think this is a reasonable prediction? Explain. _________________________________________________________________________________________ _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 286 Name ________________________________________ Date __________________ Class __________________ LESSON 16-3 Using Exponential Regression Models Practice and Problem Solving: C A pot of boiling water is allowed to cool for 50 minutes. The table below shows the temperature of the water as it cools. Use the table for the problems that follow. Temperature of Water (yd), in degrees Celsius, after cooling for x minutes x 0 5 10 15 20 25 30 35 40 45 50 yd 100 75 57 44 34 26 21 17 14 11 10 ym residual 1. Use a graphing calculator to find the exponential regression equation for this data. Round a and b to the nearest thousandth. _________________________________________________________________________________________ 2. Complete the rows labeled ym (predicted y-values) and residual above. Round your answers to the nearest tenth. 3. Fit a linear regression equation to the original data. Write the equation here. _________________________________________________________________________________________ 4. The data for the scatter plot is shown in the first two rows of the table below. Complete the next two rows of the table for the model you found in Problem 3. Temperature of Water (yd), in degrees Celsius, after cooling for x minutes x 0 5 10 15 20 25 30 35 40 45 50 yd 100 75 57 44 34 26 21 17 14 11 10 ym residual 5. Examine the residuals in each table. Which appears to be the better model—the linear or exponential equation? Explain. _________________________________________________________________________________________ _________________________________________________________________________________________ 6. Find the correlation coefficients for the two equations. Based on that information, which equation is the better model? Explain. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 287 Name ________________________________________ Date __________________ Class __________________ LESSON 16-3 Using Exponential Regression Models Practice and Problem Solving: Modified The table below shows the cost of mailing a letter in the United States at 10-year intervals since 1950. Use the table for the problems that follow. The first one is done for you. U.S. First-Class Postage Rate (yd) vs. Years Since 1950 (x) x 0 10 20 30 40 50 60 yd 3 4 6 15 25 33 44 ym residual 1. Use a graphing calculator to find the exponential regression equation for this data. Round your answers to the nearest hundreth. x y = 2.80(1.05) _________________________________________________________________________________________ 2. Use the exponential regression equation to complete the row labeled ym above. This row contains the predicted y-value for each x-value. Round your answers to the nearest hundredth. For example, when x = 10, y m = 2.80(1.05)10 = 4.56 . 3. Calculate the row labeled residual above. For each x-value, the residual equals y d − y m . For example, for x = 0, the residual is 3 − 2.80 = 0.20. 4. Analyze the residuals from your table. Does it seem like the equation is a good fit for the data? _________________________________________________________________________________________ _________________________________________________________________________________________ 5. Use your graphing calculator to find the correlation coefficient for the equation and write it below. Does the correlation coefficient make it seem like the equation is a good fit for the data? _________________________________________________________________________________________ 6. The cost of first-class postage in 2013 was raised to 46 cents. According to the exponential regression model, what was the predicted cost for 2013? Recognize that 2013 is 63 years from 1950. _________________________________________________________________________________________ _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 288 Name ________________________________________ Date __________________ Class __________________ LESSON 16-4 Comparing Linear and Exponential Models Practice and Problem Solving: A/B Without graphing, tell whether each quantity is changing at a constant amount per unit interval, at a constant percent per unit interval, or neither. Justify your reasoning. 1. A bank account started with $1000 and earned $10 interest per month for two years. The bank then paid 2% interest on the account for the next two years. _________________________________________________________________________________________ 2. Jin Lu earns a bonus for each sale she makes. She earns $100 for the first sale, $150 for the second sale, $200 for the third sale, and so on. _________________________________________________________________________________________ Use this information for Problems 3–8. A bank offers annual rates of 6% simple interest or 5% compound interest on its savings accounts. Suppose you have $10,000 to invest. 3. Express f(x), the value of your deposit after x years in the simple interest account, and g(x), the value of your deposit after x years in the compound interest account. _________________________________________________________________________________________ 4. Is either f(x) or g(x) a linear function? An exponential function? How can you tell? _________________________________________________________________________________________ 5. Find the values of your deposit after three years in each account. After three years, which account is the better choice? _________________________________________________________________________________________ 6. Find the values of your deposit after 20 years in each account. After 20 years, which account is the better choice? _________________________________________________________________________________________ 7. Use a graphing calculator to determine the length of time an account must be held for the two choices to be equally attractive. Round your answer to the nearest tenth. _________________________________________________________________________________________ 8. Use your answer to Problem 7 to write a statement that advises an investor regarding how to choose between the two accounts. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 291 Name ________________________________________ Date __________________ Class __________________ LESSON 16-4 Comparing Linear and Exponential Models Practice and Problem Solving: C Without graphing, tell whether each quantity is changing at a constant amount per unit interval, at a constant percent per unit interval, or neither. Justify your reasoning. 1. When Josh read his first book alone, his mother gave him a penny. For his second book, she gave him two cents, and for his third book, she gave him four cents. She plans on doubling the amount for each book Josh reads. _________________________________________________________________________________________ 2. The annual cost of a club membership starts at $100 and increases by $15 each year. _________________________________________________________________________________________ Use this information for Problems 3–7. A bank offers annual rates of 4% simple interest or 3.5% compound interest on its savings accounts. 3. Express the values of an initial investment of A dollars after x years. Let f(x) represent the amount in a simple interest account and let g(x) represent the amount in a compound interest account. _________________________________________________________________________________________ 4. If you planned on depositing money for three years, which rate would be a better choice? Explain. _________________________________________________________________________________________ _________________________________________________________________________________________ 5. If you planned on depositing money for 15 years, which rate would be a better choice? Explain. _________________________________________________________________________________________ _________________________________________________________________________________________ 6. Determine the length of time an account must be held for the two choices to be equally attractive. (HINT: You may want to graph the equations.) Round to the nearest tenth. _________________________________________________________________________________________ 7. Would the amount deposited affect any of the answers you gave for Problems 4–6? Justify your reasoning. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 292 Name ________________________________________ Date __________________ Class __________________ LESSON 16-4 Comparing Linear and Exponential Models Practice and Problem Solving: Modified Without graphing, tell whether each quantity is changing at a constant amount per unit of time, at a constant percent per unit of time, or neither. Justify your reasoning. The first one is done for you. 1. Carl’s hourly pay rate is $9.00 now. Each month, his hourly pay rate is scheduled to increase by $0.10. Constant amount per unit of time. The change is $0.10 per month. _________________________________________________________________________________________ 2. The population of a small town was 280 in 2008, 300 in 2009, and 320 in 2010. Then it began to increase by 5% annually. _________________________________________________________________________________________ 3. A bank account that started with $500 grew at a rate of 3% each year. _________________________________________________________________________________________ Use this information for Problems 4–8. The first one is started for you. Dana gets $1 every day but has a choice to make. From now on, that amount will be increased by $0.05 every day or by 4% every day. 4. Make a table showing f(x), the amount Dana will receive on Day x from the $0.05 (5 cents) plan, and g(x), the amount Dana will receive on Day x from the 4% plan. Day 0 1 2 3 4 f(x) $1.00 $1.05 $1.10 $1.15 $1.20 g(x) $1.00 $1.04 $1.08 $1.12 $1.17 5 6 7 8 5. Examine your table. Is f(x) a linear function or an exponential function? Is g(x) a linear function or an exponential function? _________________________________________________________________________________________ 6. Write equations for f(x) and g(x). _________________________________________________________________________________________ 7. Use a calculator to find f(20) and g(20). _________________________________________________________________________________________ 8. Dana chooses the plan that increases the amount by $0.05 each day. Did Dana make a good decision? Explain your thinking. _________________________________________________________________________________________ _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 293 Name ________________________________________ Date __________________ Class __________________ LESSON 17-1 Understanding Polynomial Expressions Practice and Problem Solving: A/B Identify each expression as a monomial, a binomial, a trinomial, or none of the above. Write the degree of each expression. 1. 6b2 −7 2. x2y − 9x4y2 + 3xy ________________________________________ ________________________________________ 3. 35r3s 4. 3p + ________________________________________ 2p − 5q q ________________________________________ 5. 4ab5 + 2ab − 3a4b3 6. st + t 0.5 ________________________________________ ________________________________________ Simplify each expression. 7. 6n3 − n2 + 3n4 + 5n2 8. c3 + c2 + 2c − 3c3 − c2 − 4c ________________________________________ ________________________________________ 9. 11b2 + 3b − 1 − 2b2 − 2b − 8 10. a4b3 + 9a3b4 − 3a4b3 − 4a3b4 ________________________________________ ________________________________________ 11. 9xy + 5x2 + 15x − 10xy 12. 3p2q + 8p3 − 2p2q + 2p + 5p3 ________________________________________ ________________________________________ Determine the polynomial that has the greater value for the given value of x. 13. 4x2 − 5x − 2 or 5x2 − 2x − 4 for x = 6 14. 6x3 − 4x2 + 7 or 7x3 − 6x2 + 4 for x = 3 ________________________________________ ________________________________________ Solve. 15. A rocket is launched from the top of an 80-foot cliff with an initial velocity of 88 feet per second. The height of the rocket t seconds after launch is given by the equation h = −16t2 + 88t + 80. How high will the rocket be after 2 seconds? __________________________________________________________________ 16. Antoine is making a banner in the shape of a triangle. He wants to line the banner with a decorative border. How long will the border be? _________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 297 Name ________________________________________ Date __________________ Class __________________ LESSON 17-1 Understanding Polynomial Expressions Practice and Problem Solving: C Simplify each expression. Then identify the expression as a monomial, a binomial, a trinomial, or none of the above. Write the degree of each polynomial. 1. 6ab2 − 3a2b − 3ab2 2. 5xy − 9x + 3xy + 2y2 ________________________________________ ________________________________________ 4. 3n2 + 9n − 6 + 2n + 6 3. − 16y 3 ________________________________________ ________________________________________ 5. 4b5 + 2b2 − 3b6 − 7b5 − b2 + 3b5 6. ________________________________________ 9x 25 ________________________________________ Simplify each expression. 7. 6mn3 − mn2 + 3mn3 + 15mn2 8. 1.6c3 + 5.6c2 + 2.5c − 3.7c3 + 7.3c2 − 4.9c ________________________________________ ________________________________________ 1 1 1 11 2 5 9. 11 b2 + 3 b − 6 − 2 b2 + 4 b + 1 3 6 4 2 3 12 ________________________________________ 10. a4b3 + 8a3b4 − 2a2b5 − 6a4b3 − 9a3b4 ________________________________________ 1 7 3 1 12. 8 p3 + pq + 5 p3 − 2 pq 2 8 4 3 11. 5.2x2 + 5.1x − 7.3xy + 6.4x2 − 2.4x + 1.8xy ________________________________________ ________________________________________ Determine the polynomial that has the greater value for the given value of x. Then, determine how much greater it is than the other polynomial. 13. 4x2 − 5x − 2 or 5x2 − 2x − 4 for x = 1.5 14. 6x3 − 4x2 + 7 or 7x3 − 6x2 + 4 for x = −3 ________________________________________ ________________________________________ Solve. 15. A rocket is launched from the top of an 80-foot cliff with an initial velocity of 88 feet per second. The height of the rocket t seconds after launch is given by the equation h = −16t 2 + 88t + 80. How high will the rocket be after 2 seconds? After 3.5 seconds? What do you notice about the heights? Explain your answer. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 298 Name ________________________________________ Date __________________ Class __________________ LESSON 17-1 Understanding Polynomial Expressions Practice and Problem Solving: Modified Identify each expression as a monomial, a binomial, a trinomial, or none of the above. Write the degree of each expression. The first one is done for you. 1. 4w 2 2. 9x3 + 2x monomial; degree 2 ________________________________________ ________________________________________ 3. 35b6 4. 4p5 − 5p3 + 11 ________________________________________ ________________________________________ 5. 12 + 3x4 − x 6. 3m + 1 ________________________________________ ________________________________________ Simplify each expression. The first one is done for you. 7. 6n2 + 3n − n2 8. 5c3 + 2c − 4c 2 5n + 3n ________________________________________ ________________________________________ 10. 7a4 − 9a3 − 3a4 − 4a 9. 3b − 1 − 2b − 8 ________________________________________ ________________________________________ 11. 5x2 + 15x − 10x − 9x2 12. 3p + 8p2 − 2p − 6 + 5p2 ________________________________________ ________________________________________ Find the value of each polynomial for the given value of x. Then determine the polynomial that has the greater value. The first one is started for you. 13. 4x2 − 5x − 2 or 5x2 − 2x − 4 for x = 3 14. 6x3 − 4x2 + 7 or 7x3 − 6x2 + 4 for x = 2 2 19; 35; 5x − 2x − 4 ________________________________________ ________________________________________ Solve. The first one is started for you. 15. A firework is launched from the ground at a velocity of 180 feet per second. Its height after t seconds is given by the polynomial −16t2 + 180t. What is the height of the firework after 2 seconds? 2 h = 16 (2) + 180 (2) _________________________________________________________________________________________ 16. The volume of one box is 4x3 + 4x2 cubic units. The volume of the second box is 6x3 − 18x2 cubic units. Write a polynomial for the total volume of the two boxes. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 299 Name ________________________________________ Date __________________ Class __________________ LESSON 17-2 Adding Polynomial Expressions Practice and Problem Solving: A/B Add the polynomial expressions using the vertical format. 1. (10g 2 + 3g − 10) 2. + (2g 2 + g + 9) ________________________________________ 3. + (3 x 3 + x 2 + 4 x ) ________________________________________ (11b 2 + 3b − 1) 4. + (2b 2 + 2b + 8) ________________________________________ 5. (4 x 3 + x 2 + 2 x ) ( c 3 + 2c 2 + 2c ) + ( −3c 3 + c 2 − 4c ) ________________________________________ (ab 2 + 13b − 4a ) 6. + (3ab 2 + a + 7b ) ________________________________________ ( −r 2 + 8 pr − p ) + ( −12r 2 − 2 pr + 8 p ) ________________________________________ Add the polynomial expressions using the horizontal format. 7. (3y2 − y + 3) + (2y2 + 2y + 9) 8. (4z3 + 3z2 + 8) + (2z3 + z2 − 3) ________________________________________ ________________________________________ 9. (6s3 + 9s + 10) + (3s3 + 4s − 10) 10. (15a4 + 6a2 + a) + (6a4 − 2a2 + a) ________________________________________ ________________________________________ 11. (−7a2b3 + 3a3b − 9ab) + (4a2b3 − 5a3b + ab) ________________________________________ 12. (2p4q2 + 5p3q − 2pq) + (8p4q2 − 3p3q − pq) ________________________________________ Solve. 13. A rectangular picture frame has the dimensions shown in the figure. Write a polynomial that represents the perimeter of the frame. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 302 Name ________________________________________ Date __________________ Class __________________ LESSON 17-2 Adding Polynomial Expressions Practice and Problem Solving: C Simplify. 1. (ab2 + 13b − 9) + (6 − 4a + 3ab2) + (a + 7b) ________________________________________ 2. (9x3 − 2x2 − x) + (3x + x3 − 4) + (x2 − 3x) ________________________________________ 3. (−r 2 + 8pr − p) + (−12r 2 − 2pr) + (8p + 3r 2) 4. (rs2 − s − 6) + (2rs2 − 3s + 1) + (s + 4rs2) ________________________________________ ________________________________________ 5. What algebraic expression must be added to the sum of 3x2 + 4x + 8 and 2x2 − 6x + 3 to give 9x2 − 2x − 5 as the result? _________________________________________________________________________________________ Give an example for each statement. 6. The sum of two binomials is a monomial. 7. The sum of two trinomials is a binomial. ________________________________________ ________________________________________ Solve. n3 n2 n + + . 3 2 6 n 4 n3 n2 The sum of the cubes of the first n positive integers is + + . 4 2 4 Write an expression for the sum of the squares and cubes of the first n positive integers. Then find the sum of the first 10 squares and cubes. 8. The sum of the squares of the first n positive integers is _________________________________________________________________________________________ 9. Vincent is going to frame the rectangular picture with dimensions shown. The frame will be x + 1 inches wide. Find the perimeter of the frame. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 303 Name ________________________________________ Date __________________ Class __________________ LESSON 17-2 Adding Polynomial Expressions Practice and Problem Solving: Modified Add. The first one is done for you. 1. 2m + 4 2. 3m + 6 ________________________ 12k + 3 + 4k + 2 ________________________ 3. + 2y 2 + 2y + 9 +m+2 4. 3y 2 − y + 3 + 2z 3 + z 2 − 3 _______________________ 5. 6s 3 + 9s + 10 ________________________ 6. + 3s 3 + 4s − 10 15a 4+ 6a 2+a + 6a 4 − 2a 2 + a _______________________ 7. (3x3 + 4) + (x3 − 10) 4z 3 + 3z 2 + 8 ________________________ 8. (10g 2 + 3g − 10) + (2g 2 + g + 9) ________________________________________ ________________________________________ 9. (12p5 + 8) + (8p5 + 6) 10. (11b 2 + 3b − 1) + (2b 2 + 2b + 8) ________________________________________ ________________________________________ Solve. The first one is started for you. 11. Rebecca is building a pen for her rabbits against the side of her house. The polynomial 4n + 8 represents the length and the polynomial 2n + 6 represents the width. a. What polynomial represents the perimeter of the entire pen? (4n + 8) + (4n + 8 ) + (2n + 6) + (2n + 6) = _________________________________________ ________________________________________ b. What polynomial represents the perimeter of the pen NOT including the side of the house. ________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 304 Name ________________________________________ Date __________________ Class __________________ LESSON 17-3 Subtracting Polynomial Expressions Practice and Problem Solving: A/B Subtract using the vertical form. 1. (5g 2 + 6g − 10) 2. − (2g 2 + 2g + 9) ________________________________________ 3. − (2 x 3 + x 2 + x ) ________________________________________ (10b 2 + 5b − 2) 4. − ( 2b 2 + b + 1) ________________________________________ 5. (8 x 3 + 4 x 2 + x ) ( 7c 3 − 5c 2 + 2c ) − ( −3c 3 + 2c 2 − 2c ) ________________________________________ (14ab 2 + 9b − 2a ) 6. − ( 4ab 2 + 2a + 5b ) ________________________________________ (6 x 3 + 2 x 2 + 3 x ) − (3 x 3 − 2 x 2 − 3 x ) ________________________________________ Subtract using the horizontal form. 7. (7y2 − 7y + 7) − (4y2 + 2y + 3) 8. (11z3 + 6z2 + 3) − (9z3 + 2z2 − 8) ________________________________________ ________________________________________ 9. (9s3 + 10s + 8) − (2s3 + 9s − 11) 10. (25a4 + 9a2 + 3a) − (24a4 − 5a2 + 3a) ________________________________________ ________________________________________ 11. (−a2b3 + a3b − ab) − (a2b3 − a3b + ab) 12. (3p4q2 + 8p3q − 2) − (5p4q2 − 2p3q − 8) ________________________________________ ________________________________________ Solve. 13. Darnell and Stephanie have competing refreshment stand businesses. Darnell’s profit can be modeled with the polynomial c2 + 8c − 100, where c is the number of items sold. Stephanie’s profit can be modeled with the polynomial 2c2 − 7c − 200. Write a polynomial that represents the difference between Stephanie’s profit and Darnell’s profit. _________________________________________________________________________________________ 14. There are two boxes in a storage unit. The volume of the first box is 4 x 3 + 4 x 2 cubic units. The volume of the second box is 6 x 3 − 18 x 2 cubic units. Write a polynomial to show the difference between the two volumes. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 307 Name ________________________________________ Date __________________ Class __________________ Subtracting Polynomial Expressions LESSON 17-3 Practice and Problem Solving: C Simplify. 1. (ab2 + 13b − 9) − (6 − 4a + 3ab2) + (a + 7b) ________________________________________ 2. (9x3 − 2x2 − x) + (3x + x3 − 4) − (x2 − 3x) ________________________________________ 3. (−r 2 + 8pr − p) − (−12r 2 − 2pr) + (8p + 3r 2) 4. (rs2 −s − 6) + (2rs2 − 3s + 1) − (s + 4rs2) ________________________________________ ________________________________________ 5. What algebraic expression must be subtracted from the sum of y 2 + 5y − 1 and 3y2 − 2y + 4 to give 2y 2 + 7y − 2 as the result? _________________________________________________________________________________________ Give an example for each statement. 6. The difference of two binomials is a binomial. 7. The difference of two binomials is a trinomial. ________________________________________ ________________________________________ Solve. 8. Ned, Tony, Matt, and Juan are playing basketball. Ned scored 2p + 3 points, Tony scored 3 more points than Ned, Matt scored twice as many points as Tony, and Juan scored 8 fewer points than Ned. Write an expression that represents the total number of points scored by all four boys. _________________________________________________________________________________________ 9. Mr. Watford owns two car dealerships. His profit from the first can be modeled with the polynomial c 3 − c 2 + 2c − 100, where c is the number of cars he sells. Mr. Watford’s profit from his second dealership can be modeled with the polynomial c 2 − 4c − 300. a. Write a polynomial to represent the difference of the profit at his first dealership and the profit at his second dealership. _________________________________________________ ____________________________________ b. If Mr. Watford sells 45 cars in his first dealership and 300 cars in his second, what is the difference in profit between the two dealerships? _____________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 308 Name ________________________________________ Date __________________ Class __________________ LESSON 17-3 Subtracting Polynomial Expressions Practice and Problem Solving: Modified Subtract. The first one is done for you. 1. 8p + 6 2. 4p + 4 ________________________ 20k + 6 − (10k + 2) ________________________ − (2z3 +3z2 − 2) _______________________ 5. 7s3 + 4s + 30 ________________________ 25a4 + 9a2 + 6a 6. − (5s3 + 2s − 10) − (10a4 − 2a2 + a) _______________________ 7. (5x3 + 14) − (2x3 − 1) 5z3 + 8z2 + 5 3. − (5y2 − 3y + 2) − (4p + 2) 4. 9y2 − 6y + 3 ________________________ 8. (15g2 + 6g − 3) − (10g2 + 2g + 2) ________________________________________ ________________________________________ 9. (7p5 + 8) − (3p5 + 6) 10. (4b2 + 8b − 1) − (2b2 + 3b + 5) ________________________________________ ________________________________________ Solve. The first problem is started for you. 11. The angle GEO is represented by 3w + 7 and angle OEM is 2w − 1. Write a polynomial that represents the difference between angle GEO and angle OEM. (3w + 7) − (2w − 1) = _______________________________________________________ 12. The polynomial 35p + 300 represents the number of men enrolled in a college and 25p + 100 represents the number of women enrolled in the same college. What polynomial shows the difference between the number of men and women enrolled in the college? _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 309 Name ________________________________________ Date __________________ Class __________________ LESSON 18-1 Multiplying Polynomial Expressions by Monomials Practice and Problem Solving: A/B Find the product. 1. 5x(2x 4y 3) 2. 0.5p(−30p3r 2) ________________________________________ ________________________________________ 3. 11ab2(2a 5b 4) 4. −6c 3d 5(−3c 2d) ________________________________________ ________________________________________ 5. 4(3a 2 + 2a − 7) 6. 9x 2(x3 − 4x 2 − 3x) ________________________________________ ________________________________________ 7. 6s 3(−2s 2 + 4s − 10) 8. 5a 4(6a 4 − 2a 2 − a) ________________________________________ ________________________________________ 9. 8pr (−7r 2 − 2pr + 8p) 10. 2mn 3(3mn 3 + n 2 + 4mn) ________________________________________ ________________________________________ 11. −3x 4y 2(2x 2 + 5xy + 9y 2) 12. 0.75 v 2w 3(12v 3 + 16v 2w − 8w 2) ________________________________________ ________________________________________ 13. −7a 2b 3(4a 2b 3 + ab − 5a 3b) 14. 2p 4q 2(8p 4q 2 − 3p 3q + 5p 2q) ________________________________________ ________________________________________ Solve. 15. The length of a rectangle is 3 inches greater than the width. a. Write a polynomial expression that represents the area of the rectangle. _____________________________________ b. Find the area of the rectangle when the width is 4 inches. _____________________________________ 16. The length of a rectangle is 8 centimeters less than 3 times the width. a. Write a polynomial expression that represents the area of the rectangle. _____________________________________ b. Find the area of the rectangle when the width is 10 centimeters. _____________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 313 Name ________________________________________ Date __________________ Class __________________ Multiplying Polynomial Expressions by Monomials LESSON 18-1 Practice and Problem Solving: C Find the product. 1. 1 3 m (6m)(2m 2) 3 2. −3x 4(12x)(0.75x 4) ________________________________________ 3. ________________________________________ 2 2 ⎛1 ⎞ xy (xy) ⎜ x ⎟ 3 ⎝2 ⎠ 4. −6c3d 5(−3c2d)(−2cd ) ________________________________________ 5. ________________________________________ 1 x(6x 2 + 10x + 5) 2 6. 0.4x(5x 3 − 8x 2 − 1.4x) ________________________________________ ________________________________________ Simplify. 7. 3 2 3 v (4v + 16v 2 − 8v) − 3v(v 4 + 4v 3 − 2) 4 8. 5a4(6a4 − 2a2 − a) − 2a(a7 + 5a5 − 3) ________________________________________ ________________________________________ 9. 6s3(−2s2 + 4s − 10) + 3s(4s4 − 8s3 + 5s2) 10. 2jk3(3jk3 + j 2 + 4jk) − jk(9j 2k 2 + jk 3) ________________________________________ ________________________________________ 11. −3x 4y 2(2x 2 + 5xy + 9y 2) − xy(2x 3y 3 − x 4y 2) 12. 8pr(7r 2 − 2pr + p) + 3r(−5pr 2 + 6p2r − 8p2) ________________________________________ ________________________________________ Solve. 13. The shaded area represents the deck around a swimming pool. Write a polynomial expression in simplest form for the following. a. the area of the swimming pool _____________________________________ b. the total area of the swimming pool and deck _____________________________________ c. the area of the deck _____________________________________ 14. Write four multiplication problems that have a product of 24a3b2 − 16a2b. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 314 Name ________________________________________ Date __________________ Class __________________ Multiplying Polynomial Expressions by Monomials LESSON 18-1 Practice and Problem Solving: Modified Multiply. The first one is done for you. 1. 4x4(8x2) 2. 5p(3p3) 6 32x ________________________________________ ________________________________________ 3. 11a2(2a5b4) 4. −6c3(−3c2d) ________________________________________ ________________________________________ 5. 9rs2(5r 3s) 6. 8x3y 2(−2x4y 3) ________________________________________ ________________________________________ Find the product. The first one is done for you. 7. 7(3a2 + 2a − 7) 8. 9(3x2 − 4x − 3) 2 21a + 14a − 49 ________________________________________ ________________________________________ 9. 6s3(−2s2 + 4s − 10) 10. 5a2(6a4 − 2a2 − 1) ________________________________________ ________________________________________ 11. 8r(−7r 2 − 2pr + 8p) 12. 2n3(3n3 +m2n2 − 4n) ________________________________________ ________________________________________ 13. −3x4y 2(8x 2 − 5xy + 9y 2) 14. 5v 2w 3(2v 3 + 4v 2w − w 2) ________________________________________ ________________________________________ Solve. The first one is done for you. 15. The length of a rectangle is 5 inches greater than the width. a. Write a variable for the width of the rectangle. w __________________________ b. Write an expression for the length of the rectangle. __________________________ c. Write a simplified expression for the area of the rectangle. (area = length × width) __________________________ d. Find the area of the rectangle when the width is 3 inches. __________________________ e. Find the area of the rectangle when the length is 9 inches. __________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 315 Name ________________________________________ Date __________________ Class __________________ LESSON 18-2 Multiplying Polynomial Expressions Practice and Problem Solving: A/B Multiply. 1. (x + 5)(x + 6) ________________________ 4. (2x − 3)(x + 4) ________________________ 7. (5k − 9)(2k − 4) ________________________ 10. (r + 2s)(r − 6s) ________________________ 13. (y + 3)(y − 3) ________________________ 16. (4w + 9)2 ________________________ 19. (x + 4)(x 2 + 3x + 5) ________________________ 2. (a − 7)(a − 3) _______________________ 5. (5b + 1)(b − 2) _______________________ 8. (2m − 5)(3m + 8) _______________________ 11. (3 − 2v)(2 − 5v) _______________________ 14. (z − 5)2 3. (d + 8)(d − 4) ________________________ 6. (3p − 2)(2p + 3) ________________________ 9. (4 + 7g)(5 − 8g) ________________________ 12. (5 + h)(5 − h) ________________________ 15. (3q + 7)(3q − 7) _______________________ 17. (3a − 4)2 ________________________ 18. (5q − 8r)(5q + 8r) _______________________ 20. (3m + 4)(m2 − 3m + 5) _______________________ ________________________ 21. (2x − 5)(4x 2 − 3x + 1) ________________________ Solve. 22. Write a polynomial expression that represents the area of the 1 ⎛ ⎞ trapezoid. ⎜ A = h ( b1 + b2 ) ⎟ 2 ⎝ ⎠ _________________________________________________________________________________________ 23. If x = 4 in., find the area of the trapezoid in problem 22. _________________________________________________________________________________________ 24. Kayla worked 3x + 6 hours this week. She earns x − 2 dollars per hour. Write a polynomial expression that represents the amount Kayla earned this week. Then calculate her pay for the week if x = 11. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 318 Name ________________________________________ Date __________________ Class __________________ LESSON 18-2 Multiplying Polynomial Expressions Practice and Problem Solving: C Multiply. 1. 2(x + 5)(x + 6) ________________________ 4. 4(2x − 3)(x + 4) ________________________ 7. 2k(5k − 9)(2k − 4) ________________________ 10. rs(r + 2s)(r − 6s) ________________________ 13. y(2y 2 + 3)(2y 2 − 3) ________________________ 16. −3w(4w + 9)2 ________________________ 19. (3x − 1)(2x2 − 3x − 7) ________________________ 2. 3(a − 7)(a − 3) 3. −5(8 + d)(4 − d) _______________________ 5. 6(5b + 1)(b − 2) ________________________ 6. −2(3p − 2)(2p + 3) _______________________ 8. m2(2m − 5)(3m + 8) ________________________ 9. −8g2(4 + 7g)(5 − 8g) _______________________ 11. 4v(3 − 2v)(2 − 5v) ________________________ 12. 6h2(5 + 9h)(5 − 9h) _______________________ 14. 3(6z − 5)2 ________________________ 15. 4c(3c + 7d)(3c − 7d) _______________________ 17. 2a(3a − 4)2 ________________________ 18. qr(5q2 − 8r 2 )(5q2 + 8r 2 ) _______________________ 20. (5z + 6)(2z + 1)(2z − 1) ________________________ 21. (x + 2)(5x − 3)2 _______________________ ________________________ Solve. 22. Write a polynomial expression that represents the volume of the cube. ______________________________________________________________ 23. Explain how you can use the polynomial expression to find the volume of the cube in problem 22 if x = 4 in. Then find the volume when x = 4 in. How can you check your answer? _________________________________________________________________________________________ _________________________________________________________________________________________ 24. Multiply (n − 1)(n + 1), (n − 1)(n2 + n + 1), and (n − 1)(n3 + n2 + n + 1). Describe the pattern of the products. Use the pattern to find (n − 1)(n4 + n3 + n2 + n + 1). _________________________________________________________________________________________ _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 319 Name ________________________________________ Date __________________ Class __________________ LESSON 18-2 Multiplying Polynomial Expressions Practice and Problem Solving: Modified Fill in the blanks by multiplying the First, Outer, Inner, and Last terms. Then simplify. The first one is started for you. first last (x + 3) (x − 2) 1. (x + 5) (x + 2) 2. (x + 4) (x − 3) inner 2 x _____ F 2x _____ O 5x _____ I 10 _____ L _____ F Simplify: ___________________________ 3. (x + 5)(x + 6) _____ I _____ L Simplify: __________________________ 4. (a − 7)(a − 3) ________________________ _____ O outer 5. (d + 8)(d − 4) _______________________ ________________________ Fill in the blanks below. The first three are started for you. 6. (x + 5)2 x2 + 2 ( 5 )( x ) + 5 2 ________________________ 9. (x + 4)2 ________________________ 7. (x − 10)2 x 2 − 2 ( 10 )( x ) + 10 2 _______________________ 10. (b − 2)2 8. (x + 7) (x − 7) 2 2 x − 7 ________________________ 11. (p − 9)(p + 9) _______________________ ________________________ Fill in the blanks below. Then simplify. 12. (x + 3) (x 2 + 4x + 7) = x (x 2 + 4x + 7) + 3(x 2 + 4x + 7) Distribute: _____ _____ _____ + _____ _____ _____ Simplify: _____________________________________ 13. (y + 2)(y 2 + 6y + 5) ________________________ 14. (p + 4)(p 2 − 3p − 2) _______________________ 15. (n − 2)(n 2 − 4n + 1) ________________________ Solve. 16. Zoe babysat for x + 3 hours yesterday. She earned x − 2 dollars per hour. Write a polynomial expression that represents the amount Zoe earned. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 320 Name ________________________________________ Date __________________ Class __________________ LESSON 18-3 Special Products of Binomials Practice and Problem Solving: A/B Find the product. 1. ( x + 2)2 2. (m + 4)2 ________________________ 4. (2 x + 5)2 _______________________ 5. (8 − y )2 ________________________ 7. (b − 3)2 8. (3 x − 7)2 13. (5 x + 2)(5 x − 2) 11. (8 + y )(8 − y ) _______________________ 14. (4 + 2y )(4 − 2y ) ________________________ ________________________ 9. (6 − 3n )2 _______________________ ________________________ ________________________ 6. (a − 10)2 _______________________ ________________________ 10. ( x + 3)( x − 3) 3. (3 + a )2 _______________________ ________________________ 12. ( x + 6)( x − 6) ________________________ 15. (10 x + 7 y )(10 x − 7 y ) ________________________ Solve. 16. Write a simplified expression for each of the following. a. area of the large rectangle _____________________________________ b. area of the small rectangle _____________________________________ c. area of the shaded area _____________________________________ 17. The small rectangle is made larger by adding 2 units to the length and 2 units to the width. a. What is the new area of the smaller rectangle? _____________________________________ b. What is the area of the new shaded area? _____________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 323 Name ________________________________________ Date __________________ Class __________________ LESSON 13-2 Absolute Value Functions and Transformations Practice and Problem Solving: C Create a table of values for f(x), graph the function, and tell the domain and range. 1. f ( x ) = −2 x − 1 + 2 x 2. f ( x ) = − x f(x) ________________________________________ 1 x +1 + 3 2 f(x) ________________________________________ Write an equation for each absolute value function whose graph is shown. 3. 4. ________________________________________ ________________________________________ Solve. 5. Suppose you plan to ride your bicycle from Portland, Oregon, to Seattle, Washington, and back to Portland. The distance between Portland and Seattle is 175 miles. You plan to ride 25 miles each day. Write an absolute value function d(x), where x is the number of days into the ride, that describes your distance from Portland and use your function to determine the number of days it will take to complete your ride. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 224 Name ________________________________________ Date __________________ Class __________________ LESSON 18-3 Special Products of Binomials Practice and Problem Solving: Modified Fill in the blanks. Then simplify. The first one is done for you. 1. (x + 5) 2 x2 + 2(x)(5) + 52 2 x + 10x + 25 ________________________ 2. (m + 3)2 2 ____ 3. (2 + a) 2 + 2(____)(____) + ____2 _______________________ 2 ____ + 2(____)(____) + ____2 ________________________ Find the product. 4. (x + 4)2 ________________________ 5. (a + 7)2 6. (8 + b) 2 _______________________ ________________________ Fill in the blanks. Then simplify. The first one is done for you. 7. (y − 4) 2 y2 − 2(y)(4) + 42 2 y − 8y + 16 ________________________ 8. (y − 6) 2 2 ____ 9. (9 − x)2 + 2(____)(____) + ____2 _______________________ 2 ____ + 2(____)(____) + ____2 ________________________ Find the product. 10. (x −10) 2 ________________________ 11. (b −11)2 12. (3 − x)2 _______________________ ________________________ Fill in the blanks. Then simplify. The first one is done for you. 13. (x + 7)(x − 7) 14. (4 + y)(4 − y) 15. (x + 2)(x − 2) x2 − 72 2 x − 49 ________________________ 2 ____ − ____2 _______________________ 2 ____ − ____2 ________________________ Find the product. 16. ( x + 8)( x − 8) ________________________ 17. (3 + y )(3 − y ) _______________________ 18. ( x + 1)( x − 1) ________________________ Solve. 19. This week Kyra worked x + 4 hours. She is paid x − 4 dollars per hour. Write a polynomial expression for the amount that Kyra earned this week. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 325 Name ________________________________________ Date __________________ Class __________________ LESSON 19-1 Understanding Quadratic Functions Practice and Problem Solving: A/B For Exercises 1–4, tell whether the graph of the function a. opens upward or downward b. has a maximum or minimum c. is a reflection across the x-axis of the parent function d. is a stretch or a compression (shrink)? 1. y = 4 x 2 2. y = −5 x 2 ________________________________________ ________________________________________ ________________________________________ ________________________________________ 3. y = −3.2 x 2 4. y = 0.4 x 2 ________________________________________ ________________________________________ ________________________________________ ________________________________________ Determine the characteristics of each quadratic function. 5. y = 1.5 x 2 6. y = −2.5 x 2 Vertex: ______________________________________ Vertex: ____________________________________ Minimum (if any): _____________________________ Minimum (if any): __________________________ Maximum (if any): ____________________________ Maximum (if any): _________________________ Parent function reflected across Parent function reflected across x-axis? ____ Stretch or shrink? _____________________________ x-axis? Stretch or shrink? _________________________ Solve. 7. A quadratic function has the form y = ax 2 for some nonzero value of a and (4, 48) is on the graph. What is the value of a? __________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 329 Name ________________________________________ Date __________________ Class __________________ LESSON 19-1 Understanding Quadratic Functions Practice and Problem Solving: C Solve. 1. The graph of a quadratic function contains (4, −64) and its vertex is at the origin. a. Write an equation for this function. ________________________________________________ b. Does the equation show a stretch or a shrink of its parent equation? ________________________________________________ 2. The diagram shows the graph of a quadratic function f. Point P is on the graph. a. Write an equation for this function. ________________________________________________ b. Does the equation show a stretch or a shrink of its parent equation? ________________________________________________ 3. The axis of symmetry of the graph of a quadratic function is x = 0. The vertex has coordinates (0, 0). Point (−2, −10) is on the graph. Write an equation for the function. __________________________________ x y −3 −31.5 Write an equation for the function. ____________________________________ 0 0 5. A quadratic function has the form y = ax2 for some nonzero value of a. Suppose that (m, n) is on the graph of the function for some nonzero n real numbers m and n. Show that a = 2 . m 3 −31.5 4. The table represents three points on the graph of a quadratic function. _________________________________________________________________________________________ _________________________________________________________________________________________ 6. Functions f and g have the form y = ax2. The graph of f contains (1, 5). The graph of g contains (1, 0.2). Which function has a graph wider than that of y = x2? Explain. _________________________________________________________________________________________ _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 330 Name ________________________________________ Date __________________ Class __________________ LESSON 19-1 Understanding Quadratic Functions Practice and Problem Solving: Modified Determine the characteristics of each quadratic function. The first one is done for you. 1. y = 1 2 x 16 2. y = −0.4 x 2 (0, 0) Vertex: _____________________________________ Vertex: _________________________________ 0 Minimum (if any): ___________________________ Minimum (if any): _______________________ none Maximum (if any): __________________________ Maximum (if any): ______________________ 2 y=x Parent function: ____________________________ Parent function: ________________________ compression Stretch or compression? ____________________ Stretch or compression? ________________ Solve. The first one is started for you. 3. An equation has the form y = ax2. The point (3, 45) is on the graph. 4. An equation has the form y = ax2. The point (−4, −48) is on the graph. Complete the work to find a. Find a. y = ax2 45 = a(3)2 45 = 9a a = ____ ______________________________________ 5. The graphs of both y = 3.5x2 and y = −3.5x2 have the same vertex. What are the coordinates of the vertex? x-coordinate: _________________________ y-coordinate: _________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 331 Name ________________________________________ Date __________________ Class __________________ LESSON 19-2 Transforming Quadratic Functions Practice and Problem Solving: A/B A parabola has the equation f(x) = 2(x − 3)2 − 4. Complete: 1. The vertex is ____________. 2. The graph opens _________________. 3. The function has a minimum value of _____________________. The following graph is a translation of y = x2. Use it for 4–6. 4. What is the horizontal translation? 5. What is the vertical translation? 6. What is the quadratic equation for the graph? Graph the following parabolas. 7. y = −2(x + 1)2 + 2 8. y = 1 ( x − 2 )2 − 3 2 A ball follows a parabolic path represented by f(x) = −2(x − 5)2 + 9. Use this equation for 9–12. 9. What is the vertex? ____________ 10. What is the axis of symmetry? ______________ 11. Find two points on either side of the axis. _____________ and _______________ 12. Graph the parabola. Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 334 Name ________________________________________ Date __________________ Class __________________ LESSON 19-2 Transforming Quadratic Functions Practice and Problem Solving: C A parabola has the equation f(x) = −2(x − 3)2 + 4. Complete: 1. The vertex is ____________. 2. The graph opens _________________. 3. The function has a minimum value of _____________________. The following graph is a translation of y = x2. Use it for 4–7. 4. What is the horizontal translation? 5. What is the vertical translation? 6. What is the sign of a? 7. What is the quadratic equation for the graph? Graph the following parabolas. 8. y = −2(x + 2)2 + 5 9. y = 1 ( x − 3 )2 − 2 2 A ball follows a parabolic path represented by f(x) = −2(x − 4)2 + 8. Use this equation for 10–12. 10. What is the vertex? ____________ 11. Graph the parabola. 12. Why does the graph stop at x = 2 and x = 6? _______________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 335 Name ________________________________________ Date __________________ Class __________________ LESSON 19-2 Transforming Quadratic Functions Practice and Problem Solving: Modified A parabola has the equation f(x) = (x − 3)2 − 4. Complete. The first one is done for you. 3 to the right 1. What is the horizontal translation? 2. What is the vertical translation? 3. What is the vertex? The following graphs are translations of y = x2. Use them for 4–9. The first one is done for you. 4. What is the horizontal translation? 5. What is the vertical translation? ________________________________________ −4 ________________________________________ 6. What is the quadratic equation for the left graph? 7. What is the horizontal translation for the right graph? ________________________________________ ________________________________________ 8. What is the vertical translation for the left graph? 9. What is the quadratic equation for the right graph? ________________________________________ ________________________________________ Graph the following equations for parabolas. 10. y = (x + 1)2 − 2 11. y = (x − 3)2 + 2 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 336 Name ________________________________________ Date __________________ Class __________________ Interpreting Vertex Form and Standard Form LESSON 19-3 Practice and Problem Solving: A/B Determine if each function is a quadratic function. 1. y = 2 x 2 − 3 x + 5 ________________________ 2. y = 2 x − 4 3. y = 2 x + 3 x − 4 _______________________ ________________________ Write each quadratic function in standard form and write the equation for the line of symmetry. 4. y = x + 2 + x 2 ________________________ 5. y = −1 + 2 x − x 2 6. y = 2 x − 5 x 2 − 2 _______________________ ________________________ Change the vertex form to standard quadratic form. 7. y = 2( x + 3)2 − 6 8. y = 3( x − 5)2 + 4 ________________________________________ ________________________________________ Use the values in the table to write a quadratic equation in vertex form, then write the function in standard form. 9. The vertex of the function is (1, −3). 10. The vertex of the function is (−3, −2). x y x y −1 17 −1 14 0 2 −2 2 1 −3 −3 −2 2 2 −4 2 3 17 −5 14 ________________________________________ ________________________________________ ________________________________________ ________________________________________ 11. The graph of a function in the form f(x) = a(x − h)2 + k is shown. Use the graph to find an equation for f(x). _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 339 Name ________________________________________ Date __________________ Class __________________ Interpreting Vertex Form and Standard Form LESSON 19-3 Practice and Problem Solving: C Determine if each function is a quadratic function. 1. y = 0.5 x 2 − 3 ________________________ 2. y = 2( x − 4)2 − 5 3. y = 2 x + 3 x + 24 _______________________ ________________________ Write each quadratic function in standard form and write the equation for the line of symmetry. 4. y = 3 x + 2 + 2 x 2 ________________________ 5. y = −0.5 + 1.5 x − 2 x 2 6. y = −2 − 5 x 2 _______________________ ________________________ Change the vertex form to standard quadratic form. 3 7 8. y = − (2 x − 5)2 + 2 2 7. y = 3( x + 0.5)2 − 2.4 ________________________________________ ________________________________________ Use the values in the table to write a quadratic equation in vertex form, then write the function in standard form. 9. The vertex of the function is (1, 2). 10. The vertex of the function is (3, 5). x y x y −1 8 1 −23 0 3.5 2 −2 1 2 3 5 2 3.5 4 −2 3 8 5 −23 ________________________________________ ________________________________________ ________________________________________ ________________________________________ 11. The graph of a function in the form f(x) = a(x − h)2 + k is shown. Use the graph to find an equation for f(x). _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 340 Name ________________________________________ Date __________________ Class __________________ Interpreting Vertex Form and Standard Form LESSON 19-3 Practice and Problem Solving: Modified Determine if each function is a quadratic function. The first one is done for you. 2. y = x − 4 1. y = x 2 − 5 Quadratic function ________________________ 3. y = x 2 + 2 x _______________________ ________________________ Write each quadratic function in standard form and write the equation for the line of symmetry. The first one is done for you. 4. y = x 2 + 2 + x 5. y = −1 + 2 x 2 − x y = x + x + 2, x = −0.5 ________________________ 2 6. y = 2 x + 5 x 2 − 2 _______________________ ________________________ Change the vertex form to standard quadratic form. The first one is done for you. 8. y = 7. y = ( x + 1)2 + 2 y = x + 2x + 3 ________________________________________ 2 1 ( x − 0)2 + 1 9 ________________________________________ Use the values in the table to write a quadratic equation in vertex form, y = a( x − h )2 + k and then write the function in standard form, y = ax 2 + bx + c. The first one is done for you. 9. The vertex of the function is (0, 0). 10. The vertex of the function is (0, 2). x y x y −2 4 −2 6 −1 1 −1 3 0 0 0 2 1 1 1 3 2 4 2 6 y = 1( x − 0)2 + 0 ________________________________________ ________________________________________ y=x ________________________________________ ________________________________________ 2 11. The graph of a function in the form f(x) = a(x − h)2 + k is shown. Use the graph to find an equation for f(x). _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 341 Name ________________________________________ Date __________________ Class __________________ LESSON 20-1 Connecting Intercepts and Zeros Practice and Problem Solving: A/B Solve each equation by writing the related function, creating a table of values, graphing the related function, and finding its zeroes. 1. x2 + 1 = 2x y = ____________________________________ x −1 0 1 2 3 y ________________________________________ 2. −2x2 − 2x = 2x y = ____________________________________ x −3 −2 −1 0 1 y ________________________________________ Create a quadratic equation then solve the equation with a related function. You can use a table, graph, or graphing calculator. 3. A skydiver jumps out of a plane 5,000 feet above the ground and her parachute opens 3,000 feet above the ground. The function h(t) = −16t 2 + 5,000, where t represents the time in seconds, gives the height h, in feet, of the skydiver as she falls. When does her parachute open? Round to the nearest second. _________________________________________________________________________________________ 4. An astronaut on the moon drops a tool from the door of the landing ship. The quadratic function f(x) = −2x2 + 10 models the height of the tool,in meters, after x seconds. How long does it take the tool to hit the surface of the moon? Round your answer to the nearest tenth. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 345 Name ________________________________________ Date __________________ Class __________________ LESSON 20-1 Connecting Intercepts and Zeros Practice and Problem Solving: C Solve each equation by writing the related function, creating a table of values, graphing the related function, and finding its zeroes. Graph both functions on the same set of axes. 1. x2 + 1 = 2x x −1 0 1 2 3 y y = ____________________________________ ________________________________________ 2. 4x − 2 = 2x2 y = ____________________________________ x −1 0 1 2 3 y ________________________________________ 3. Can two different quadratic functions have the same zeroes? Explain. _________________________________________________________________________________________ _________________________________________________________________________________________ Create a quadratic equation then solve the equation with a related function using a graphing calculator. 4. A skydiver jumps out of a plane 5,000 feet above the ground and her parachute opens 3,000 feet above the ground. A second skydiver jumps out of the same plane at the same time, but does not open his parachute until 2,000 feet above the ground. The function h(t) = −16t 2 + 5,000, where t represents the time in seconds, gives the height h, in feet, of the skydivers as they fall. How much longer does the second skydiver fall, neglecting air resistance? Round to the nearest tenth of a second. _________________________________________________________________________________________ 5. An archway has vertical sides 10 feet high. The top of an archway can be modeled by the quadratic function f(x) = −0.5x2 + 10 where x is the horizontal distance, in feet, along the archway. How far apart are the walls of the archway? Round your answer to the nearest tenth of a foot. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 346 Name ________________________________________ Date __________________ Class __________________ LESSON 20-1 Connecting Intercepts and Zeros Practice and Problem Solving: Modified Solve each equation by writing the related function, creating a table of values, graphing the related function, and finding its zeroes. The first one is started for you. 1. x2 = 4 y = x2 − 4 ____________________________________ x −2 −1 y 0 −3 0 1 2 x = −2 and x = ____________________________________ 2. x2 − x = 6 y = ____________________________________ x −2 −1 0 1 2 y ________________________________________ Create a quadratic equation then solve the equation with a related function. You can use a table, graph, or graphing calculator. The first one has been started for you. 3. A competitive diver stands at the end of a 30-foot platform and falls forward into a dive. The function h(t) = −16t 2 + 30, where t represents the time in seconds, gives the height h, in feet, of the diver as he falls. How long is the diver in the air? Round your answer to the nearest tenth of a second. 0 = −16t 2 + 30 _________________________________________________________________________________________ 4. A basketball player successfully makes a free throw. The height of the ball above the ground can be modeled by the quadratic function f(t) = −16t 2 + 27t + 6 where t is the time the ball is in the air. The basket is 10 feet above the ground. How long does it take the ball to get to the basket? Round your answer to the nearest tenth of a second. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 347 Name ________________________________________ Date __________________ Class __________________ LESSON 20-2 Connecting Intercepts and Linear Factors Practice and Problem Solving: A/B Graph each quadratic function and each of its linear factors. Then identify the x-intercepts and the axis of symmetry of each parabola. 1. y = ( x − 1)( x − 5) 2. y = ( x − 3)( x + 2) ________________________________________ ________________________________________ ________________________________________ ________________________________________ Write each function in standard form. 3. y = 5( x + 3)( x − 2) 4. y = −2( x − 3)( x − 1) ________________________________________ ________________________________________ Graph the axis of symmetry, the vertex, the point containing the y-intercept, and another point. Then reflect the points across the axis of symmetry. Connect the points with a smooth curve. 5. y = ( x − 1)( x + 3) 6. y = ( x + 1)( x − 3) Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 350 Name ________________________________________ Date __________________ Class __________________ LESSON 20-2 Connecting Intercepts and Linear Factors Practice and Problem Solving: C Graph each quadratic function and each of its linear factors. Then identify the x-intercepts and the axis of symmetry of each parabola. 1. y = 2( x − 1)( x − 5) 2. y = −( x − 3)( x + 2) ________________________________________ ________________________________________ Write each function in standard form. 3. y = ( −3 x − 4)(2 x − 1) 4. y = ________________________________________ 2 (3 x − 2)(3 x + 6) 3 ________________________________________ Graph the axis of symmetry, the vertex, the point containing the y-intercept, and another point. Then reflect the points across the axis of symmetry. Connect the points with a smooth curve. 5. y = 1 ( x − 8)( x + 2) 2 6. y = ( x + 1)( x − 3) Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 351 Name ________________________________________ Date __________________ Class __________________ LESSON 20-2 Connecting Intercepts and Linear Factors Practice and Problem Solving: Modified Graph each quadratic function and each of its linear factors. Then identify the x-intercepts and the axis of symmetry of each parabola. The first one is done for you. 1. y = ( x − 2)( x + 2) 2. y = ( x + 5)( x + 1) x intercepts −2 and 2 Axis of symmetry x = 0 ________________________________________ ________________________________________ Write each function in standard form. The first one is done for you. 3. y = ( x + 3)( x + 2) 4. y = ( x − 3)( x − 1) y = x2 + 5x + 6 ________________________________________ ________________________________________ Graph the axis of symmetry, the vertex, the point containing the y-intercept, and another point. Then reflect the points across the axis of symmetry. Connect the points with a smooth curve. The first one is done for you. 5. y = ( x − 2)( x − 2) 6. y = ( x + 2)( x + 2) Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 352 Name ________________________________________ Date __________________ Class __________________ LESSON 20-3 Applying the Zero Product Property to Solve Equations Practice and Problem Solving: A/B Find the zeros of each function. 1. f(x) = (x − 3)(x + 5) 2. f(x) = x(x − 1) ________________________________________ ________________________________________ 3. f(x) = (x + 1)(x + 1) 4. f(x) = (x − 5)(x + 1) ________________________________________ ________________________________________ 6. f(x) = (x − 6)(x + 1) 5. f(x) = x(x − 3) ________________________________________ ________________________________________ 7. f(x) = (x − 11)(x − 1) 8. f(x) = (x + 13)(x + 5) ________________________________________ ________________________________________ 9. f(x) = (x + 5)(x − 8) 10. f(x) = (x − 7)(x + 2) ________________________________________ ________________________________________ Use the Distributive Property and the Zero Product Property to solve the equations. 11. f(x) = 2x(x − 2) + 14(x − 2) 12. f(x) = x(x − 4) − 2(x − 4) ________________________________________ ________________________________________ 13. f(x) = 5x(x − 3) + 25(x − 3) 14. f(x) = 3x(x − 7) + 7(x − 7) ________________________________________ ________________________________________ Solve. 15. The height of a javelin after it has left the hand of the thrower can be modeled by the function h = 3(4t − 2)(−t + 4), where h is the height of the javelin and t is the time in seconds. How long is the javelin in the air? _________________________________________________________________________________________ 16. The height of a flare fired from the deck of a ship can be modeled by h = (−4t + 24)(4t + 4) where h is the height of the flare above water in feet and t is the time in seconds. Find the number of seconds it takes the flare to hit the water. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 355 Name ________________________________________ Date __________________ Class __________________ LESSON 20-3 Applying the Zero Product Property to Solve Equations Practice and Problem Solving: C Find the zeros of each function. 1. f(x) = (3x − 1)(2x + 3) 2. f(x) = (x + 12)(x − 8) ________________________________________ ________________________________________ 3. f(x) = (x − 12)(x − 9) 4. f(x) = x(x − 1)(x − 1) ________________________________________ ________________________________________ 5. f(x) = (x + 6)(x − 5) 6. f(x) = (x − 3)(x + 2) ________________________________________ ________________________________________ 7. f(x) = (x + 9)(x − 2) 8. f(x) = (x − 1)(x − 1) ________________________________________ ________________________________________ 9. f(x) = (x − 1)(x + 1)(x + 2)(x − 2) 10. f(x) = 4(x + 7)(x − 1) ________________________________________ ________________________________________ Use the Distributive Property and the Zero Product Property to find the zeros of each function. 11. f(x) = 2x(x + 3) − 4(x + 3) 12. f(x) = 3x(x + 7) − 2x − 14 ________________________________________ ________________________________________ 13. f(x) = x2 + 4x − 3(x + 4) 14. f(x) = 2x(x + 4) + 3x + 12 ________________________________________ ________________________________________ Solve. 15. The height of an arrow after it has left the bow can be modeled by the function h = 2t(3t − 9), where h is the height of the arrow and t is the time in seconds. How long is the arrow in the air before it hits the target? _________________________________________________________________________________________ 16. The height of a person after he has left the trampoline in a jump can be modeled by the function h = −3t(−4t + 8), where h is the height of the person and t is the time in seconds. How long is the person in the air before he lands back on the trampoline? _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 356 Name ________________________________________ Date __________________ Class __________________ LESSON 20-3 Applying the Zero Product Property to Solve Equations Practice and Problem Solving: Modified Complete. 1. If ab = 0, then ___________________________ or ___________________________. Use the Zero Product Property to solve each equation. Check your answers. The first one is done for you. 2. (x − 7)(x + 2) = 0 x−7=0 or 7 x = ______ or 3. (x − 5)(x − 1) = 0 x+2=0 x−5=0 −2 x = ______ x = ______ 4. x(x − 5) = 0 or x−1=0 or x = ______ 5. (x + 2)(x + 1) = 0 x = 0 or (__________) = 0 x + 2 = 0 or (_________) = 0 x = 0 or x = _____ x = −2 6. (x − 9)(x + 3) = 0 or x = _____ 7. (x + 5)(x + 3) = 0 (_________) = 0 or (__________) = 0 (__________) = 0 or x = ______ or x = _____ x = _____ or x = _____ 8. (x + 2)(x + 6) = 0 (_________) = 0 9. (3x − 4)(x − 3) = 0 ________________________________________ ________________________________________ 10. (x − 5)(x − 1) = 0 11. (x − 6)(x + 2) = 0 ________________________________________ ________________________________________ Solve. The first one is started for you. 12. The product of two consecutive positive integers, 56, can be modeled by the function f(x) = (x − 8)(x − 7). Find the integers. x = 8 or _________________________________________________________________________________________ 13. The product of two consecutive positive integers, 110, can be modeled by the function f(x) = (x − 10)(x − 11). Find the integers. _________________________________________________________________________________________ 14. Jan is 3 years younger than Ari. Bea is 3 years older than Ari. The product of Jan’s age and Bea’s age is 55. Use the function f(x) = (x + 8)(x − 8) to find Ari’s age. Then, find Jan and Bea’s ages. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 357 Name ________________________________________ Date __________________ Class __________________ LESSON 21-1 Solving Equations by Factoring x2 + bx + c Practice and Problem Solving: A/B What factors are shown by the algebra tiles? 1. 2. ________________________________________ ________________________________________ Factor. 3. x2 − 3x − 4 ________________________ 6. x2 + 11x + 24 ________________________ 9. x2 − 11x − 42 ________________________ 4. x2 + 4x + 3 5. x2 − 14x + 45 ________________________ 7. x2 − 12x + 32 _________________________ 8. x2 − 15x + 36 ________________________ 10. x2 − 18x + 81 _________________________ 11. x2 − 7x − 44 ________________________ _________________________ Solve by factoring. 12. x2 = 5x ________________________ 15. x2 = −4x + 21 ________________________ 13. x2 = 9x − 18 14. x2 − 15x + 50 = 0 ________________________ 16. x2 + 7x = 8 _________________________ 17. x2 = −2x + 15 ________________________ _________________________ Solve. 18. The product of two consecutive integers is 72. Find all solutions. _________________________________________________________________________________________ 19. The length of a rectangle is 8 feet more than its width. The area of the rectangle is 84 square feet. Find its length and width. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 361 Name ________________________________________ Date __________________ Class __________________ LESSON 21-1 Solving Equations by Factoring x2 + bx + c Practice and Problem Solving: C What polynomials are shown by the algebra tiles? 1. 2. ________________________________________ ________________________________________ 3. A trinomial is in the form x2 + bx + c, where b < 0 and c > 0. What do you know about the two factors? ___________________________________________________________________________________ Solve by factoring. 4. x2 − 25 = 0 ________________________ 7. x2 − 9 + 2x + 1 = 0 ________________________ 10. x + 3 = x2 − 3 ________________________ 13. x2 = −3x + 28 ________________________ 5. x2 − 2x + 1 = 0 6. x2 − 5x + 4 = 0 ________________________ 8. x2 + x = 30 _________________________ 9. x2 = 36 ________________________ 11. x2 + 3x − 11 = 43 _________________________ 12. x2 − 3x = 40 ________________________ 14. x2 + 8x = −63 − 8x _________________________ 15. x2 − 20 = x ________________________ _________________________ Solve. 16. The product of two consecutive integers is five less than five times their sum. Find all possible solutions. _________________________________________________________________________________________ 17. The sum of the first n positive integers can be found using the formula n(n + 1) . How many integers must be added to get 253 as the sum? 2 _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 362 Name ________________________________________ Date __________________ Class __________________ LESSON 21-1 Solving Equations by Factoring x2 + bx + c Practice and Problem Solving: Modified Complete the table to find the correct factors. The first one is started for you. 1. x2 + 8x + 15 2. x2 + 5x + 6 factors of 15 sum of factors 3, 5 8 −3, −5 −8 1, 15 16 −1, −15 −16 factors of 6 factors: ___________________________ sum of factors factors: ___________________________ Factor each trinomial. The first one is done for you. 3. x2 + x − 2 (x + 2)(x − 1) ________________________ 6. x2 − x − 12 ________________________ 9. x2 − x − 6 ________________________ 4. x2 + x − 6 5. x2 + 2x + 1 _______________________ ________________________ 7. x2 − 6x + 5 8. x2 + 6x + 9 _______________________ 10. x2 − 8x + 15 ________________________ 11. x2 + 7x + 12 _______________________ ________________________ Solve each equation by factoring. The first one is done for you. 12. x2 − 3x + 2 = 0 x = 1, 2 ________________________ 15. x2 + 10x + 25 = 0 ________________________ 13. x2 + 2x − 3 = 0 _______________________ 16. x2 + 10x + 21 = 0 _______________________ 14. x2 + 6x + 8 = 0 ________________________ 17. x2 − 11x + 24 = 0 ________________________ Write and solve an equation for each problem. The first one is started for you. 18. The product of two consecutive positive integers is 30. Find the integers. x(x + 1) = x2 + x; x2 + x = 30; x2 + x − 30 = 0 _________________________________________________________________________________________ 19. The product of two consecutive positive integers is 110. Find the integers. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 363 Name ________________________________________ Date __________________ Class __________________ LESSON 21-2 Solving Equations by Factoring ax2 + bx + c Practice and Problem Solving: A/B Solve the equations by factoring. 1. 2x2 − 3x = 2x − 2 2. 3x2 − 4x = 6x − 3 ________________________________________ ________________________________________ 3. 3x2 − 7x = x − 4 4. 5x2 + 6x = −5x − 2 ________________________________________ ________________________________________ 5. 4x2 + 16x − 48 = 0 6. 2x2 − 32 = 0 ________________________________________ ________________________________________ 7. 2x2 − 7 = 14 − 11x 8. 7x2 − 12x = 36 + 7x ________________________________________ ________________________________________ 9. 5x2 = 45 10. 2x2 − 7x = 15 − 6x ________________________________________ ________________________________________ 11. 4x2 − 20x = −25 12. 5x2 − 20x + 20 = 0 ________________________________________ ________________________________________ 13. 3x2 + 5x = 6 − 2x 14. 2x2 + 3x + 6 = 4x ________________________________________ ________________________________________ 15. 3x2 = 9x 16. 9x2 − 13x = 8x − 10 ________________________________________ ________________________________________ 17. 4x2 − 50x + 49 = 50x 18. 4x2 + 21x = 6x − 14 ________________________________________ ________________________________________ 19. 24x2 − x = 10x − 1 20. 3x2 + 12x − 15 = 0 ________________________________________ ________________________________________ Solve. 21. The height of a flare fired from the deck of a ship in distress can be modeled by h = −16t 2 + 104t + 56, where h is the height in feet of the flare above water and t is the time in seconds. Find the time it takes the flare to hit the water. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 366 Name ________________________________________ Date __________________ Class __________________ LESSON 21-2 Solving Equations by Factoring ax2 + bx + c Practice and Problem Solving: C Simplify the equation. Then solve the equation by factoring. 1. 2x(x + 1) = 7x − 2 2. 3x(x − 2) = 4x − 3 ________________________________________ ________________________________________ 3. 3x2 = 4(2x − 1) 4. 5x(x + 1) = −2(3x + 1) ________________________________________ ________________________________________ 5. 4x(x + 4) = 48 6. 2(x + 3)(x − 3) = 14 ________________________________________ ________________________________________ 7. 2x(x + 4) − 7 = 14 − 3x 8. 7x(x − 1) = 12(x + 3) ________________________________________ ________________________________________ 9. 8x2 = 3(x2 + 15) 10. 2x(x − 3) = 5(3 − x) ________________________________________ ________________________________________ 11. 2x(3x − 10) = 2x2 − 25 12. 5x2 − 2(x − 10) = 18x ________________________________________ ________________________________________ 13. 3x(x + 2) = 6 − x 14. 2x(x − 2) = −3(x − 2) ________________________________________ ________________________________________ 15. 0.3x2 + x = 0.1x 16. 0.9x2 − 1.3x = 0.8x − 1 ________________________________________ ________________________________________ 17. 0.4x2 − 5x + 4.9 = 5x 18. 1.5x2 + 6x = 7.5 ________________________________________ 19. 6x2 − ________________________________________ 7 1 x=x− 4 4 20. 8x2 − ________________________________________ 2 1 x= + 7x2 3 3 ________________________________________ Solve. 21. The height of a ball thrown upward on the moon with a velocity of 8 meters per second can be modeled by h = −0.8t 2 + 8t, where h is the height of the ball in meters and t is the time in seconds. At what times will the height of the ball reach 19.2 meters? _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 367 Name ________________________________________ Date __________________ Class __________________ LESSON 21-2 Solving Equations by Factoring ax2 + bx + c Practice and Problem Solving: Modified Use the Zero Product Property to solve each equation. The first one is done for you. 1. 2x2 − 5x + 2 = 0 2. 3x2 + 12x − 15 = 0 (2x − 1)(x − 2) = 0 3(x2 + ___ x − ___) = 0 2x − 1 = 0 or x − 2 = 0 3(______)(______) = 0 2x = 1 or x = 2 (______) = 0 or (_____) = 0 x= 1 or x = 2 2 x = ___ or x = ____ Factor each quadratic expression. Then use the Zero Product Property to solve each equation. The first one is done for you. 3. 3x2 − 8x + 4 = 0 4. 5x2 + 11x + 2 = 0 2 x= , 2 3 ________________________________________ ________________________________________ 5. 4x2 + 16x − 48 = 0 6. 2x2 − 32 = 0 ________________________________________ ________________________________________ 7. 2x2 − 11x + 14 = 0 8. 7x2 − 19x − 36 = 0 ________________________________________ ________________________________________ 9. 5x2 − 45 = 0 10. 2x2 − x − 15 = 0 ________________________________________ ________________________________________ 11. 4x2 − 20x + 25 = 0 12. 5x2 − 20x + 20 = 0 ________________________________________ ________________________________________ 13. 3x2 + 7x − 6 = 0 14. 2x2 − x − 6 = 0 ________________________________________ ________________________________________ 15. A package is dropped from a helicopter at 1600 feet. The height of the package can be modeled by h = −16t 2 + 1600, where h is the height of the package in feet and t is the time in seconds. How long will it take for the package to hit the ground? a. Write the equation. ____________________________________ b. Solve the equation. ____________________________________ c. Answer the question. ____________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 368 Name ________________________________________ Date __________________ Class __________________ LESSON 21-3 Using Special Factors to Solve Equations Practice and Problem Solving: A/B Factor using the perfect-square technique. 1. x2 + 10xy + 25y2 2. 32x2 + 80xy + 50y2 ________________________________________ ________________________________________ Factor using the difference of squares technique. 3. 81x2 − 121y2 4. 75x3 − 48x ________________________________________ ________________________________________ Solve each equation with special factors. 5. 50x2 = 72 6. 18x3 + 48x2 = − 32x ________________________________________ ________________________________________ Solve. 7. A projectile is launched from a hole in the ground one foot deep. Its height follows the equation h = −16t2 + 8t −1. Use factoring by perfectsquares to find the time when the projectile lands back on the ground. (Hint: Landing on the ground means projectile height is zero.) _________________________________________________________________________________________ 8. Which of the following are solutions to 4x3 − 16x = 0? A −2 B −1 C0 D1 E2 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 371 Name ________________________________________ Date __________________ Class __________________ LESSON 21-3 Using Special Factors to Solve Equations Practice and Problem Solving: C Factor using the perfect-square technique. 1. 27x2 + 72xy + 48y2 2. 25x3 − 60x2y + 36xy2 ________________________________________ ________________________________________ Factor using the difference of squares technique. 3. x4 − 81 4. 36x4 − 16x2y2 ________________________________________ ________________________________________ Solve each equation with special factors. 5. −7x3 + 100x = −75x 6. x3 + 8x2 + 4x = −x3 − 4x ________________________________________ ________________________________________ Solve. 7. A projectile is launched from an underground silo 81 feet deep. Its height follows the equation h = −16t2 + 72t − 81. Use factoring by perfect-squares to find the time when the projectile lands back on the ground. _________________________________________________________________________________________ 8. Which of the following are solutions to 81x3 = 256x? A − 16 9 B − 4 3 C0 D 16 9 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 372 Name ________________________________________ Date __________________ Class __________________ LESSON 21-3 Using Special Factors to Solve Equations Practice and Problem Solving: Modified To factor each perfect-square trinomial, use a = term 1 and b = term 2. Then use the form (a ± b )2 . The first one is done for you. 1. 4x2 − 20x + 25 2. 9x2 + 12x + 4 5 b = ______ 2x a = ______ a = _______ negative Middle term’s sign: ______________ b = _______ Middle term’s sign: ______________ 2 (2x − 5) Factored form: ______________ Factored form: ______________ 3. 25x2 − 30x + 9 4. 36x2 + 24x + 4 a = _______ a = _______ b = _______ b = _______ Middle term’s sign: ______________ Middle term’s sign: ______________ Factored form: ______________ Factored form: ______________ Factor each difference of squares. Use a = term 1 and b = term 2. Then use the form (a + b)(a − b). The first one is done for you. 5. 49x2 − 16 7x a = ______ 6. 36 − 25x2 4 b = ______ a = _______ (7x − 4)(7x + 4) Factored form = ________________ b = _______ Factored form = ______________ Solve each equation with special factors. 7. 49x2 − 14x + 1 = 0 8. 121 = 36x2 (__________)(__________) = 0 (__________)(__________) = 0 x = __________; x = __________ x = __________; x = __________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 373 Name ________________________________________ Date __________________ Class __________________ LESSON 22-1 Solving Equations by Taking Square Roots Practice and Problem Solving: A/B Solve. If the equation has no solution, give that as your answer. 1. x2 − 25 = 0 ________________________ 4. −3x2 + 27 = 0 ________________________ 7. x 2 − 121 = 0 ________________________ 10. ( x + 5)2 − 6 = 43 ________________________ 13. 2(x − 3)2 + 1= 73 ________________________ 2. x2 + 25 = 0 3. 6x2 − 6 = 0 ________________________ 5. −2x2 − 1 = 0 _________________________ 6. 4x2 − 100 = −100 ________________________ 8. x 2 − 49 = 0 _________________________ 9. x 2 − 16 = 20 ________________________ 11. (x − 1)2 − 19 = 81 _________________________ 12. ( x − 14)2 + 13 = 14 ________________________ 14. (x − 1)2 + 15 = 14 _________________________ 15. −2(x + 1)2 − 5 = −55 ________________________ _________________________ Solve. Express square roots in simplest form. 16. 2(x + 1)2 − 1= 9 ________________________ 17. 2(x − 3)2 + 7 = 19 18. 5(x − 7)2 + 10 = 25 ________________________ _________________________ Solve. 19. An auditorium has a floor area of 20,000 square feet. The length of the auditorium is twice its width. Find the dimensions of the room. _________________________________________________________________________________________ 20. A ball is dropped from a height of 64 feet. Its height, in feet, can be modeled by the function h(t) = −16t 2 + 64, where t is the time in seconds since the ball was dropped. After how many seconds will the ball hit the ground? ______________________________________________________________ 21. A plot of land is in the shape of a square. The shaded square inside is covered with gravel. The rest of the square plot is covered in grass. Its area is 1400 square feet. How long are the sides of the square? ______________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 377 Name ________________________________________ Date __________________ Class __________________ LESSON 22-1 Solving Equations by Taking Square Roots Practice and Problem Solving: C 1. Let ax2 + b = c, where a, b, and c are real numbers and a is nonzero. How many real roots the equation has is determined by the relationship among a, b, and c. How are a, b, and c related if the equation has no real roots, one real root, or two real roots? Explain your reasoning. _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ 2. Show, using the graph, that 0.5(x − 1)2 + 3 = 0 has no real roots. Then write an algebraic argument to support your conclusion. ______________________________________________ ______________________________________________ ______________________________________________ ______________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ 3. Let a(x − h)2 = p, where a, h, and p are positive real numbers. Show that this equation has two real roots. Then determine the sum of the roots. Explain your reasoning. _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 378 Name ________________________________________ Date __________________ Class __________________ LESSON 22-1 Solving Equations by Taking Square Roots Practice and Problem Solving: Modified Solve. The first one is done for you. If the equation has no solution, give that as your answer. 1. x 2 100 0 2. x 2 64 0 3. x2 + 144 = 0 x 2 = 100 x = ± 100 x = 10 or − 10 ________________________ _______________________ ________________________ Solve. The first one is done for you. 4. 2x2 + 4 = 22 5. 3x2 − 5 = 103 2 x 2 + 4 = 22 6. 2x2 + 1 = 99 x2 = 36 2 x = 18 2 2x2 = 98 x2 = 9 x = 3 or − 3 ________________________ _______________________ ________________________ Solve. The first one is done for you. 7. (x 2)2 25 8. (x 9)2 4 9. (x 6)2 16 ( x + 2)2 = 25 x + 2 = ± 25 x = 3 or − 7 ________________________ _______________________ ________________________ Write an equation and solve. The first one is started for you. 10. The length of a rectangular garden is three times its width. The area of the garden is 300 square feet. Find the length and width of the garden. (3w) (w) = 300; 3w2 = 300 __________________________________________________________ 11. The square of a number is increased by 27 and the result is 148. Find all possible solutions for the number. Show your work. _________________________________________________________________________________________ _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 379 Name ________________________________________ Date __________________ Class __________________ LESSON 22-2 Solving Equations by Completing the Square Practice and Problem Solving: A/B Solve each equation by completing the square. The roots are integers. 1. x2 + 4x = 5 ________________________ 4. x2 + 2x = 15 ________________________ 2. x2 − 2x = 8 _______________________ 5. x2 − 10x = 24 _______________________ 3. x2 − 10x = −25 ________________________ 6. x2 + 4x = 32 ________________________ Solve each equation by completing the square. Express square roots in simplest form. 7. x2 − 2x = 1 ________________________ 10. 2x2 − 4x = 8 ________________________ 13. 3x2 − 6x = 21 ________________________ 8. x2 − 6x = −6 _______________________ 11. x2 + 4x = −1 _______________________ 14. 3x2 − 12x = 69 _______________________ 9. x2 − 4x = −1 ________________________ 12. 3x2 − 12x = 3 ________________________ 15. 5x2 − 50x = −85 ________________________ Solve. 16. A rectangular deck has an area of 320 ft2. The length of the deck is 4 feet longer than the width. Find the dimensions of the deck. Solve by completing the square. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 382 Name ________________________________________ Date __________________ Class __________________ LESSON 22-2 Solving Equations by Completing the Square Practice and Problem Solving: C Solve each problem. 1. For some real number b, the equation x2 + bx = −4 has exactly one root. Determine that value of b and show your work. _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ 2. The equation below is true for all real numbers x and only one real number b. x2 + 4x + 9 = (x + b)2 + 5 Determine the value of b. Show your work. _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ 3. Consider the function below. y = x2 − 6x + 14 For what value of x will y = 5? Determine this value of x and show your work. _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ 4. Let y = x2 + 4x − 21. Use completing the square to show that the graph has two x-intercepts. What are they? Show your work. _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 383 Name ________________________________________ Date __________________ Class __________________ LESSON 22-2 Solving Equations by Completing the Square Practice and Problem Solving: Modified Solve each equation by completing the square. The first one is partially done for you. 1. x2 + 6x = −8 6 The coefficient of x is ______________. 3 One half of 6 is _______________. 9 The square of one half of this coefficient is ________. Add this number to each side of the equation. x 2 + 6 x + 9 = −8 + 9 Factor the left side and simplify the right side. (x + 3)2 = 1 Take the square root of each side. ( x + 3)2 = 1 x + 3 = ±1 Finish. x+3=1 → 2. x2 − 8x = 20 ________________________ 5. x2 − 10x = −24 ________________________ 8. 2x2 − 4x = 8 ________________________ x= x + 3 = −1 3. x2 + 12x = 13 _______________________ 6. x2 − 16x = 17 _______________________ 9. x2 + 4x = −1 _______________________ → x= 4. x2 − 2x = 35 ________________________ 7. x2 + 10x = −16 ________________________ 10. 3x2 − 12x = 3 ________________________ Solve. The first one is partially done for you. 11. A rectangular patio has an area of 91 square feet. The length is 6 feet greater than the width. Find the dimensions of the patio. Solve by completing the square. w and w + 6 a. Find the width and the length in terms of w. b. Write an equation for the total area. c. Find the square of the coefficient of w. d. Find the dimensions. Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 384 Name ________________________________________ Date __________________ Class __________________ LESSON 22-3 Using the Quadratic Formula to Solve Equations Practice and Problem Solving: A/B Solve using the quadratic formula. 1. x2 + x = 12 2. 4x2 − 17x − 15 = 0 ________________________________________ ________________________________________ 3. 2x2 − 5x = 3 4. 3x2 + 11x + 5 = 0 ________________________________________ ________________________________________ 5. x2 − 11x + 28 = 0 6. x2 − 49 = 0 ________________________________________ ________________________________________ 7. 6x2 + x − 1 = 0 8. x2 + 8x − 20 = 0 ________________________________________ ________________________________________ Find the number of real solutions of each equation using the discriminant. 9. x2 + 25 = 0 ________________________ 10. 3 x 2 − x 7 − 3 = 0 _______________________ 11. x2 + 8x + 16 = 0 ________________________ Solve. 12. In the past, professional baseball was played at the Astrodome in Houston, Texas. The Astrodome has a maximum height of 63.4 m. The height in meters of a baseball t seconds after it is hit straight up in the air with a velocity of 45 m/s is given by h = −9.8t 2 + 45t + 1. Will a baseball hit straight up with this velocity hit the roof of the Astrodome? Use the discriminant to explain your answer. _________________________________________________________________________________________ _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 387 Name ________________________________________ Date __________________ Class __________________ LESSON 22-3 Using the Quadratic Formula to Solve Equations Practice and Problem Solving: C Determine the number of real solutions for each equation. Then solve each equation that has one or more real solutions by using the quadratic formula. 1. 4x2 + 7x = 10 ________________________ 4. 14 − 3x2 = 2x ________________________ 7. 3x2 − 9 = 7x2 − 12x ________________________ 10. 7x2 − 5x + 4 = 5x2 − 2 ________________________ 2. 3x2 − 4 = 4x ________________________ 5. 5x2 + 4 = 3x + 2 ________________________ 8. 9x2 − 12x + 9 = 5x − 4x2 ________________________ 11. 6x2 − 49 + 34x = 6x + 10x2 ________________________ 3. 2x2 = 6x + 3 _________________________ 6. 3x2 − 12x = 8 − 15x _________________________ 9. 3x2 + 9x + 5 = 1 − 2x2 _________________________ 12. 9 − 8x2 = 6x + 14 _________________________ 13. Explain what happens in the quadratic formula when there are no real roots for a quadratic equation. _________________________________________________________________________________________ 14. The length and width of a rectangular patio are (x + 7) feet and (x + 9) feet, respectively. If the area of the patio is 190 square feet, what are the dimensions of the patio? _________________________________________________________________________________________ 15. A model rocket is launched from a platform 12 meters high at a speed of 35 meters per second. Its height h can be modeled by the equation h = −4.9t 2 + 35t + 12, where t is the time in seconds. At what time will the rocket be at an altitude of 60 meters? _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 388 Name ________________________________________ Date __________________ Class __________________ LESSON 22-3 Using the Quadratic Formula to Solve Equations Practice and Problem Solving: Modified Solve using the quadratic formula. The first one is done for you. 1. x2+ 6x + 5 = 0 a: x = 1 b: 2. x2 − 9x + 20 = 0 6 − 6 ± c: 6 2 5 −4 1 a: 5 x = 2 1 −1, −5 ________________________________________ b: − c: 2 ± −4 2 ________________________________________ 3. 2x2 + 9x + 4 = 0 a: b: 4. x2 − 3x − 18 = 0 c: a: ________________________________________ b: c: ________________________________________ 5. x2 + 4x − 32 = 0 6. 2x2 + 9x − 5 = 0 ________________________________________ ________________________________________ Find the number of real solutions of each quadratic equation using the discriminant. The first one is done for you. 7. x2 + 3x + 5 = 0 b2 − 4ac = 3 8. x2 + 10x + 25 = 0 2 −4 1 • 5 b2 − 4ac = −11 = ___________ no real solutions ________________________ 2 −4 9. x2 − 6x − 7 = 0 • b2 − 4ac = _________ = ___________ _______________________ ________________________ Solve using the quadratic formula. 10. x2 − 64 = 0 11. x2 + 2x + 36 = 0 ________________________________________ ________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 389 Name ________________________________________ Date __________________ Class __________________ LESSON 22-4 Choosing a Method for Solving Quadratic Equations Practice and Problem Solving: A/B Solve each quadratic equation by any means. Identify the method and explain why you chose it. Express irrational answers in radical form and use a calculator to approximate your answer rounded to two decimal places. 1. 4 x 2 = 64 2. 4( x − 3)2 = 25 ________________________________________ ________________________________________ ________________________________________ ________________________________________ 3. x 2 − 3 x − 28 = 0 4. x 2 − x = 6 ________________________________________ ________________________________________ ________________________________________ ________________________________________ 5. 2 x 2 − 4 x − 3 = 0 6. x 2 + 10 x − 3 = 0 ________________________________________ ________________________________________ ________________________________________ ________________________________________ 7. 1.5 x 2 − 4.3 x = −1.2 8. x 2 − 1 =0 4 ________________________________________ ________________________________________ ________________________________________ ________________________________________ Use any method to solve each quadratic equation. Identify the method and explain why you chose it. Convert irrational answers and fractions to decimals and round to the hundredths place. 9. The formula for height, in feet, of a projectile under the influence of gravity is given by h = −16t 2 + vt + s, where t is the time in seconds, v is the upward velocity at the start, and s is the starting height. Marvin throws a baseball straight up into the air at 70 feet per second. The ball leaves his hand at a height of 5 feet. When does the ball reach a height of 75 feet? _________________________________________________________________________________________ _________________________________________________________________________________________ 10. Use the projectile motion formula and solve the quadratic equation. Melissa drops a tennis ball from the roof of a building that is 256 feet high. How long does it take the tennis ball to hit the ground? _________________________________________________________________________________________ _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 392 Name ________________________________________ Date __________________ Class __________________ LESSON 22-4 Choosing a Method for Solving Quadratic Equations Practice and Problem Solving: C Solve each quadratic equation by any means. Identify the method and explain why you chose it. Express irrational answers in radical form and use a calculator to approximate your answer rounded to two decimal places. 1. 1 2 1 x = 2 8 2. 2 x 2 − 15 x − 8 = 0 ________________________________________ ________________________________________ ________________________________________ ________________________________________ 2 1⎞ 1 ⎛ 3. 2 ⎜ x + ⎟ = 2 2 ⎝ ⎠ 4. 2 x 2 + 7 x − 15 = 0 ________________________________________ ________________________________________ ________________________________________ ________________________________________ 5. 3 x 2 + 4 x = 1 6. x 2 − 24 x + 128 = 0 ________________________________________ ________________________________________ ________________________________________ ________________________________________ 7. 0.16 x 2 + 0.08 x + .01 = 0.16 8. 0.36 x 2 − 0.25 = 0 ________________________________________ ________________________________________ ________________________________________ ________________________________________ Use any method to solve each quadratic equation. Identify the method and explain why you chose it. Convert irrational answers and fractions to decimals and round to the hundredths place. 9. The formula for height, in feet, of a projectile under the influence of gravity is given by h = −16t 2 + vt + s, where t is the time in seconds, v is the upward velocity at the start, and s is the starting height. Andrea launches a bottle rocket filled with water under pressure straight up into the air from the ground at a velocity of 48 feet per second. How long is the rocket in the air? _________________________________________________________________________________________ _________________________________________________________________________________________ 10. Use the projectile motion formula and solve the quadratic equation. Eric launches a water balloon straight up into the air from a platform five feet high at a velocity of 20 feet per second. Will the balloon hit the target suspended at a height of 50 feet? Explain how you know. _________________________________________________________________________________________ _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 393 Name ________________________________________ Date __________________ Class __________________ LESSON 22-4 Choosing a Method for Solving Quadratic Equations Practice and Problem Solving: Modified Solve each quadratic equation by any means. Identify the method and explain why you chose it. Express irrational answers in radical form and use a calculator to approximate your answer rounded to two decimal places. The first one is done for you. 1. x 2 = 16 2. 9 x 2 = 81 x = 4 or x = −4; taking the square ________________________________________ ________________________________________ root because b = 0. ________________________________________ ________________________________________ 3. ( x − 4)2 = 36 4. x 2 + 7 x + 6 = 0 ________________________________________ ________________________________________ ________________________________________ ________________________________________ 5. x 2 + 4 x − 12 = 0 6. 2 x 2 + 5 x + 2 = 0 ________________________________________ ________________________________________ ________________________________________ ________________________________________ 7. 9 x 2 − 16 = 0 8. 3 x 2 − 6 x = 0 ________________________________________ ________________________________________ ________________________________________ ________________________________________ Use any method to solve each quadratic equation. Identify the method and explain why you chose it. Convert irrational answers and fractions to decimals and round to the hundredths place. The first one is done for you. 9. The formula for height, in feet, of a projectile under the influence of gravity is given by h = −16t 2 + vt + s, where t is the time in seconds, v is the upward velocity at the start (t = 0), and s is the starting height. Darin launches a bottle rocket from the ground at a velocity of 32 feet per second. How long is the rocket in the air? −16t + 32t + 0 = 0, − 16t (t − 2) = 0, t = 0 or t = 2; The rocket is in the air for _________________________________________________________________________________________ 2 2_________________________________________________________________________________________ seconds.; factoring because c = 0 and the terms have a common factor. 10. Use the projectile motion formula and solve the quadratic equation. Stephanie drops an egg from the top of a cliff that is 324 feet high. How many seconds until the egg hits the ground? _________________________________________________________________________________________ _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 394 Name ________________________________________ Date __________________ Class __________________ LESSON 22-5 Solving Nonlinear Systems Practice and Problem Solving: A/B Solve each system represented by the functions graphically. ⎧y = x 2 − 2 1. ⎨ ⎩y = 5 x − 8 ⎧y = x 2 − 4x + 6 2. ⎨ ⎩y = − x + 4 ________________________________________ ________________________________________ Solve each system algebraically. ⎧y = x 2 − 3 3. ⎨ ⎩y = − x + 3 ⎧y = x 2 − 2x − 3 4. ⎨ ⎩ y = −2 x − 5 ________________________________________ ________________________________________ ⎧y = 2x 2 + x − 3 5. ⎨ ⎩−3 x + y = 1 ⎧ y = x 2 − 25 6. ⎨ ⎩y = x + 5 ________________________________________ ________________________________________ ⎧y = x 2 − 1 7. ⎨ ⎩2 x − y = −2 ⎧y = x 2 + 4x + 3 8. ⎨ ⎩ x − y = −1 ________________________________________ ________________________________________ Use a graphing calculator to solve. 9. A ball is thrown upward with an initial velocity of 40 feet per second from ground level. The height of the ball, in feet, after t seconds is given by h = −16t 2 + 40t. At the same time, a balloon is rising at a constant rate of 10 feet per second. Its height, in feet, after t seconds is given by h = 10t. Find the time it takes for the ball and the balloon to reach the same height. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 397 Name ________________________________________ Date __________________ Class __________________ LESSON 22-5 Solving Nonlinear Systems Practice and Problem Solving: C Solve each system. If necessary, use the Quadratic Formula. 1. y = x 2 + 13 x − 46; y = 5 x − 13 2. y = x 2 + 4; y = 2 x − 9 ________________________________________ ________________________________________ 3. y = 2 x 2 + 7 x + 12; y = 2 x + 15 4. y = x 2 + 4 x + 2; y = 1 − x ________________________________________ ________________________________________ 5. y = 4 x 2 + 28 x − 11; y = 3 x + 10 6. y = 5 x 2 + 9 x + 7; y = 7 − 6 x ________________________________________ ________________________________________ 7. y = 3 x 2 − 4 x − 1; y = − 4 x + 59 8. y = x 2 − 12 x ; y = − x 3 ________________________________________ ________________________________________ 9. y = 2 x 2 + 5 x + 1; y = 3 x 2 − x + 10 10. y = 15( x 2 + 2) − 19 x; y = 15( x + 1) ________________________________________ ________________________________________ A ball is thrown directly upward from a height of h0 feet with an initial velocity of v0 feet per second. The ball’s height after t seconds is given by the formula h(t) = −16t 2 + v0t + h0. Use this information for Problems 11 and 12. 11. Suppose a ball is thrown directly upward from a height of 7 feet with an initial velocity of 50 feet per second. Use the Quadratic Formula or a graphing calculator to find the number of seconds it takes the ball to hit the ground. Round to the nearest tenth of a second. _________________________________________________________________________________________ 12. a. A helium balloon released from a height of h0 feet rises at a constant rate of k feet per second. Its height after t seconds is given by the formula h(t) = kt + h0. Suppose a helium balloon, released from a height of 25 feet at the same time as the ball in Problem 11, rises at 9 feet per second. After how many seconds will the ball and the balloon reach identical heights? _____________________________________________________________________________________ b. Examine your answer to Part a. Explain how it is physically possible for the ball and the balloon to have the same height twice. _____________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 398 Name ________________________________________ Date __________________ Class __________________ LESSON 22-5 Solving Nonlinear Systems Practice and Problem Solving: Modified Solve each system represented by the functions graphically. The first one is done for you. 1. y = x2; y = x + 2 2. y = −x2 + 3; y = x + 1 (−1, 1), (2, 4) ________________________________________ ________________________________________ 3. y = 2x2− 4; y = −2x 4. y = ________________________________________ 1 (x + 1)2 − 3; y = x + 2 2 ________________________________________ Solve each system algebraically. The first one is started for you. 5. y = x2; y = 2x + 8 2 6. y = x2 + 9x + 12; y = 6x + 30 2 x = 2x + 8; x − 2x − 8 = 0 ________________________________________ ________________________________________ Use a graphing calculator to solve. 7. The height of a toy rocket shot upward can be found using the formula h = −16t 2 + 90t. The height of a rising balloon follows the formula h = 10t. Here, t is time in seconds and h is measured in feet. If they are released together, find the time it takes for the toy rocket and the balloon to reach the same height. ________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 399 Name ________________________________________ Date __________________ Class __________________ Modeling with Quadratic Functions LESSON 23-1 Practice and Problem Solving: A/B Determine if the function in the table is quadratic by finding the second differences. Write “is” or “is not”. Justify your response. 1. x 1 2 3 4 5 6 f(x) −2 7 22 43 70 103 The function _______________ a quadratic function. _________________________________________________________________________________________ x 1 2 3 4 5 6 f(x) 6 22 42 72 110 156 2. The function _______________ a quadratic function. _________________________________________________________________________________________ Each table can be represented by a quadratic function, g(x) = ax2 + bx + c. Determine the values of a, b, and c to the nearest tenth. Write the equation for g(x), the quadratic that is the best fit. x 1 2 3 4 5 6 f(x) −3 9 29 57 93 137 3. a = ________ b = ________ c = ________ g(x) = ____________________________ x 1 2 3 4 5 6 f(x) 4 14 30 52 80 114 4. a = ________ b = ________ c = ________ g(x) = ____________________________ x 1 2 3 4 5 6 f(x) 7 12 24 37 55 77 5. a = ________ b = ________ c = ________ g(x) = ____________________________ Solve. 6. The table represents plant height measured in inches over a six-week period. Write an equation for g(x), the quadratic function that best fits the data. Round coefficients to the nearest tenth. x 1 2 3 4 5 6 f(x) 1.4 2.4 3.8 5.4 7.4 10.1 _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 403 Name ________________________________________ Date __________________ Class __________________ Modeling with Quadratic Functions LESSON 23-1 Practice and Problem Solving: C 1. The table below represents data that can be modeled by a quadratic equation, g(x) = ax2 + bx + c. x 1 2 3 4 5 6 f(x) 4.0 16.8 40.0 73.4 116.0 168.9 a. Verify this by examining second differences. _____________________________________________________________________________________ b. Find the values of a, b, and c, rounded to the nearest tenth, and write the equation. _____________________________________________________________________________________ c. Consider the table of values below. x 3 4 5 6 7 8 f(x) 4.0 16.8 40.0 73.4 116.0 168.9 Without using a graphing calculator, find an equation for g’, the quadratic model that is the best fit for this table. Explain. _____________________________________________________________________________________ _____________________________________________________________________________________ 2. The graph shown at the right can be modeled by a quadratic function. a. Verify by examining second differences. _________________________________________ _________________________________________ b. Use a quadratic model to estimate y given x = 2.5. Show the work in obtaining the model and finding the estimate. Write the coefficients to the nearest tenth. _____________________________________________________________________________________ _____________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 404 Name ________________________________________ Date __________________ Class __________________ Modeling with Quadratic Functions LESSON 23-1 Practice and Problem Solving: Modified Verify that the table of values can be modeled by a quadratic function. Then find an equation of the form g(x) = ax2 + bx + c as the model. 1. Fill in each blank box. x 1 2 3 4 5 6 f(x) −2 7 22 43 70 103 first differences second differences Complete this sentence. The function _______________ is a quadratic function. Use a graphing calculator to write an equation that models the data. _____________________________ For each table of values, write the first differences, the second differences, and a quadratic model for the data. Round coefficients to integers. The first one is started for you. 2. x 1 2 3 4 5 6 f(x) 3 15 35 63 99 143 first differences 12, 20, 28, 36, 44 second differences _____________ equation g(x) = 3. x 1 2 3 4 5 6 f(x) 1 6 15 28 45 66 first differences second differences _____________ equation g(x) = 4. x 1 2 3 4 5 6 f(x) −1 3 9 19 35 55 first differences second differences _____________ equation g(x) = Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 405 Name ________________________________________ Date __________________ Class __________________ LESSON 23-2 Comparing Linear, Quadratic, and Exponential Models Practice and Problem Solving: A/B Complete the following to determine if each function is linear, quadratic, or exponential. −2 −2 −2 −1 −1 −1 0 0 0 1 1 1 2 2 2 3 3 3 2. End behavior as x increases: 5. End behavior as x increases: ratio f(x) 2nd difference x 1st difference f(x) 7. f(x) = 3x ratio x 2nd difference ratio 2nd difference f(x) 1st difference x 1st difference 4. f(x) = (x + 1)2 − 3 1. f(x) = 2x + 1 8. End behavior as x decreases: f(x) ____________________ f(x) ____________________ f(x) ____________________ 3. f(x) is: _________________ 6. f(x) is: _________________ 9. f(x) is: _________________ Use the following information for 10–11. Todd had a piggy bank holding $384. He began taking out money each month. The table shows the amount remaining, in dollars, after each of the first four months. Month Amount 0 1 2 3 4 384 192 96 48 24 10. Does the data follow a linear, quadratic, or exponential model? How can you tell? _________________________________________________________________________________________ _________________________________________________________________________________________ 11. How much will be left in the piggy bank at the end of the fifth month? _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 408 Name ________________________________________ Date __________________ Class __________________ LESSON 23-2 Comparing Linear, Quadratic, and Exponential Models Practice and Problem Solving: C Complete the following to determine if each function is linear, quadratic, or exponential. −1 −1 −1 0 0 0 1 1 1 2 2 2 3 3 3 4 4 4 2. End behavior as x increases: 5. End behavior as x increases: ratio f(x) 2nd difference x 1st difference f(x) 7. f(x) = 10x + 1 ratio x 2nd difference ratio 2nd difference f(x) 1st difference x 1st difference 4. f(x) = 5x + 4 − x2 1. f(x) = −10x + 1 8. End behavior as x decreases: f(x) ____________________ f(x) ____________________ f(x) ____________________ 3. f(x) is: _________________ 6. f(x) is: _________________ 9. f(x) is: _________________ Solve. 10. The functions f(x) = x2 and g(x) = 2x both approach infinity as x approaches infinity. Write the function h(x) = f(x) − g(x). Then determine the end behavior of h(x) as x approaches infinity. _________________________________________________________________________________________ 11. An exponential function approaches 10 as x approaches infinity. Write a possible equation for the function. _________________________________________________________________________________________ 12. A function’s second differences are constant but not 0. Can you conclude whether the function is exponential, linear, or quadratic? _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 409 Name ________________________________________ Date __________________ Class __________________ Comparing Linear, Quadratic, and Exponential Models LESSON 23-2 Practice and Problem Solving: Modified Determine if each function is linear, quadratic, or exponential. The first one is done for you. quadratic 1. f(x) = 5x2 ______________ 2. f(x) = x + 3 ______________ 3. f(x) = 4x ______________ Complete the following to determine if each function is linear, quadratic, or exponential. −1 −1 −1 0 0 0 1 1 1 2 2 2 3 3 3 4 4 4 ratio f(x) 2nd difference x 1st difference f(x) ratio x 2nd difference ratio 2nd difference f(x) 1st difference x 10. f(x) = 2x 1st difference 7. f(x) = x2 − 2 4. f(x) = −3x + 1 5. End behavior as x increases: increases without bound f(x) ____________________ 8. End behavior as x increases: 11. End behavior as x increases: f(x) ____________________ f(x) ____________________ 6. f(x) is: _________________ 9. f(x) is: _________________ 12. f(x) is: _________________ Use the following information for 13–14. The first one is done for you. Flavia had $125 in an account and began adding money each month. The table shows the amount in Flavia’s account in dollars after each of the first four months. Month Amount 0 1 2 3 4 125 140 155 170 185 13. Does the data follow a linear, quadratic or linear model exponential model? __________________ 14. Let x stand for the number of months Flavia adds money, and let f(x) stand for the number of dollars she has. Write an equation for f(x). Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 410 Name ________________________________________ Date __________________ Class __________________ LESSON 24-1 Graphing Polynomial Functions Practice and Problem Solving: A/B Identify whether the polynomial f(x) is of odd or even degree and whether the leading coefficient is positive or negative. 1. 2. degree: degree: __________________ leading coefficient: __________________ __________________ leading coefficient: __________________ Identify each function as odd, even, or neither, and whether the leading coefficient is positive or negative. 3. 4. function type: function type: __________________ leading coefficient: __________________ __________________ leading coefficient: __________________ Identify: i the degree of the function i whether the function is even, odd, or neither i whether the leading coefficient is positive or negative 5. 6. degree: __________________ degree: __________________ function type: __________________ function type: __________________ leading coefficient: __________________ leading coefficient: __________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 414 Name ________________________________________ Date __________________ Class __________________ LESSON 24-1 Graphing Polynomial Functions Practice and Problem Solving: C Answer the following questions. 1. Only one of these graphs represents a polynomial function of degree 4, is an even function, and has a positive leading coefficient. Which is it? Explain why the other graphs do not represent the function. _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ 2. Show, by using the definition of an even function, that f(x) = 2x4 − 3x is not an even function. _________________________________________________________________________________________ _________________________________________________________________________________________ 3. Polynomial function f is defined by the following facts. i f is defined for all real numbers. i f is an odd function. i The graph of f has only one turning point to the left of the vertical axis in a coordinate system. What is the degree of f ? Justify your response with a sketch. What can be said of the leading coefficient of the polynomial? __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 415 Name ________________________________________ Date __________________ Class __________________ LESSON 24-1 Graphing Polynomial Functions Practice and Problem Solving: Modified In 1–10, use the graph of f, a polynomial function, shown at the right. The first one is done for you. 1. On the graph, place dots at the turning points. 2. How many turning points does the graph have? _________________ 3. On the graph, sketch arrows that show the trend in the graph as you trace along it from left to right. 4. By looking at the left end and the right end of the graph, you can tell whether the leading coefficient of the polynomial defining f is positive or negative. The leading coefficient is _________________. 5. By counting the number of turning points, you can tell the degree of the polynomial. The degree of the polynomial is _________________. 6. Is the graph its own reflection in the vertical axis? _________________ 7. Is the function an even function? _________________ 8. Is the graph its own reflection in the origin? _________________ 9. Is the function an odd function? _________________ 10. Is function f an even function, an odd function, or neither of these? ________________ Use the graph of f, a polynomial function. Identify the degree of the polynomial, whether the function is even, odd or neither, and whether the leading coefficient is positive or negative. The first one is done for you. 11. 12. degree: 3; function type: odd; leading coefficient: positive ________________________________________ ________________________________________ 13. 14. ________________________________________ ________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 416 Name ________________________________________ Date __________________ Class __________________ LESSON 24-2 Understanding Inverse Functions Practice and Problem Solving: A/B Graph the relation and connect the points. Then create an inverse table and graph the inverse. Identify the domain and range of each relation. 1. x −3 −2 −1 0 1 y −1 5 7 1 3 2. Domain ________, Range __________ x −2 −1 0 1 2 y 0 4 5 7 1 Domain ________, Range __________ x x y y Domain ________, Range __________ Domain ________, Range __________ Use inverse operations to find each inverse. Use a sample input for x to check. 2x 4. f ( x ) = −3 3. f ( x ) = 3 x + 2 5 ________________________________________ ________________________________________ ________________________________________ ________________________________________ Graph each function. Then write and graph each function’s inverse. x −x 5. f ( x ) = + 3 _____________________ − 2 _____________________ 6. f ( x ) = 3 2 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 419 Name ________________________________________ Date __________________ Class __________________ LESSON 24-2 Understanding Inverse Functions Practice and Problem Solving: C Use inverse operations to find each inverse. Check your solution with a sample input. 1. f ( x ) = 4x + 3 3 2. f ( x ) = 3x −6 5 ________________________________________ ________________________________________ ________________________________________ ________________________________________ Graph each function. Then write and graph each function’s inverse. 3. f ( x ) = 2x +2 5 4. f ( x ) = ________________________________________ −3 x −3 4 ________________________________________ Solve. 5. Sandy wants to know how many miles he drove on the interstate toll road. The charge to enter the toll road is $4, and the per-mile rate is $0.13. The total charge when he exited the toll road was $35.20. Write a function to model the situation, and use the inverse to find the number of miles he drove. Make sure to check your answer. _________________________________________________________________________________________ 6. In March 2014, the currency exchange rate between the U.S. dollar and the Euro was 1.3858 dollars per Euro, plus a 5 dollar fee to exchange currency. Write a function to model the situation, and use the inverse to determine the number of Euros that would be received in exchange for 250 dollars. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 420 Name ________________________________________ Date __________________ Class __________________ LESSON 24-2 Understanding Inverse Functions Practice and Problem Solving: Modified Graph the relation and connect the points. Then create an inverse table and graph the inverse. Identify the domain and range of each relation. The problem is started for you. 1. x –2 –1 0 1 2 y 1 3 4 5 2 { x | −2 ≤ x ≤ 2} Domain: _________________ { y | 1 ≤ y ≤ 5} Range: _________________ x y Domain: _____________________________________ Range: _____________________________________ Use inverse operations to find the inverse of y = 2x − 3. The first one is done for you. 2. Undo subtraction: x+3 ________________________________________ 3. Undo multiplication: ________________________________________ 4. Inverse function: ________________________________________ 5. Check: ________________________________________ ________________________________________ Graph each function. Then write and graph each function’s inverse. The first one is started for you. 6. f ( x ) = 2 x + 3, f −1( x ) = x −3 2 7. f ( x ) = 3 x − 4, _________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 421 Name ________________________________________ Date __________________ Class __________________ LESSON 24-3 Graphing Square Root Functions Practice and Problem Solving: A/B Identify the translation of the parent function. Tell whether each is a stretch or compression, and give the factor if applicable. Then find the domain of each function. 1. y = x−6 2. y = 10 x − 9 ________________________________________ ________________________________________ ________________________________________ ________________________________________ 3. y = 1− x 4. y = 1 x −2 2 ________________________________________ ________________________________________ ________________________________________ ________________________________________ Graph each square root function. 5. y = x − 2 6. y = 3 x + 4 + 2 The function d = 4.9t 2 gives the distance, d, in meters, that an object dropped from a height will fall in t seconds. Use this for Problems 9–10. 7. Express t as a function of d. _________________________________________________________________________________________ 8. Find the number of seconds it takes an object to fall 100 feet. Round to the nearest tenth of a second. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 424 Name ________________________________________ Date __________________ Class __________________ LESSON 24-3 Graphing Square Root Functions Practice and Problem Solving: C Find the domain of each function. 1. y = 3 − x + 3 2. y = ________________________________________ 3. y = 2 3−x 5 ________________________________________ 1 x −9 −3 4 4. y = 2 x + 3 x − 1 ________________________________________ ________________________________________ Graph each square root function. Then describe the graph as a transformation of the graph of the parent function y = x , and give its domain and range. 5. y = 10 − 4 x − 1 6. y = 1 + 2 x + 9 ________________________________________ ________________________________________ ________________________________________ ________________________________________ ________________________________________ ________________________________________ Solve. 7. The relation y 2 = x is not a function. Explain why. Then write the relation as two functions that can be graphed together on a graphing calculator to represent the original relation. _________________________________________________________________________________________ _________________________________________________________________________________________ 8. Examine the function f ( x ) = x + 4 − x on a graphing calculator. Explain why its range is all positive numbers less than or equal to 2. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 425 Name ________________________________________ Date __________________ Class __________________ LESSON 24-3 Graphing Square Root Functions Practice and Problem Solving: Modified Evaluate each expression for x = 2. The first one is done for you. 1. x +7 2. 3 ________________ 3. 8x ________________ x + 2 −1 _______________ 2x − 4 + 8 4. ________________ Find the domain and range of each function. The first one is done for you. 5. y = x + 2 6. y = x − 10 Domain: x ≥ −2 ________________________________________ Domain: _______________________________ Range: y ≥ 0 ________________________________________ Range: ________________________________ 7. y = 1 x 3 8. y = x − 8 + 3 Domain: __________________________ Domain: _______________________________ Range: ________________________________ Range: ________________________________ Complete the table. Then graph each square root function. The first one is started for you. 9. y = x − 2 x 2 10. y = x + 1 y= x −2 (x, y) x 2−2 =0 (2, 0) 0 6 4 11 9 x +1 (x, y) Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 426 Name ________________________________________ Date __________________ Class __________________ LESSON 24-4 Graphing Cube Root Functions Practice and Problem Solving: A/B Find the inverse of each cubic function. 1. f(x) = x3 2. f(x) = ________________________________________ 1 3 x 8 ________________________________________ 3. f(x) = −27x3 4. f(x) = 5x3 ________________________________________ ________________________________________ 5. f(x) = 125x3 − 7 6. f(x) = x3 + 8 ________________________________________ ________________________________________ Graph the cube root function. 7. y = 3 2 x 8. y = 3 − x 3 In a square cylinder, height, h, equals diameter, d. The function V = π 4 d 3 gives the volume, V, of a square cylinder. Use this for 9–10. 9. Express d as a function of V. _________________________________________________________________________________________ 10. Find the diameter of a square cylinder with a volume of 300 cubic inches. Round to the nearest tenth of an inch. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 429 Name ________________________________________ Date __________________ Class __________________ LESSON 24-4 Graphing Cube Root Functions Practice and Problem Solving: C Find the inverse of each cubic function. 1. f(x) = 8x3 − 1 2. f(x) = (x + 3)3 + 2 ________________________________________ 3. f ( x ) = − ________________________________________ 1 x 3 + 27 1000 4. f(x) = 6 − 5(x − 1)3 ________________________________________ ________________________________________ Write the equation of the cube root function whose graph is shown. 5. 6. ________________________________________ ________________________________________ Use the information below for 7–9. According to the Third Law of Johannes Kepler (1571–1630), the square of the orbital period of a planet is proportional to the cube of its distance from the Sun. This is expressed in the formula T 2 = a3, where T is measured in years and a is measured in astronomical units (1 astronomical unit is the mean distance of Earth from the Sun). 7. Express T as a function of a. Express a as a function of T. _________________________________________________________________________________________ 8. Mercury’s mean distance from the Sun is approximately 38.7% that of Earth’s. Estimate Mercury’s orbital period. Show your work. _________________________________________________________________________________________ 9. Jupiter’s orbital period is approximately 11.9 times that of Earth’s. Estimate Jupiter’s mean distance from the Sun. Show your work. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 430 Name ________________________________________ Date __________________ Class __________________ LESSON 24-4 Graphing Cube Root Functions Practice and Problem Solving: Modified Evaluate each expression for x = 5. The first one is done for you. 1. 3 x +3 2. 2 ________________ 3 x−4 3. ________________ 3 5x + 2 4. _______________ 3 25 x − 4 ________________ Use f(x) = 4x3 for Problems 5–8. The first one is done for you. y = 3 0.25 x 5. Write the parent cube root function. ___________________________ 6. Complete the function table for f(x). x 0 0.5 −0.5 1 −1 f(x) 7. Use the parent cube root function for the function table. x f(x) 8. Graph f(x) and the cube root function on the coordinate grid below. The function V = e3 gives the volume, V, of a cube with edges of length e. Use this for 9–11. The first one is done for you. 9. Express e as a function of V. e_________________________________________________________________________________________ = 3V 10. Use the function from above to find e for a cube whose volume is 216 cubic millimeters. Show your work. _________________________________________________________________________________________ 11. Use the same function to estimate e for a cube whose volume is 100 cubic millimeters. Show your work. _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 431 UNIT 1 Quantities and Modeling 6. greater than, + MODULE 1 Quantitative Reasoning 7. +, 18 8. 3, 6, greater, 3 LESSON 1-1 9. 6, greater, 3; 6 + 3 Practice and Problem Solving: A/B 10. 6, 3, 9 1. x = 3 11. h = 3 2. y = 5 12. b = 36 3. x = 13 13. d = −3 4. y = 2 14. y = 15 5. c = 7 15. 7% 6. a = 34 Reading Strategies 7. y = − 5 12 1. Roy delivers 45 water bottles to an office building. The building had 28 bottles before the delivery. How many water bottles does the building have now? 8. w = 16 9. 8.8 in. 2. subtract 10. 21 min 3. = 184 11. 75 min 4. How many stamps did he have? Practice and Problem Solving: C 5. 6s = 180; s = 30 1. t = 45 Success for English Learners 2. w = −41 1. to isolate the variable 3. n = 5 2. m 4. y = 6 3. 15 + 38 = 53 5. k = 3 LESSON 1-2 6. m = 2 Practice and Problem Solving: A/B 7. b = −3 8. x = 624 1. 4.8 ft 9. $1310 2. 50 m by 40 m 3. 20 m by 10 m 10. $100,000 11. 4. 60 m by 48 m 1 11 1 t + t = 1 ; 2 hr 5 12 7 5. 2880 m2 6. 40 min Practice and Problem Solving: Modified 7. 3168 ft Practice and Problem Solving: C 1. 5, no 2. 6, yes 1. 160 mi 3. 6, no 2. 100 mi 4. 8, yes 3. 2.2 hr 5. greater than 4. 345 mi Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 435 5. 12. 6 1 in. = 130 mi 2 13. inch, foot, yard, mile 14. milligram, centigram, gram, kilogram Practice and Problem Solving: Modified 1. 15. 12 lb 16. 2 m 4m 2m 17. 34.00 mi 18. 82 m 1m 2. x 19. 26 beads 4. 2 m 20. No. The measurement of 786 mm is more precise because it is measured to the nearest millimeter, which is a smaller unit than a centimeter. 5. 180 min Practice and Problem Solving: C 3. 2 m 1m = 4m x 6. 20 mi 1. 7.0 cm 2. 32 mm Reading Strategies 1. No, because a ratio compares two numbers by division. 3. 0.1 oz 3 4 3. No, because both ratios have to compare the same units. 5. 6 4. 1 2. Possible answer: 6. 1 7. 0.090 mL 8. 5000 ft 4. y = 1 9. 9.01 g 5. x = 4 10. Sample answer: In mathematics, you would write 17 = 17.0. But in measurement, 17.0 is a more precise measurement than 17. So, 17 and 17.0 are sometimes not the same. 6. t = 2.02 Success for English Learners 1. 45 2. All denominators are 1. 11. The student recorded the combined mass incorrectly. 3.4 is the least precise digit. So, the combined mass should be recorded as 4.0 g. LESSON 1-3 Practice and Problem Solving: A/B 12. The dimensions of a two-by-four are 1.5 inches by 3.5 inches. The cross-sectional area is therefore 5.3 square inches. This is 34% less than the cross-sectional area of a “true” two-by-four (8 square inches). 1. 6 in. 2. 2 mL 3. 4 pt 4. 7.05 mg Practice and Problem Solving: Modified 5. 2.25 cm 6. 12 oz 7. 3 1. 32 ft 8. 1 2. 4.3 lb 9. 4 3. 23 mm 10. 1 4. 2 11. 3 5. 3 6. 3 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 436 2. Convert all of the distances to feet. Aisha 5280 feet rode 25 miles, or 25 miles i = 1 mile 132,000 feet. Brian rode 100,000 feet. Wei rode 45,000 yards, or 3 feet = 135,000 feet. 45,000 yards i 1 yard 7. milliliter, liter, kiloliter 8. ounce, pound, ton 9. cup, quart, gallon 10. centigram, gram, kilogram 11. 580 mi 12. 25 oz 13. 6.2 sec Wei rode the farthest. 14. 0.5; 2.5; 0.5; 3.5 3. Convert the rates to feet per second. Max drove 55 miles per hour, or 1 hour 1 minute 55 miles i i i 1 hour 60 minutes 60 seconds 5280 feet = 81 feet per second. Nadya 1 mile drove 85 feet per second. Pavel drove 1600 yards per minute, or 1600 yards 1 minute 3 feet = 80 i i 1 minute 60 seconds 1 yard 15. 0.05; 2.35; 0.05; 2.45 16. 2.5 m × 2.35 m = 5.875 m2; 3.5 m × 2.45 m = 8.575 m2 17. 64.5 oz Reading Strategies 1. 9, 2, and 1 are nonzero digits = 3 2. the 0 after the 1 = 1 3. There is a 0 between 9 and 2 = 1. 4. 3 + 1 + 1 = 5 feet per second. Nadya drove the fastest. 5. 3 significant digits MODULE 2 Algebraic Models 6. 5 significant digits 7. 4 significant digits LESSON 2-1 Success for English Learners Practice and Problem Solving: A/B 1. The size of the unit. The smaller unit is more precise. 1. terms: 4a, 3c, 8; coefficients: 4, 3 2. terms: 9b, 6, 2g; coefficients: 9, 2 2. Scale 1 goes to tenths, Scale 2 goes to thousandths, and Scale 3 goes to hundredths. Thousandths are the smallest measurement, so Scale 2 is the most precise. 3. terms: 8.1f, 15, 2.7g; coefficients: 8.1, 2.7 4. terms: 7p, −3r, 6, −5s; coefficients: 7, −3, −5 5. terms: 3m, −2, −5n, p; coefficients: 3, −5, 1 MODULE 1 Challenge 6. terms: 4.6w, −3, 6.4x, −1.9y; coefficients: 4.6, 6.4, −1.9 1. Convert all of the times to seconds. Ann read the book in 14 hours, or 60 minutes 60 seconds 14 hours i i 1 hour 1 minute = 50,400 seconds. Ben read the book in 855 minutes, or 60 seconds 855 minutes i 1 minute = 51,300 seconds. Carly read the book in 50,000 seconds. Carly finished the book first. 7. the total cost of the oranges and apples 8. the difference between the cost of the grapes and the cost of the kiwi 9. 400 + 15c 10. 18 − 2h 11. 4a − 3 Practice and Problem Solving: C 1. terms: 5b, 6d, −5c, 19a; coefficients: 5, 6, −5, 19 2. terms: 4w, 12x, 18; coefficients: 4, 12 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 437 3. terms: 8r, −3s, 27, 15t; coefficients: 8, −3, 15 5. 2 + m 4. terms: 9g, 2h, −6j, 7, −8k; coefficients: 9, 2, −6, −8 7. 3.5 ÷ y 5. Sample answer: Bill is 6 years older than 3 times Sally’s age. 9. 48 ÷ b 6. 9k 8. 7 − i 10. 3.7h 6. Sample answer: If Shawn had scored two more points, Ron would have exactly five times as many points as Shawn. LESSON 2-2 Practice and Problem Solving: A/B 7. 3(a + 5) + 2; 41 years old 1. 4x = 16 8. n + n + 2 + n + 4 + n + 6 = 4n + 12 2. y −11 = 12 9 3. x + 6 = 51 10 ⎛1 ⎞ 4. 3 ⎜ m + 8 ⎟ = 11 ⎝3 ⎠ ⎛5⎞ 9. (F − 32) ⎜ ⎟ ; 35°C ⎝9⎠ Practice and Problem Solving: Modified 2. 7, 5p, 4r, 6s 5. a + 2a − 3 = 30; Tritt is 11 years old; Jan is 19 years old 3. 7.3w, 2.8v, 1.4 4. 12m, 16n, 5p, 16 8. 4, 3, −6 1 i 7 i f = 470; fee = $50, rate = $60 hr 5 7. 2x − 10 + 3x − 10 = 90; 34° and 56° 9. 4, 2, −7, 5 8. 35 + 0.1d = 20 + 0.15d; 200 mi 6. f + 1 6. 7f, −2g, −6h, 8 9. 10. 3, −4, −6, 9 13. p + 3 Volunteer Volunteer Hours per week 14. 5p + 6 Katie Reading Strategies 12. p − 7 1. 10 × y, 10y, 10(y), 10 i y Volunteer Hours over 3 weeks x+3 3(x + 3) Elizabeth x 3x Siobhan x−1 3(x − 3) Katie 7 hours, Elizabeth 4 hours, Siobhan 3 hours 2. the quotient of k and 6; k divided by 6; k separated into 6 equal groups 3. b − 4 Practice and Problem Solving: C 4. 4 − b 1. 8(m − 2) = 3(m +3) 5. 20m 2. −7w +8(w + 1) = w − 7 6. 58 + t n − 24 n = 8 6 7. 1.19p 3. 8. 100 − s 4. 0.10c + 0.25(c + 3) = 10.90; 29 dimes and 32 quarters Success for English Learners 5. (n − 2)180 = 1,980; The polygon has 13 angles. 1. ÷ 2. + 6. F = 1.8C + 32; −40° 3. two less than y; y minus two 7. Amanda 10 books, Bryan 5 books, Colin 11 books 4. nine multiplied by z; nine times z Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 438 4. Sample answer: Elena has 2 fewer candles than five times the number of candles Amie has. Together they have 40 candles. How many candles does each girl have? Practice and Problem Solving: Modified 1. x + 9 = 15 2. a − 11 = 3 3. 2n = 12 LESSON 2-3 k 4. = 15 5 5. d = 17 Practice and Problem Solving: A/B x 3 1. y = 6. m = −8 7. p = 5 2. m = p − 5n 8. z = 7 3. r = t + 6s 12 4. d = 21 − e c 5. j = h 15 9. w = −5 10. v = 6 11. 4s = 4; 11 cm 12. 100 = 63 + d; $37 13. p + 0.08p = 378; $350 6. f = gh + 7 14. h + 28 = 32; 4 hr 15. 8c = 25.52; $3.19 16. 9l = 117; 13 cm Reading Strategies 7. b = P − 2a 2 8. r = C 2π 9. C = 180 − (A + B) 1. Sample answer: A campground rents canoes for $8 an hour plus a $25 security deposit. Hank paid $57 to rent a canoe. For how long did he rent the canoe? 2. Sample answer: Jeff earns $15 allowance per week. After saving his allowance for a number of weeks, he spends $25 and has $35 left. For how many weeks did Jeff save his allowance? 3. Sample answer: To play a card game, a deck of 72 cards is divided equally among the players. Each player gets 8 cards. How many people can play the game? 4. Sample answer: The temperature rose 5.8° to 2.8 °F. What was the temperature before it rose? 10. h = V πr 2 11. E = 15h + 5; h = E −5 15 12. T = mn + r 13. m = T −r ; m = $2.50 per hr. n I would be more useful pr if you needed to determine the amount of time needed to earn a certain amount of interest. 14. The formula t = Practice and Problem Solving: C Success for English Learners 1. 2v + 9 2. Subtraction Property of Equality, Division Property of Equality 3. Sample answer: Jaime bought 4 sweaters and a pair of gloves for $79. The gloves cost $7. What was the cost of each sweater? 1. x = 8 y −4 3 2. a = 2 b−c 3. h = − (7 − 5 j ) 3 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 439 Reading Strategies 4. m = 2n + 3 1. Possible answer: 2x − 3y = 10 e2 − e 1 − 3e q 6. r = q+6 5. d = 7. m = 2. The equation contains only one variable, b. 3. Yes, because it has two or more variables. Fr v2 4. Divide both sides by b. 5. r = 3V 8. r = 3 4π 6. a. p = 2V 9. r = πh 10. U = s+6 2 b. $150 FV F −V Success for English Learners 1. Time 11. a = c 2 − b 2 2. Rate 12. s = S − πr πr 13. K = 1 mv 2 ; v = 2 2 or s = S −r πr 3. 2 steps 4. less 2K m LESSON 2-4 Practice and Problem Solving: A/B 2A 14. C = r 1. 9 i 2 + t ≤ $25 15. $175 2. x ≥ 3 Practice and Problem Solving: Modified 3. a < 5 4. x ≥ −1 1. x = y + 7 5. z ≤ 2 2. L = K − 9 c 3. d = 12 r 4. s = 0.75 6. x ≤ −2 7. b < 3 8. 18n ≤ 153; n ≤ 8.5; Sarah can buy from 0 to 8 CDs. 9. 5. v = 6w 6. j = I rt G−4 3 85 + 60 + s ≥ 70; s ≥ 65: Ted needs at 3 least a grade of 65 on his third test. 10. p − 0.15p + 12 < p; p > 80; the stereo is cheaper online if the regular price is greater than $80 8. d = vt 2A h V 10. w = lh p p 12. r = ; w = w r Practice and Problem Solving: C 9. b = 1. 10(c + 1.50) ≤ 75 2. x ≥ −3 3. a > 12 4. x ≤ 5 5. k > 9 13. P = 3s; s = 24 inches Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 440 6. True for all real y. 14 7. w ≤ 3 Success for English Learners 1. w ≤ 165; 165 ≥ w 2. It means to explain what the variable represents. 8. 40 + 0.1m < 30 + 0.25m, m > 66.6. So, you must drive 67 or more miles. 3. “No more than” is the same as “less than or equal to.” 9. Hal is partially correct. He did not take account of the possibility that x + 8 could be negative. The correct solution is x > 9 or x < −8. LESSON 2-5 Practice and Problem Solving: A/B 10. 3 ≥ 5 − 2 x 0 ≥ 2 − 2x 1. 2x ≥ 2 − 2x + 2x 2. 2x ≥ 2 x ≥1 3. Practice and Problem Solving: Modified 1. F < 7 4. 2. p + s ≤ 14 3. x ≥ 3 5. 4. a < 5 5. p ≥ 5 6. m < −9 6. 7. n ≤ 6 8. x ≥ 0 7. x ≥ 2 AND x < 9 9. 2n ≤ 9; n ≤ 4.5; Perdita can buy 4 avocadoes. 8. x ≤ 2 OR x > 9 10. The minimum grade is 70. 9. x > 1.2 AND x < 2.0 11. 20g ≥ 100; g ≥ 5; the car needs 5 gallons of gas to travel 100 miles. 10. x ≥ 4 AND x ≤ 10.0 Reading Strategies 1. x is greater than or equal to 5; x is at least 5; x is no less than 5. 11. x < 2 OR x > 5.5 2. p is greater than 8; Sample answer: Jack has more than 8 pens. 3. m is less than or equal to −2; Sample answer: The temperature will fall to −2° or below. 4. a. g ≥ 85 b. 85% c. Sample answers: 87%, 92% Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 441 5. open Practice and Problem Solving: C 6. open 1. 64 ≤ x ≤ 74 7. or 8. x < 20 AND x > 80 2. 4 ≤ x ≤ 6 OR 8 ≤ x ≤ 10 Success for English Learners 3. Sample answer: The job will take me at least 3 hours, but no more than 6. 1. 4. Sample answer: The temperature was either below −20 or at least 30. 2. x < 80 OR x ≥ 95 Practice and Problem Solving: Modified 3. 1. open overlapping closed MODULE 2 Challenge 1. Set Perimeter A equal to Perimeter B: 2[(3 x − 3) + ( x − 3)] = (2 x − 1) + (2 x − 1) + 2 x 2. open combined closed 2(4 x − 6) = 6 x − 2 8 x − 12 = 6 x − 2 2 x = 10 x =5 3. x ≥ 6 AND x < 7; closed overlapping open 4. x < 4 OR x > 6; open combined open Simplify Simplify Subtract 6 x Divide by 2 Evaluating for x = 5: 2[(3 x − 3) + ( x − 3)] = 2[(15 − 3) + (5 − 3)] = 2[12 + 2] = 28 5. The graphs are line segments—they have ends that are either open circles or closed circles. 6. The graphs look like a complete line with a break in it. The breaks have either open circles or closed circles at one end. Reading Strategies 1. The temperature on the camping trip seemed like it was either less than 20 degrees or more than 80 degrees. 2. More than 80 degrees 3. x < 20 (2 x − 1) + (2 x − 1) + 2 x = (10 − 1) + (10 − 1) + 10 = 9 + 9 + 10 = 28 2. Let x = the number of days that Celia and Ryan are on the diet. Celia consumes 1200 + 100x calories. Ryan consumes 3230 − 190x calories per day. Set the two expressions equal to find x: 1200 + 100 x = 3230 − 190 x Set the expressions equal 1200 + 290 x = 3230 290 x = 2030 4. x > 80 x =7 Add 190 x Subtract 1200 Divide by 290 They will consume the same number of calories after 7 days. Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 442 4. Let x = the number of months. The first bank charges 2500 + 150x the second bank charges 3000 + 125 x. Set the expressions equal to find when the loan payments are the same: 2500 + 150 x = 3000 + 125 x Set the expressions equal 3. Let x = the number of hours. The first moving company charges 800 + 16 x. The second company charges 720 + 21x. Set the two expressions equal to find x: 800 + 16 x = 720 + 21x Set the expressions equal 800 = 720 + 5 x 80 = 5 x x = 16 Subtract 16 x Subtract 720 2500 + 25 x = 3000 25 x = 500 Divide by 5 x = 20 The two companies will charge the same price after 16 hours. Subtract 125 x Subtract 2500 Divide by 25 After 20 months, Aaron will have paid the same amount for the loan. Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 443 UNIT 2 Understanding Functions MODULE 3 Functions and Models 3. Sample answer: LESSON 3-1 Practice and Problem Solving: A/B 1. The rain stopped, so the rainfall does not increase. 2. c, because the line is steepest 3. discrete 4. continuous 5. continuous 6. discrete 4. Sample answer: Counting the number of people who visited the library each day for a week, or counting the number of cars sold during one week by a dealer. 7. b; continuous; D: {0 ≤ x < 8}; R: {0 ≤ y ≤ 2} Practice and Problem Solving: C 1. Sample answer: 5. Sample answer: 2. Sample answer: 6. Sample answer: It snowed for 5 hours and reached 8 inches in depth. Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 444 2. Practice and Problem Solving: Modified x y 5 3 4 3 3. b 3 3 4. a 2 3 5. f 1 3 1. d 2. g 6. input 7. output 8. D: {0, 1, 2, 3, 4, 5}; R: {0, 1, 2, 3, 4, 5} 9. D: {1, 2, 3, 4, 5, 6, 7, 8}; R: {3, 6, 9} 10. D: {0 ≤ t ≤ 5}; R: {0 ≤ d ≤ 10} Reading Strategies 3. {0 ≤ x ≤ 4}; {0 ≤ y ≤ 4}; yes; each domain value is paired with exactly one range value. 4. {8, 9}; {−3, −4, −6, −9}; no; both domain values are paired with more than one range value. 1. Graph B 2. Graph D 3. Graph A 4. Sample answer: Paolo blew up a balloon. Then the balloon popped. 5. {0, 1, 2}; {4, 5, 6, 7, 8}; no; two domain values are paired with two range values. Success for English Learners Problem 2 Practice and Problem Solving: C discrete 1. It is a function because each input has exactly one output. Problem 3 A. D: {0 ≤ x ≤ 8}; R: {0 ≤ y ≤ 6} B. D: {0≤ x ≤ 8}; R: {0 ≤ y ≤ 80} LESSON 3-2 Practice and Problem Solving: A/B 1. x y −2 5 −1 1 3 1 −1 −2 2. It is not a function because 3 is paired with two different outputs. 3. Sample answer: Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 445 3. Because the domain value 1 is paired with more than one range value. 4. no 5. no 6. yes Success for English Learners 1. Change one of the x-coordinates of either (7, 0) or (7, −1) to another number that isn’t already being used in the table. A possible function is the following: Sample explanation: I know that a domain value cannot have two different range values, so I changed one domain value to match another domain number. This means one domain member has two range values. 4. The first graph is not a function because a vertical line passes through the curve more than one time. The second graph is a function because a vertical line only passes through the curve once. 5. INT(4.6) = 4, INT(−2.3), INT(SQRT{2}) = 1 Domain and range in general for INT(x): D = All real numbers, R = All integers Domain and range for INT(x) for given values of x: D = {4.6, −2.3, SQRT{2}}, R = {−3, 1, 4} x y 3 0 7 −1 9 −7 12 −1 15 0 2. No; because −5 is less than −4, it does not fall within the numbers of the domain, which is all real numbers between, and including, −4 and 4. Practice and Problem Solving: Modified LESSON 3-3 1. Domain: {0, 3, 5}; Range: {−1, 1} 2. Domain: {−1, 0, 2, 3}; Range: {−4, 2, 4, 5} 3. It is not a function because 9 is paired with two outputs. 4. It is a function. 5. It is a function. 6. It is not a function because 5 is paired with three outputs. 7. {30, 40, 50, 60} 8. {1, 2, 3, 4, 5, 6, 7} 9. It is a function because each year is paired with exactly one number of members. Practice and Problem Solving: A/B 1. cost; size 2. earnings; number of hours worked 3. total cost; number of pounds bought 4. y = 8x; f(x) = 8x 5. y = 1.59x; f(x) = 1.59x 6. independent: number of hours; dependent: total cost; function: f(x) = 70x + 40; solution: $320 7. independent: number of bags; dependent: total cost; function: f(g) = 4g + 10; solution: $66 Reading Strategies 1. Possible answer: Practice and Problem Solving: C x 1 2 3 4 y 1 2 3 4 1. D: {2, 5, 7, 8}; Sample answer: I substituted each value of the range in the function for f(x) and worked backward to solve the equation to find the value of x. 2. Possible answer: 2. D: {1, 3, 5, 6}; Sample answer: I substituted each member of the range into the function to find each member of the domain. Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 446 3. D: {2, 4, 7, 8} Sample answer: I substituted each member of the range into the function to find each member of the domain. 4. The number of gallons of gas is 1/32 of the distance traveled. Independent variable: number of gallons; dependent variable: total distance traveled 4. D: {3, 5, 6, 7}; Sample answer: I substituted each member of the range into the function to find each member of the domain. 5. The number of loads of laundry is 1/3 times the number of ounces of detergent. Independent variable: number of loads of laundry; dependent variable: total detergent used 5. Set d(m) = 0 and solve for m; 0 = −5m + 320; −320 = −5m; 64 = m. It takes 64 minutes to cut all the dough. D: {whole numbers from 0 to 64}; R: {multiples of 5 from 0 to 320} Success for English Learners 1. Independent variable: number of pies baked; dependent variable: total number of apples Practice and Problem Solving: Modified 2. R: {3, 13, 23} LESSON 3-4 1. amount earned; number of hours worked; number of hours worked; amount earned Practice and Problem Solving: A/B 2. total cost; number of pounds bought; number of pounds bought; total cost 1. 3. f(x) = x + 4 4. f(x) = −5x + 16 5. f(x) = 2x − 4 6. total cost; number of hours worked; independent variable: number of hours worked; dependent variable: total cost; equation: y = 60x; function: f(x) = 60x x y −3 −11 1 1 5 13 7. total cost; number of extra propellers bought; independent variable: number of extra propellers bought; dependent variable: total cost; equation: y = 25x + 300; function: f(x) = 25x + 300 Reading Strategies 1. The total earned is 45 times the hours worked. Independent variable: hours worked; dependent variable: total earned 2. The total charge is $1.25 times the number of pounds shipped. Independent variable: number of pounds shipped; dependent variable: total charge 2. 3. The total number of cards is 52 times the number of decks. Independent variable: number of decks; dependent variable: number of cards x y 2 8 3 6 4 4 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 447 2. D = {0 ≤ x ≤ 8} 3. x y 0 −3 8 5 3 0 3. D = {0 ≤ x ≤ 4 } 3 4. f(d) = 25h; After 3.5 hours, the car had traveled 87.5 miles. 5. R = {16, 64, 144, 256, 400, 576}; It takes the object about 4.3 seconds to fall 300 feet since 300 is between 256 and 400 and 16 times (4.3)2 = 295.84. 4. 7.3 ft Practice and Problem Solving: Modified Practice and Problem Solving: C 1. 1. D = {2, 4, 6, 8} Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 448 2. The graph would be a line from (0, 2) to (4, 6) instead of the points. 3. Reading Strategies 1. x y 1 4 2 5 11 3 6 15 4 7 x y 3 5 5 9 6 8 Ordered pairs: (1, 4), (2, 5), (3, 6), and (4, 7) 2. 4. x y x y 2 3 2 1 4 7 3 5 6 11 4 9 7 13 5 13 Ordered pairs: (2, 3), (4, 7), (6, 11), and (7, 13) Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 449 Success for English Learners 1. x y 2 1 3 3 4 5 5 7 2. Ordered pairs: (1, 4), (2, 5), (3, 6), (4, 7) The ordered pairs are (2, 1), (3, 3), (4, 5), (5, 7). MODULE 3 Challenge 1. The completed table is here: g(x) f(x) = x2 − 4 f(g(x)) −2 g(−2) = −2(−2) = 4 4 f (4) = 4 2 − 4 = 12 12 −1 g(−1) = −2(−1) = 2 2 f (2) = 22 − 4 = 0 0 x g(x) = −2x 0 g(0) = −2(0) = 0 0 f (0) = 0 2 − 4 = −4 −4 1 g(1) = −2(1) = −2 −2 f (4) = (−2)2 − 4 = 0 0 2 g(2) = −2(2) = −4 −4 f (4) = (−4)2 − 4 = 12 12 2. To find the one-step rule for y = g(f (x)), b. To find the one-step rule for y = g(h(f (x))), you can first replace you can first replace f (x) with x − 4 . 2 f (x) with x 2 − 4 . Next, evaluate Next, evaluate g(x 2 − 4) using the rule h(x 2 − 4) using the rule for h. for g. g(x 2 − 4) = −2(x 2 − 4), which equals x2 − 4 − 1 x2 − 5 . Next = 4 4 ⎛ x2 − 5 ⎞ evaluate g ⎜ ⎟ using the rule ⎝ 4 ⎠ −2x 2 + 8 The one-step rule for y = g(f (x)) is g (f ( x )) = −2 x 2 + 8. h(x 2 − 4) = 3. a. To calculate f (g(h(−3))) first find h(−3) . Using the rule, −3 − 1 −4 h( −3) = = = −1. Now we 4 4 need to calculate f (g(−1)) . g ( −1) = −2( −1) = 2, so f (g ( −1)) = f (2), ⎛ x2 − 5 ⎞ ⎛ x2 − 5 ⎞ 5 − x2 for g. g ⎜ = −2 ⎜ 4 ⎟= 2 . ⎟ ⎝ 4 ⎠ ⎝ ⎠ The one-step rule for y = g(h(f (x))) is g(h(f (x))) = or f (2) = 22 − 4 = 0. 5 − x2 . 2 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 450 8. MODULE 4 Patterns and Sequences LESSON 4-1 Practice and Problem Solving: A/B 1. 3; 5; 24; 48; 72; Domain: 1, 2, 3, 4, 5, 6; Range: 12, 24, 36, 48, 60, 72 2. 2, 5, 8, 11 9. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 3. 8, 13, 20, 29 10. 1, 0.5, 0.667, 0.6, 0.625, 0.615, 0.619, 0.618; Possible explanation: The terms alternate going up and then down, but they seem to be approaching some number near 0.617 or 0.618. 4. 0, 0, 2, 6 5. 0, 1, 2, 3 6. 2; 2; 10; 40; 40; 3; 3;10; 55; 55;4; 4; 10; 70; 70; 5;5; 10; 85; 85; ordered pairs (1, 25), (2, 40), (3, 55), (4, 70), (5, 85) Practice and Problem Solving: Modified 7. 1. 1, 2, 3, 4, 5; 3, 6, 9, 12, 15; Domain: 1, 2, 3, 4, 5; Range: 3, 6, 9, 12, 15 2. 1, 2, 3, 4, 5; 10, 20, 30, 40, 50; Domain: 1, 2, 3, 4, 5; Range: 10, 20, 30, 40, 50 3. 2, 2, 8, 8; 3, 3, 10, 10; 4, 4, 12, 12; 5, 5, 14, 14; ordered pairs (1, 6), (2, 8), (3, 10), (4, 12), (5, 14) Practice and Problem Solving: C 1. 1, 5, 19, 49 2. 4. 1 1 1 1 , , , 2 6 12 20 3. 1, 5, 14, 30 4. 0, 5. − 3 4 15 , , 5 5 17 7 1 5 1 ,− ,− ,− 12 2 12 3 6. 9, 16, 17, 13 + 17 Reading Strategies 7. 1. 7, 10, 13, 16, 19 2. 5, −2, −9, −16, −23 3. 2, 4, 8, 16, 32 4. −1, −2, −5, −14, −41 Success for English Learners 1. (6, 24) 2. Locate the first number on the x-axis. Locate the second number on the y-axis. Find where these numbers intersect and place a point. Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 451 LESSON 4-2 5. f ( n ) = −8 − 6.5( n − 1) Practice and Problem Solving: A/B 6. f ( n ) = −2 + 3( n − 1) 7. $60; Month 15 8. −1, 1, −1, 1, −1, 1; it is not an arithmetic sequence because there is not a common difference. 1. f (n) = 8 + 4(n −1); f (1) = 8, f (n) = f (n −1) + 4 for n ≥ 2 2. f (n ) = 11 − 4(n − 1); f (1) = 11, f (n ) = f (n − 1) − 4 for n ≥ 2 9. f (1) = 24.6, f ( n ) = f ( n − 1) + 5.6 for n ≥ 2 3. f ( n ) = −20 + 7( n − 1); f (1) = −20, f ( n ) = f ( n − 1) + 7 for n ≥ 2 10. $60,000 Practice and Problem Solving: Modified 4. f ( n ) = 2.7 + 1.6( n − 1); f (1) = 2.7, f ( n ) = f ( n − 1) + 1.6 for n ≥ 2 1. 5 2. 14 3. −6 5. f(n) = 45 + 5(n − 1); f(1) = 45, f(n) = f(n − 1) + 5 for n ≥ 2 4. 19, 21, 23 6. f(n) = 94 −7(n − 1); f(1) = 94, f(n) = f(n − 1) − 7 for n ≥ 2 5. −4, −7, −10 6. 40, 51, 62 7. f(n) = 12 + 14(n − 1); f(1) = 12, f(n) = f(n − 1) + 14 for n ≥ 2 7. 8. f(n) = 83 − 40(n − 1); f(1) = 83, f(n) = f(n − 1) − 40 for n ≥ 2 f ( n ) = 1 + 2( n − 1); f (1) = 1, f (n ) = f ( n − 1) + 2 for n ≥ 2 8. f (n ) = 15 − 2(n − 1); f (1) = 15, f (n ) = f (n − 1) − 2 for n ≥ 2 9. 13, 19, 25, 31 10. f(n) = 100 + 50(n − 1) 9. f (n ) = 16 + 5(n − 1); f (1) = 16, f (n ) = f (n − 1) + 5 for n ≥ 2 Practice and Problem Solving: C 1. f ( n ) = −3.4 + 1.3( n − 1); f (1) = −3.4, f ( n ) = f ( n − 1) + 1.3 for n ≥ 2 10. f ( n ) = 10 − 0.5(n − 1); f (1) = 10, f (n ) = f (n − 1) − 0.5 for n ≥ 2 11. 80 12. f(n) = 30 + 30(n − 1) 1 1 + (n − 1); 6 12 1 f (1) = , 6 1 f ( n ) = f ( n − 1) + for n ≥ 2 12 2. f ( n ) = Reading Strategies 1. The terms do not all differ by the same number. 2. The same number is being multiplied, not added, to each term. 3. f ( n ) = 82 − 0.5( n − 1); f (1) = 82, f ( n ) = f ( n − 1) − 0.5 for n ≥ 2 3. Possible answer: 1, 6, 11, 16, 21, … 4. f ( n ) = −22 + 8( n − 1); f (1) = −22, f ( n ) = f ( n − 1) + 8 for n ≥ 2 5. 130 1 1 1 4. − ; 5 , 5, 4 2 2 2 6. 88 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 452 Success for English Learners 6. 24 1. A sequence is an arithmetic sequence if it has a common difference. 7. 18 8. 36 2. The 22nd term of Problem 1 is −72 Reading Strategies f(22) = 12 − (22 − 1)(4) 1. 17 f(22) = 12 − (21)(4) 2. 23 laps f(22) = 12 − 84 Success for English Learners f(22) = −72 1. 45 LESSON 4-3 2. f(n) = 45 + 45 (n − 1) Practice and Problem Solving: A/B 3. 360 hours 1. 55, 110, 165, 220; 55 MODULE 4 Challenge 2. $1.20, $2.40, $3.60, $4.80; $1.20 1. The recursive formula is f (1) = 1, f (2) = 1 and f (n ) = f (n − 1) + f (n − 2). The first ten terms of the Fibonacci sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. 3. 90; 110; 130; 150; 170; 190; 210; 230; 250 4. 20 5. f(n) = 30 + 20 (n − 1) 2. The number that each sum is equal to is a term of the Fibonacci sequence. That number is skipped when writing each successive equation. The last equation equals 21, so 21 will be skipped in the next equation, which is 1 + 2 + 5 + 13 + 34 = 55. 6. The amount of money Riley spends through December for her pool cost. 7. $250 Practice and Problem Solving: C 1. 6 sides; no 2. Figure 1 2 3 4 5 6 Number of Sides 3 3 4 4 5 5 3. The ratios of successive terms of the f (2) 1 Fibonacci sequence are: = = 1, f (1) 1 f (3) 2 f (4) 3 = = 2, = = 1.5 , f (2) 1 f (3) 2 f (5) 5 f (6) 8 = = 1.666... , = = 1.6 , f (4) 3 f (5) 5 f (7) 13 f (8) 21 = = 1.625 , = = 1.61538 , f (6) 8 f (7) 13 f (9) 34 f (10) 55 = = 1.61904 , = = 1.61764 , f (8) 21 f (9) 34 which rounds to 1.618. 3. 13 sides, yes 4. The sequence is not arithmetic. There is no common difference. 5. $213.40 6. $339.95 7. 9 Practice and Problem Solving: Modified 1. 6; 48 2. 7; 56 3. 12; 84 4. 3; 85 5. 30 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 453 UNIT 3 Linear Functions, Equations, and Inequalities MODULE 5 Linear Functions 4. 13 LESSON 5-1 1 2 5. Practice and Problem Solving: A/B 1. not linear 2. linear 3. linear 4. 4/5, 1/5, −2/5; yes; −3/5 5. 0, −12; no 6. −5, 1, 7, 13; yes, 6 7. 6. 8. 7. y = 2x + 6 has a line as its graph. y −8 = 2 has almost the same graph as x −1 y = 2x + 6. The point (1, 8) is not a part of its graph because the denominator of a fraction cannot equal 0. The graph of y −8 = 2 is a line with a hole in it at x −1 (1, 8). 9. A charges $300 and B charges $400. 10. B charges according to a linear function. A does not. 8. (0, 0) 11. 8 hours 9. 12. continuous; any fractional part of an hour is represented on the graph. Practice and Problem Solving: Modified Practice and Problem Solving: C 1. − 3 4 1. linear 2. not linear 2. −5 3. C − 3B A 3. not linear 1 7 4. 8 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 454 5. 3 3. No; a constant change of +2 in x does not correspond to a constant change in y. 6. 5 7. x 0 1 2 3 x 6 4 2 0 −2 y 0 2 4 6 y −3 −1 0 2 3 4. x + y = 4 Success for English Learners 1. The x and y are not allowed to be multiplied together in the first equation. The x is not allowed to have an exponent of 3 in the second equation. The y is not allowed to be in the denominator in the third equation. 8. x 0 1 2 3 y 5 4 3 2 2. The function y = x2 is not a linear function because the graph is not a line and the exponent on x is not a 1. LESSON 5-2 Practice and Problem Solving: A/B 1. x-int: 4; y-int: 2 2. x-int: −1; y-int: 4 3. x-int: −3; y-int: 3 4. 9. rectangle 5. Reading Strategies 1. x + 4y = 9 2. Yes; each domain value is paired with exactly one range value. Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 455 6. 5. Possible answer: y = x − 1, y = 2x − 1, y = 3x − 1 6. The x-intercept represents the number of days after which the tank would become empty, if not refilled. Since the x-intercept is 34.375, the tank would run out on the 35th day. The tank will be half-full after 34.375 ÷ 2 = 17.1875 days. So, the tank is half-full on the 18th day. 7. x + 2y = 10 8. It must be the vertical line, x = 0. Practice and Problem Solving: Modified a. x-int: 10; y-int: 20 1. y-intercept; 0 b. x-int: the number of hamburgers they can buy if they buy no hot dogs. y-int: the number of hot dogs they can buy if they buy no hamburgers. 2. x-intercept; 0 3. x-intercept: 2; y-intercept: 4 4. x-intercept: 4; y-intercept: −3 5. x-intercept: 1; y-intercept: −2 Practice and Problem Solving: C 1. x-intercept: −4; y-intercept: 2 6. x-intercept: 3; y-intercept: 3 2. x-intercept: 3; y-intercept: 1.8 7. x-intercept: 10; y-intercept: 6 3. x-intercept: −2.5; y-intercept: 1 8. x-intercept: 7; y-intercept: −14 9. The y-intercept, 50, represents the number of vitamins that were in the jar when Jaime bought it. The x-intercept, 25, represents the number of days the vitamins will last until the jar is empty. 4. x-intercept: 6; y-intercept: 2 Reading Strategies 1. x-int: 2; y-int: −3 2. x-int: 3; y-int: −3 3. x-int: −3; y-int: 5 Success for English Learners 1. Because x = 0 at the y-intercept. 2. Because y = 0 at the x-intercept. Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 456 13. A 7% grade means that a road rises 7%, 7 7 . So, the slope of the road is . or 100 100 For a driver, this means that the road rises (or falls) 7 feet for every 100 feet in horizontal distance. LESSON 5-3 Practice and Problem Solving: A/B 4 5 2. rise = −6, run = 3, slope = −2 1. rise = 4, run = 5, slope = 14. If the two intercepts had represented two different points, Ariel could graph the points and find the slope. Since that was impossible, the intercepts must have represented the same point. This can only happen if both intercepts are 0. The line passes through (0, 0). 3 3. rise = 3, run = 4, slope = 4 3 4. slope = ; hourly salary increases $3 2 every 2 years, or $1.50 per year. 400 ; the number of people 5. slope = − 3 remaining decreases by 400 every 3 hours, or about 133 per hour. Practice and Problem Solving: Modified 1. rise = 1, run = 3, slope = 6. The slope would be 58 since $58 is added to the total cost as the number of tickets bought increases by 1. 2. rise = 2, run = 1, slope = 2 3. rise = −3, run = 2, slope = − Practice and Problem Solving: C 1. 4 5. negative 6. undefined 7. 35 mph; 12 mph; 11 mph; 39 mph; 0.5 mph 4. 2 Reading Strategies 5. −2 1. 4 3 5 2. Possible answer: (−1, 6) 7. −9 3. 2 8. 0 5. 9. 1 4 10. 1 11 11. 2 3 3 2 4. zero 17 2. − 7 11 3. 3 6. 1 3 2 5 4. (−4, 1) and (6, 5) 6. 0 7. horizontal Success for English Learners 1. 4 2. −5 3. Sample Answer: Graph the two given points and connect them with a line. Find the y-coordinate when the x-coordinate is 80. There will be 1,500,000 ft3 of water in the reservoir. 12. According to the formula, 6−2 4 = . But division by 0 is not slope = 3−3 0 possible. So, slope is undefined for a vertical line. Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 457 6. MODULE 5 Challenge 1 a. $30,000 b. $165,000 c. $1,515,000 2. The fixed costs are spread over a greater number of units. 3 a. $54,500 b. $545,000 c. $5,450,000 4. 28 units 7. slope is 3, y-intercept is −5 5 a. no b. yes 8. y = 0.25x − 11 c. yes 9. f ( x ) = 30,000 − 500 x 6. for 100 units, $24,500; for 1000 units, $380,000 Practice and Problem Solving: C MODULE 6 Forms of Linear Equations LESSON 6-1 1. y = 2 2 x + 1; slope: ; y-intercept: 1 9 9 2. y = 7 7 x − 7; slope: ; y-intercept: −7 5 5 3. Sample equation: y = −2x − 2 Practice and Problem Solving: A/B 1. y = −4x + 7; slope: −4; y-intercept: 7 2. y = 2 2 x − 3; slope: ; y-intercept: −3 3 3 3. y = 5 3 5 3 x − ; slope: ; y-intercept: − 4 2 4 2 4. y = − 1 1 x + 4; slope: − ; y-intercept: 4 2 2 5. 4. Sample equation: y = 1 2 x+ 3 3 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 458 5. Suppose x is any real number. Since m is also a real number, mx can be found. And, since b is a real number, mx + b can then be found. So, all real numbers are in the domain of f. 14. slope: − 2 ; y-intercept: 3 5 f (x) − b . Then m suppose that f ( x ) is any real number. You can subtract b from it to get f ( x ) − b, and you can then divide f ( x ) − b by m to f (x) − b , as long as m ≠ 0. get m 6. Rewrite the function as x = So, the range of the function is all real numbers as long as m ≠ 0. If m = 0, then the function becomes f ( x ) = b and its range is the single real number b. 15. The graph is a line. Its slope would increase from 2 to 2.5, making the line steeper. Reading Strategies Practice and Problem Solving: Modified 1. y = −2x + 4 1. With a fraction, you have a “rise” and “run” for graphing. 2. y = −8x + 17 2. (0, −8) 3. 5; 12 3. y = 3x − 11 4. −3; 0 1 6. ; 3 3 5. 1; −4 4. y = − 5 x +1 4 5. y = 1 x −3 2 6. y = 1 1 x− 4 10 7. 7. −10 8. −1 9. 4 10. −8 11. 2 8. 12. −14 13. slope: 3; y-intercept: −5 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 459 6. y − 1 = 6 (x − 3); or y + 5 = 6 (x − 2) Success for English Learners 7. The slope of the line containing the first two points is 2. The slope of the line containing the last two points is 1. Since the slopes are different, the three points cannot lie on one line. 1. It is the starting point. 2. You use the slope to determine the rise and run. 3. m is the slope. 4. The slope = 2 and the y-intercept = 4. 8. Let a be the x- and y-intercept. Then the line contains the points (a, 0) and (0, a) a−0 a = = −1. and its slope is 0 − a −a LESSON 6-2 Practice and Problem Solving: A/B 1. y − 5 = 2 (x − 3) 9. $80. The equation that represents the total fee as a function of sale price, p, is 0.2p + 20. 2. y − 7 = −3 (x + 1) 3. y − 3 = (x − 4); or y − 3 = (x + 10) 4. y − 2 = 5. y = Practice and Problem Solving: Modified 2 2 (x − 5); or y = (x) 5 5 1. −3 9 9 (x); or y − 9 = (x − 2); or y − 18 = 2 2 2. 4 9 (x − 4) 2 3. 6. y − 18 = − 4. y − 5 = 3( x − 0) 9 9 (x + 2); or y − 9 = − (x + 1); 3 3 5. y − 6 = 2( x − 1) 9 or y = − (x − 4) 3 6. Possible answers: y − 6 = 4( x ); y − 10 = 4( x − 1); y − 14 = 4( x − 2) 1 (x); or y − 3 = 2 −1 y −3= (x − 4) 2 7. y − 5 = − 8. y + 3 = 7. Possible answers: y + 2 = −( x + 4); y + 7 = −( x − 1); y + 10 = −( x − 4) 8. Possible answers: y − 1 = 2( x ); y − 5 = 2( x − 2) 1 1 (x); or y + 2 = (x − 6) 6 6 9. f ( x ) = − 9. y − 400 = 50 (x − 4); $700 1 x −3 3 10. 5 Practice and Problem Solving: C 11. Day 4 8 3 1. y + = − (x + 2) 5 4 2. y − 39 = 1 2 12. y − 15 = 5( x − 1) Reading Strategies 2 (x + 35) 5 1. 2x + y = 10 1⎞ ⎛ 3. y + 5 = − 15 ⎜ x − ⎟ , or y + 10 = −15 6⎠ ⎝ 1⎞ ⎛ ⎜x − 2⎟ ⎝ ⎠ 2. The slope is 2. 3. x + y = 3 2 2 = −8 (x − 1) 4. y − 2 = − 8 (x − ); or y + 3 3 x −3 −1 0 2 3 y 6 4 2 0 −2 4. For an equation to be written in standard form, A and B ≠ 0. In x = 3, B = 0. 8 8 5. y + 4 = − (x − 4); or y − 4 = − (x + 5) 9 9 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 460 Success for English Learners Practice and Problem Solving: C 1. You cannot begin with slope-intercept form because the y-intercept is not given. 1. x y + = 1; (8, 0) and (0, 6) 8 6 5x + y = 10 2. Subtracting a number is the same as adding the opposite of the number. So, subtracting a negative number is the same as adding a positive number. 4. 6x − y = 11 x y + = 1; (2, 0) and (0, 10) 2 10 1 7 x y 3. + = 1; x+− y= 15 15 15 15 − 7 15 ⎞ ⎛ (15, 0) and ⎜ 0, − 7 ⎟⎠ ⎝ 9 1 x y ⎛ 4 ⎞ 4. − x + y = + = 1; ⎜ − , 0 ⎟ ; 4 6 4 6 ⎝ 9 ⎠ − 9 and (0,6) 5. x + y = 7 5. 2. 3. Use the slope formula. LESSON 6-3 Practice and Problem Solving: A/B 1. not standard; 3x − y = 0 2. not standard; 5x + y = −4 3. not standard; 2x + 2y = 8 6. 9x − y = −47 7. 6. 8. 7. y = 2x + 6 has a line as its graph. y −8 = 2 has almost the same graph as x −1 y = 2x + 6. The point (1, 8) is not a part of its graph because the denominator of a fraction cannot equal 0. The graph of y −8 = 2 is a line with a hole in it at x −1 (1, 8). 9. 200x − y = −50 10. x − 4y = −4 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 461 8. (0, 0) C − 3B 9. A Reading Strategies 1. x + 4y = 9 2. 2x − y = 4 Practice and Problem Solving: Modified 3. No; a constant change of +2 in x does not correspond to a constant change in y. 1. standard 1 2. x − y = 8 2 x 6 4 2 0 −2 3. 2x − y = −2 y −3 −1 0 2 3 4. point-slope 4. x + y = 4 5. slope-intercept Success for English Learners 6. standard 7. x 0 1 2 3 1. No. Using the standard form x − y = −3, −4 − 1 ≠ −3 y 0 2 4 6 2. Sample answers: (0, 3) and (−3, 0) LESSON 6-4 Practice and Problem Solving: A/B 1. y = 6x + 11 2. y = −5x − 1 3. y = 2x − 4 4. y = 6x − 1 5. y = x − 1 6. y = 2x 7. y = 3x − 1 8. x 0 1 2 3 y 5 4 3 2 8. y = 4x + 2 9. g(s) = 4000 + 0.05s 10. h(s) = 8000 + 0.15s 11. k(s) = 2000 + 0.3s 12. Practice and Problem Solving: C 9. B; C; D; A 1. y = 2x − 3 10. It is not the correct standard form. To convert to standard form, you would need to multiply both sides by 2, then subtract 3x from each side so the correct standard form of the given equation should be −3x + 2y = 6. 2. 3x + 20y = 1 3. y = 2x − 20 4. y = 4x + 5 5. y = 12x − 12 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 462 6. y = −4x − 20 8. 7. 8x + y = 1 has slope of −8 and −8x + y = 2 has slope of 8. Since −8 = 8 = 8, they are equally steep. 8. Since it is an increasing function, f(x) increases as x increases. That means that slope is positive and m > 0. And, since the function passes through (4, 0), it must have risen from the point on its graph where x = 0. That means that the y-intercept must be negative. So, b < 0. 9. 9. 10. The original deal is a better choice if the salesperson has more than $4000 in weekly sales. You can see this on the graph. For sales greater than 4000, the graph for the original deal is higher than the graph for the new deal. Practice and Problem Solving: Modified 1. down 2. down 3. up Reading Strategies 4. more steep 1. 5; 2 − (−3) = 5 5. less steep 2. f(x); f(x) 3. y-axis; opposite 6. more steep 4. rotation (less steep) about (0, −3) 7. 5. reflection about y-axis 6. translation 6 units up Success for English Learners 1. If b is positive, the function is translated up and if b is negative, the function is translated down. 2. The slope of g(x) is 2 and is steeper than f(x) because the slope of f(x) is 1. 3. A translation moves every point the same distance in the same direction. A rotation is a transformation about a point. Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 463 LESSON 6-5 Practice and Problem Solving: Modified 1. 2 ≤ x ≤ 9 Practice and Problem Solving: A/B 2. 2 ≤ x ≤ 9 1. 2 ≤ x ≤ 9 3. 1 ≤ f(x) ≤ 10 2. 1 ≤ x ≤ 8 4. 3 ≤ g(x) ≤ 10 3. 1 ≤ y ≤ 10 5. f(2) = 1 4. 0 ≤ g(x) ≤ 35 6. g(2) = 3 9 7. 7 5. f(2) = 1 6. g(1) = 35 9 7. 7 8. 1 9. Possible answer: f(x) and g(x) have the same domain. They have different slopes, different initial values and different ranges. 8. −5 9. Possible answer: The graphs are not alike at all. They have a different slope, different initial values and different domain and ranges. 10. They would intersect once, at the point (9, 10). Reading Strategies 10. Possible answer: The temperature from 2 o’clock to 9 o’clock rose from 1 to 10 9 degrees at a rate of degrees per hour. 7 11. Possible answer: A tank has 35 gallons of water at 1 o’clock. The tank loses 5 gallons each hour until there is no water left. −11 12. f(x) would have y-intercept of and 7 g(x) would have y-intercept of 40. 1. They are already in the same form. 2. Domains appear to be the same with x ≥ 0. 3. Range for f(x) is y ≥ 0, range for g(x) is y ≥ 4. 4. The initial value for f(x) is 0, and the initial value for g(x) is 4. 5. f(x) is more steep than g(x) and starts higher, and they cross at approximately (5, 7). Practice and Problem Solving: C Success for English Learners 1. f(x): −2 ≤ x ≤ 7; g(x): 1 ≤ x ≤ 10; h(x): 1 ≤ x ≤ 10; k(x): 1 ≤ x ≤ 10 1. Both compare the domain and range, and both show (5, 25) as equal. 2. 5 ≤ f(x) ≤ 20; g(x): 20 ≤ g(x) ≤ 35; h(x): 2.5 ≤ x ≤ 25; k(x): 12 ≤ x ≤ 34.5 MODULE 6 Challenge 3. f(−2) = 20; g(1) = 20; h(1) = 2.5; k(1) = 12 5 5 5 5 4. f(x): − ; g(x): ; h(x): ; k(x): 3 3 2 2 5. Possible answer: For g(x), the temperature was 20 degrees at 1 o’clock, and it rose 5 degrees every 3 hours until 10 o’clock. For h(x), fabric costs $2.50 per yard, up to 10 yards. 50 55 ; g(x): ; h(x): 0; k(x): 9.5 6. f(x): 3 3 7. (−0.5, 17.5) 1. 3, 2, 1, 0, 1, 2, 3 2. Yes, each input (x) has exactly one output (y). 3. Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 464 3. a. Variables will vary. 15t + 6s = 900 4. b. c. 25 snacks MODULE 7 Linear Equations and Inequalities Practice and Problem Solving: C 1. a. Variables will vary. 200c + g = 600 LESSON 7-1 b. Practice and Problem Solving: A/B 1 a. Variables will vary. 0.05 p + 0.10c = 500 b. c. 1.6 cords of wood 2. a. Variables will vary. 72m + 150r = 5400 b. c. 2000 aluminum cans 2. a. Variables will vary. 2g + s = 300 b. c. 12 tennis rackets c. 130 games Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 465 c. 3. a. Variables will vary. 3.5b + 2.8r = 420 b. c. 25 rye loaves d. 160 baskets of blueberries Practice and Problem Solving: Modified Reading Strategies 1 a. 1. $35, $80, $1120 Hamburgers Hot Dogs 0 100 3. 35s + 80g = 1120 40 0 4. Possible answer: 14 65 2. 35s, 80g b. 2d + 5b = 200 c. Scientific Graphing 0 14 32 0 16 7 5. d. 65 hot dogs 2 a. Cherries Blueberries 0 240 180 0 30 200 6. 7 Success for English Learners 1. Set one variable equal to zero and solve for the other. The intercepts are (0, 60) and (50, 0) b. Variables will vary. 4c + 3b = 720 2. 42 taco specials 3. Variables will vary. 4b + 6h = 1200 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 466 4. LESSON 7-2 Practice and Problem Solving: A/B 1. f(x) = 20x + 90 2. g(x) = 30x + 50 3. x f(x) g(x) 0 90 50 1 110 80 2 130 110 3 150 140 4 170 170 5 190 200 5. 4 hours Practice and Problem Solving: Modified 1. f(x) = 4x + 15 2. g(x) = 2x + 25 4. x f(x) g(x) 0 15 25 1 19 27 2 23 29 3 27 31 4 31 33 5 35 35 4. 5. 4 hours 6 a. f(x) = 600 − 75x b. 8 months Practice and Problem Solving: C 1. f(x) = 12.5x + 75.30 2. g(x) = 18.10x + 52.90 3. X f(x) g(x) 0 75.30 52.90 1 87.80 71.00 2 100.30 89.10 1. x = −5 3 112.80 107.20 2. x = 3 4 125.30 125.30 3. x = 2 5 137.80 143.40 4. x = −1.5 5. 5 hours Reading Strategies Success for English Learners 1. f(3) = 3(3) + 7 = 16 and g(3) = 7(3) − 5 = 16 2. 2 3. 2 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 467 LESSON 7-3 9. y ≤ − Practice and Problem Solving: A/B 1. no 1 x 2 Practice and Problem Solving: C 2. yes 1. 3. no 4. 2. y≤x+3 5. 3. . 5 x + 8 y ≤ 40 y > −3x − 1 6. a. x + y ≤ 8 b. c. Possible answer: 2 peach, 6 blueberry or 4 peach, 3 blueberry 7. y ≥ x − 2 8. y < 2x + 4 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 468 8. 4. 1000x + 2000y > 1,000, 000, or x + 2y > 1000 9. 5. The symbol ≤ tells you that y is less than or equal to 2x + 8. So, you shade the region below the line. 6. y ≥ x means y > x or y = x. So, you make the line solid to include y = x when you graph y ≥ x. You leave the line dashed when you graph y > x to indicate that y = x is not part of the graph. Practice and Problem Solving: Modified 10. 1. no 2. no 3. yes 4. yes 5. yes 6. no 7. 11. 5 p + 2m ≤ 45 Reading Strategies 1. solid; below 3. solid; above 2. dashed; above Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 469 4. MODULE 7 Challenge 1. solution: (4, 1) not: (−2, 2) x 5. −6 −5 −4 −3 −2 −1 solution: (−3, −3) not: (2, 7) Think: y = (x + 3)2 − 4 2 (x, y) 2 y = (−6 + 3) − 4 = (−3) − 4 = 9−4=5 y = (−5 + 3)2 − 4 = (−2)2 − 4 = 4−4=0 y = (−4 + 3)2 − 4 = (−1)2 − 4 = 1 − 4 = −3 y = (−3 + 3)2 − 4 = (0)2 − 4 = 0 − 4 = −4 y = (−2 + 3)2 − 4 = (1)2 − 4 = 1 − 4 = −3 y = (−1 + 3)2 − 4 = (2)2 − 4 = 4−4=0 (−6, 5) (−5, 0) (−4, −3) (−3, −4) (−2, −3) (−1, 0) 2. Success for English Learners 1. You would shade over the line because the symbol is >. 2. The boundary line is dashed in Problem 2 because the symbol is < which means the values on the line are not solutions. x + y ≤ 5 Solved for y :y ≤ 5 − x x Think: y = 5 − x (x, y) −2 y = 5 − −2 = 5 − 2 = 3 (−2, 3) −1 y = 5 − −1 = 5 − 1 = 4 (−1, 4) 0 y =5− 0 =5−0=5 (0, 5) 1 y = 5 − 1 = 5 −1= 4 (1, 4) 2 y =5− 2 =5−2=3 (2, 3) 3 y =5− 3 =5−3=2 (3, 2) Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 470 UNIT 4 Statistical Models MODULE 8 Multi-Variable Categorical Data Practice and Problem Solving: C 1. The first clue provides the grand total: 200. The second clue provides the totals for boys and for girls: 100 each. LESSON 8-1 Practice and Problem Solving: A/B The third clue provides the numbers for the girls two choices: 50% of 100 is 50. 1. Bag Containing: Frequency plastic 8 glass 6 metal 6 The fourth clue provides the numbers for the choices for the boys. Let x represent the number of boys who favor the change. Then: x+ 5 x = 100 3 3 × 100 x= 5 x = 60 2. Field Trip Preferences Gender Science History Total Boys 46 56 102 Girls 54 44 98 Total 100 100 200 2 x = 100 3 Sixty boys favor the change and 40 boys do not. The completed table is below. 3. Favor or Disfavor the Change Science Gender Math Yes No Total Yes 21 38 59 No 34 7 41 Total 55 45 100 Yes No Total Girls 50 50 100 Boys 60 40 100 Total 110 90 200 2. This diagram shows the eight regions in the Venn diagram. One of the regions is outside the collection of circles. 4. Cat Dog Yes No Total Yes 0 100 100 No 62 38 100 Total 62 138 200 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 471 2. 49 Sample explanation: In the Total column, 150 − 78 = 72; so there are 72 boys in all. In the Boy row, 72 − 23 = 49; so 49 boys do not play a musical instrument. Practice and Problem Solving: Modified 1. 14; 13; Add the two numbers in the Carrots column to get 34; Add the totals in the far right column or add the totals along the bottom row of the table to get 100. Preferred Vegetable Grade Carrots Celery Cucumber Total 14 18 13 45 9 20 22 13 55 10 34 40 26 100 Total 3. No Sample explanation: The given data has only one characteristic, which is the type of ticket (adult or student). A two-way frequency table is used when the data has two characteristics. LESSON 8-2 Practice and Problem Solving: A/B 2. Preferred Fruit Grade Apple Orange Berries Total 9 21 18 13 52 10 24 19 19 62 Total 45 37 32 114 1. 25 125 2. 73 125 3. Calculate frequencies with a row total as the denominator. Likes reading and likes board games: 48 , 73 about 0.658 3. Likes reading but does not like board 25 games: , about 0.342 73 Foreign Language Gender Italian Spanish French Total 6 10 12 28 Boys 19 6 7 32 Girls 25 16 19 60 Total Those who like reading are more likely to like board games. 4. Calculate frequencies with a column total as the denominator. Reading Strategies 1. A, C, D Likes board games and likes reading: 2. B, E, F 48 , 91 about 0.527 3. Sample answer: Questions B, E, and F ask about preferences between and among the men and women who were polled. The two-way frequency table contains data for a second category, which is the gender of the people who were polled. Likes board games but does not like 43 reading: , about 0.472 91 Those who like board games are more likely to like reading. 5. Success for English Learners 15 200 6. Calculate frequencies with a row total as the denominator. 1. Sample answer: The table would have another row of frequencies between the Large row and the Total row. The phrase Extra Large would be in the left cell of the row. Grade 9 and travels by bus: 64 , about 107 0.598 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 472 Grade 10 and travels by bus: 43 , about 93 7. 0.462 15 or 0.075 200 8. Calculate frequencies with a row total as the denominator. Grade 9 students are more likely to travel by bus than not. Grade 10 students are more likely not to travel by bus. Adult and travels by bus: 0.598 Practice and Problem Solving: C 1. Child and travels by bus: Like Televised Sports Gender Yes No Total Men 48 25 73 Women 43 9 52 Total 91 34 125 Adults are more likely to travel by bus than not. Children are more likely not to travel by bus. Practice and Problem Solving: Modified 3. 72.8% 1. 28; 140; 4. Calculate frequencies with a row total as the denominator. 28 140 2. 56; 140; 56 140 48 , 73 3. about 65.8% Male but does not like televised sports: 25 , about 34.2% 73 5. 1. 48 , 91 Likes televised sports but is not male: 43 , about 47.2% 91 6. Type of Transportation 15 28 64 107 Child 20 30 43 93 Total 35 58 107 200 42 21 = = 0.42 = 42% ; marginal relative 100 50 frequency 3. Sample answer: What is the conditional relative frequency that a student plays a team sport, given that the student plays chess? Sample solution: chess and sport 12 3 = = "Yes" chess row total 16 4 = 0.75 = 75% Those who like televised sports are more likely to be male. Adult 8 66 2. The given condition is that the student plays a team sport, so you are only looking for a relationship among numbers in that column. about 52.7% On Foot 28 74 Success for English Learners 5. Calculate frequencies with a column total as the denominator. Age 30 140 4. 28; 74; Men are more likely to like televised sports. Likes televised sports and is male: 43 , about 93 0.462 2. 38.4% Male and likes televised sports: 64 , about 107 By Car By Bus Total Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 473 spread out in Club B. Its range, interquartile range, and standard deviation are each three times as great or more as the corresponding statistics for Club A. Reading Strategies 1. joint; 8 4 = , 0.16, or 16% 50 25 2. conditional; 3. marginal; 3 , ∼0.429, or ∼42.9% 7 9. Possible answer: Club A has the lower mean and median age, and so it could claim to be the “younger” club. Club B has three of its eight members in their 20s, while Club A has no member below age 34. So, Club B could also make the same claim. 16 8 = , 0.32, or 32% 50 25 MODULE 8 Challenge 1. Practice and Problem Solving: C Color of Hair: Color of Black Brown Red Eyes: Brown 11.5% 20% Blonde TOTAL 1. Mean: $2.162; median: $2.285; range: $1.21; interquartile range: $0.73 4.5% 1.5% 37.5% 2. Mean: $2.057; median: $2.04; range: $0.95; interquartile range: $0.73 3. 2011: $0.39; 2012: $0.34 Blue 3.5% 14% 3% 15% 35.5% Hazel 2.5% 9% 2.5% 2% 16% Green 1% 5% 2.5% 2.5% 11% TOTAL 18.5% 48% 12.5% 21% 100% 4. Possible answer: Prices fell in 2012 and became a bit more steady. The mean price fell from $2.162 to $2.057 and the standard deviation fell from $1.24 to $1.07. 2. brown eyes and brown hair 5. 97 3. green eyes and black hair 6. C can be any of the nine integers from 56 to 64. 4. Answers will vary. Possible answer: Blondes are most likely to have blue eyes. 7. Mean and median increase by 4. Range, interquartile range, and standard deviation do not change. MODULE 9 One-Variable Data Distributions Practice and Problem Solving: Modified LESSON 9-1 1. 85 Practice and Problem Solving: A/B 2. 85 1. Mean: 24.5; median: 25; range: 12 3. 85 2. Mean: 8.6; median: 9; range: 8 4. 85 3. Mean: 84; median: 85.5; range: 21 5. 28 4. Mean: 1.3; median: 1.2; range: 2.0 6. 10 5. Mean: 40; median: 39; range: 14; interquartile range: 7 7. First quartile: 77; third quartile: 94 6. Mean: 43; median: 43.5; range: 43; interquartile range: 36 9. 17 8. First quartile: 83; third quartile: 87 10. 4 7. Club A: 4.5; Club B: 16.8 11. Possible answer: Their means are equal and their medians are equal. 8. Possible answer: Club A has a slightly lower average age in its club. Both its mean and median ages are lower than those of Club B. The ages are much more 12. Possible answer: Brad’s scores are much more widely spread out than Jin’s scores. His range is almost three times as great Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 474 showing the data over time, it makes clear that the player’s home run totals have fallen back during the past two seasons to the 18−22 zone. and his interquartile range is more than four times as great. 13. Jin is the more consistent test taker. Her grades show a much smaller range. 8. The mean would decrease to 26.4 and the median would decrease to 28. The range would increase to 30 and the interquartile range would increase to 11. 14. Jin has the standard deviation of 3 and Brad has the standard deviation of 9.6. You can tell because Brad’s test scores are so much more spread out. Practice and Problem Solving: C Reading Strategies 1. 100 is not an outlier because Q3 = 89 and IQR = 17.5. Then 100 < Q3 + 1.5(IQR). 1. when there is an even number of data values 2. mean: 8, median: 8.5, range: 12 2. 100 is not an outlier because Q1 = 110 and IQR = 14. Then 100 > Q1 − 1.5(IQR). Success for English Learners 1. Possible answer: Problem 1 has an odd amount of numbers in the set, so there is a middle. Problem 2 has an even amount of numbers in the set. 3. mean = 54.15; median = 54 4. range = 27; interquartile range = 7 5. 2. by adding the numbers and then dividing by how many numbers are present 3. Mean and median would be 5. 4. Patricia forgot to put the numbers in order from least to greatest first. 6. Possible answer: The line plot makes clear that there is a cluster of data in the 51−56 age range. Eleven of the 20 Presidents were in this range upon taking office. This pattern cannot be seen as clearly by just looking at the original data set. LESSON 9-2 Practice and Problem Solving: A/B 1. 100 is not an outlier because Q3 = 85 and IQR = 16. Then 100 < Q3 + 1.5(IQR). 7. Q1 = 51, Q3 = 58, and IQR = 7. So, if a President’s age upon taking office was less than 51 − 1.5(7) = 40.5 or greater than 58 + 1.5(7) = 68.5, there is an outlier. From the line plot, the only outlier is the President who took office at age 69. 2. 100 is an outlier because Q3 = 85 and IQR = 9. Then 100 > Q3 + 1.5(IQR) 3. mean = 27.57; median = 28.5 4. range = 22; interquartile range = 10 8. Possible answer: Because of the cluster of data in the low 50s and since the mean and median are close to 54, my guess is that Cleveland was 54 years old. 5. 6. Possible answer: The dot plot makes it appear as if 40 is an outlier. However, since Q3 = 32 and IQR = 10, 40 is not an outlier since 40 < 32 + 1.5(10) Practice and Problem Solving: Modified 1. skewed to the left 2. symmetric 7. Possible answer: The dot plot does not help predict. It makes it appear that there are two “zones” where this player tends to hit home runs: from 18 to 22 and from 28 to 34. The table may help predict. By 3. symmetric 4. skewed to the right Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 475 5. mean: 72.8; median: 74 LESSON 9-3 6. range: 8; interquartile range: 5 Practice and Problem Solving: A/B 7. 1. Frequency 6 4 8. skewed to the left 4 9. The mean would increase to 74.8. The median would not change. The range would increase to 27. The interquartile range would not change. 3 4 2. 10. For the 11 scores, the third quartile is 75 and the interquartile range is 5. Since 95 > 75 + 1.5(5), 95 would be an outlier. Reading Strategies 1. yes; 25 2. no 3. yes; 21 and 59 4. yes; 11 5. yes; 158 6. no 3. 4−7, 8−11, 16−19 7. yes; cluster at 211 and outlier at 278 4. Estimate of mean: 1.5(6) + 5.5(4) + 9.5(4) + 13.5(3) + 17.5(4) 21 179 ≈ 8.52 . ≈ 8.55. The actual mean is 21 The estimate is very close to the actual answer. 8. yes; cluster at 325; no outlier Success for English Learners 1. by adding all the numbers in the data set and then dividing by the amount of numbers 2. Possible answer: They do not need to be added, because anything plus 0 is itself. But they do need to be considered in the division as points of data. 5. 38 °F 6. 20 °F 7. 50 °F is not an outlier. Q3 = 46 and IQR = 12. So, 50 < Q3 + 1.5(IQR). 3. Possible answer: 1, 3, 3, 3, 5. Practice and Problem Solving: C 4. The mode is the number that happens the most often. In a dot plot, it will have the most dots or X’s. 1. 100,000 2. 10−19 and 50−59; 60−69 and 70−79 3. Use the midpoint values of each interval (4.5, 14.5, and so on) to estimate the mean. Multiply each midpoint value by the population of that interval and then find the sum of those products. The mean is approximately the quotient of that sum and the total population: 4140 ÷ 100 = 41.4 years of age Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 476 4. The distribution is not symmetric. It skews to the right, with its population tailing off as people get older. Reading Strategies 1. 30–39 and 50–59; Possible answer: because the bars are the tallest on the graph. 5. 2. 80−89; Possible answer: because it has the shortest bar. 3. 6 + 4 + 6 + 5 + 4 + 3 = 28 6. Killebrew: mean = 26.0 and median = 27; Mays: mean = 30 and median = 31.5 4. 6; by finding the middle numbers. The middle numbers are both 6, so the median is 6. 5. 9; I found the median between the second 6 and 10. 6. 1 and 10, 1 is the lowest value and 10 is the highest value. Practice and Problem Solving: Modified 1. Player’s Heights Heights (in.) Frequency 72−76 4 77−81 11 1. Scores between 68 and 70 in a golf tournament 82−86 7 2. 74–76 87−91 2 3. Minimum = 2, Q1 = 4, Q2(median) = 7, Q3 = 10, maximum = 12 Success for English Learners 4. They both have the same interval and the same scale. 2. LESSON 9-4 Practice and Problem Solving: A/B 1. 68.3% 2. 99.7% 3. 47.8% 4. 34.1% 3. 77−81 inches 5. 50% 4. Skewed to the right 6. 47.8% 5. 74(4) + 79(11) + 84(7) + 89(2) ≈ 80.5 24 inches 7. 81.9% 8. 2.3% 9. 97.7% 6. 28 10. 84.1% 7. 22 11. 0.13% 8. 12 Practice and Problem Solving: C 9. 40 n ⎛ 1⎞ 1. ⎜ ⎟ ; 3.125% ⎝2⎠ 2. Possible answer: Since it is a fair coin, the best prediction is that it will land showing Heads 50% of the time. That would be 500 times. 3. 68.3% Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 477 4. 15.9% 2. The normal distribution is a random distribution and there is no guarantee that each interval should have perfect agreement between projected and actual number of points falling within the range, anymore than a single sample should always hit exactly the mean value. 5. 0.13% 6. 95.4% 7. Possible answer: You cannot conclude that the coin is not fair. According to the normal distribution, the mean number of Heads is 200 with standard deviation of 10. That means that obtaining 221 Heads represents an event that is more than two standard deviations above the mean. According to the normal distribution, there is a 2.5% chance that the number of Heads obtained could be more than two standard deviations above the mean. So, a probability of 2.5% is great enough to keep you from concluding that the coin is not fair. MODULE 9 Challenge 1. The highest point of the curve is translated to the left (if the mean is less than 0) or right, but the shape stays the same. 2 As the standard deviation increases, the shape of the curve is flatter and more stretched out. MODULE 10 Linear Modeling and Regression Practice and Problem Solving: Modified 1. LESSON 10-1 Practice and Problem Solving: A/B 1. 2. The sum is 100%. It makes sense because the sum represents the total area, or 100% of the area, under the curve. 3. a. 68.3% b. 95.4% c. 99.7% 4. 34.1% 5. 15.9% 6. 2.3% 2. time, number of drinks 7. 81.9% 3. positive; r is close to 1. 4. Sample answer: No; the temperature will probably influence both. Reading Strategies 1. 83; 7 5. a. 2. 64; 11 Success for English Learners 1. mean − standard variation; mean + standard variation Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 478 Practice and Problem Solving: Modified b. y = 176x + 854 c. 3494; Students’ answers should be correct for the equations they found in the step above. 1. positive 2. none 3. negative Practice and Problem Solving: C 4. negative 1. 5. positive 6. negative 7. no correlation 8. Possible answer: There is a negative correlation because as children grow, their legs get longer and they get stronger. This enables them to run more quickly. There is causation since getting older leads to better running. 2. Possible answer: Reading Strategies 1. number of children in a family; monthly cost of food; increases; positive 2. population of cities in the U.S.; average February temperature of those cities; shows no pattern; none y = 1.44x + 4; r = 1 3. y = 1.568x + 2.787; r = 0.925 3. number of practice runs; finish time; decreases; negative 4. The answers are close. Estimating, I got a y-intercept of 4 and the calculator found 3 (rounded). Estimating, my slope was 1.44, and the calculator found 1.568, a difference of about 8%. The calculator is more accurate; it provides the best fit. Estimating, I can only get close. Success for English Learners 1. If the points on the scatterplot go down from left to right, then they have a negative slope then the correlation is negative. 2. It would have a positive correlation because as the number of empty seats increases, the number of students absent from class would also increase. LESSON 10-2 Practice and Problem Solving: A/B 1. Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 479 2. A, age in years 2 3 3 4 4 4 5 5 5 6 H, height in inches 30 33 34 37 35 38 40 42 43 42 Predicted Values 30 33.5 33.5 37 37 37 40.5 40.5 40.5 44 Residuals 0 −0.5 0.5 0 −2 1 −0.5 1.5 2.5 −2 3. Practice and Problem Solving: C 1. Height = 3.5 i Age + 23 2. The man would be 93 inches, or 7 feet 9 inches tall. Since this is not reasonable, the linear model seems unsuitable for extrapolating much beyond the data. By teenage years, growth in height generally stops. 3. The distribution seems suitable between positive and negative values. But the residuals seem to be increasing as age increases. This could be an issue. 4. Possible answer: The line fits pretty well but the residuals seem to be increasing in size as age increases. This could be a problem. 5. The man’s height would be 93 inches, or 7 feet 9 inches. This is unreasonable. The linear model cannot be extrapolated that far. 4. A 2 3 3 4 4 4 5 5 5 6 H 30 33 34 37 35 38 40 42 43 42 AH 60 99 102 148 140 152 200 210 215 252 4 9 9 16 16 16 25 25 25 36 2 A 5. Sum(A) = 41; Sum(H) = 374; Sum(A2) = 181; Sum(AH) = 1578; rounded to three decimal places, m = 3.457 and b = 23.225 2. Practice and Problem Solving: Modified 1. Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 480 3. 4. Possible answer: Of the three lines, 1 y = − x + 6 best fits the data. It cuts 3 down the middle, and has four points both above and below it. 5. The data shows negative correlation since y tends to decrease as x increases. 6. x 1 2 3 4 5 6 8 8 y 7 5 4 6 3 5 3 4 Predicted Values 5.5 5 4.5 4 3.5 3 2 2 Residuals 1.5 0 −0.5 2 −0.5 2 1 2 This is not a great fit because there are more positive residuals than negative and the positive residuals are larger. 3. a. Each y-value is reduced by half, so the points are closer to the x-axis. Reading Strategies 1. The slope and r have the same sign. b. The relationship between points would not change. 2. No; possible answer: a value or r close to 0 means that the two variables have almost no correlation. c. I would expect a change in the slope and the y-intercept 3. Possible answer: the correlation coefficient would be positive; the more hours I spend studying, the higher my grade will be. d. No; Possible answer: because the relationship between points has not changed, the correlation is the same. 4. a. The x-coordinates are the same, but the y-coordinates are multiplied by −1. 4. Possible answer: the correlation coefficient would be negative; the more rain there is, the fewer the people who want to go to the beach. b. The relationship between points would not change. c. The slope and the y-intercept of the line of best fit are numerically the same, but both are now negative. Success for English Learners 1. It would have a greater negative slope. 2. Closer to −1. The line has a negative slope, so r is negative. d. The value of r should be the same, but it is now negative since the data now shows a negative correlation. MODULE 10 Challenge 1. 2. Slope: 1.568; y-intercept: 2.787; r = 0.925 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 481 UNIT 5 Linear Systems and Piecewise Functions Practice and Problem Solving: C MODULE 11 Solving Systems of Linear Equations 1. Sample answer LESSON 11-1 Practice and Problem Solving: A/B 1. one, consistent, independent 2. infinite number, consistent, dependent 3. none, inconsistent 4. (1, 2) 2. Sample answer 5. same slope, different y-intercept, no solution 3. Sample answer ⎧ y = 8x + 5 6. ⎨ ⎩2y = 16 x + 10 ⎛3 ⎞ 4. ⎜ , 1⎟ ⎝2 ⎠ Jill and Samantha earn the same for any time. Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 482 9. Same line, infinite number of solutions 5. Same slope, no solution 6. Same line, infinite number of solutions 10. x + y = 12 , graph both equations and find x−y =6 the intersection of the lines; (9, 3) Practice and Problem Solving: Modified 1. one solution, consistent 2. infinite number of solutions, consistent 3. no solution, inconsistent 4. 9, 9, 1, −1; (5, 4) 7. Same slope, no solution 5. 4, 8; 7, −7; (5, −2) 8. (2, −2) Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 483 6. 2, −6; 2, −6; infinite number of solutions 10. 27 years old 11. length = 37.5 inches and width = 12.5 inches 12. 25 hours at the weekday job and 15 hours at the weekend job 13. Graphs will vary. The solution found by graphing should agree with the solution by substitution. Practice and Problem Solving: C 1. (−2, −1) 7. −4, 2; 6, −3; no solution 2. (36, −16) ⎛1 3. ⎜ , ⎝4 1⎞ 8 ⎟⎠ 4. 30 years old 5. 60 quarters and 40 dimes 6. $110 7. 51 8. 520 pennies and 80 nickels Reading Strategies 9. $2.50 1. Possible answer 4x + y = 6, −2x + y = 3 Practice and Problem Solving: Modified 2. Disagree. Two lines can intersect at only one point, are the same line and intersect at an infinite number of points, or are parallel and don’t intersect. 2. y 3. y 5. 5y + 5 3. one 6. 6y − 16 4. Both equations have slope of 3 and different y-intercepts so the lines are parallel and don’t intersect. 8. (5, 4) 9. (17, −24) 10. (5, 1) 5. infinite number of solutions 11. (4, −7) Success for English Learners 12. (−1, −2) 1. The ordered pair makes both equations true. 14. 9 dimes 15. $15 2. You find the point where the two lines intersect. Reading Strategies LESSON 11-2 1. y = 5 − x or y = −x + 5 Practice and Problem Solving: A/B 2. The first equation is already solved for y. 1. (−1, −3) 2. (3, −1) 3. (2, 7) 4. (1, −4) 5. (5, −2) 6. (3, −2) 7. (1, −2) 8. (−1, −2) 3. The solution of a system must satisfy both equations. 4. (6, 4) 5. (−3, 5) 9. (4, 9) Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 484 6. (3, −5) Success for English Learners 7. (1, 2) 1. You have to substitute the value you found for m into one of the equations and find T. 8. (2, 5) 9. no solution 2. 5 months 10. (47, 23) LESSON 11-3 11. y + x = 30 and y + 5x = 42.; (3, 27) Practice and Problem Solving: A/B Reading Strategies 1. (5, −1) 1. No, it is not the solution. 2. (2, −12) 2. Yes, it is the solution. 3. (−2, 1) Success for English Learners 4. (−12, 4) 1. When the variables with the same coefficient have opposite signs, add. When they are exactly the same, subtract. 5. (−3, 3) 6. infinitely many solutions 7. (0, −1) LESSON 11-4 8. (8, −7.2) Practice and Problem Solving: A/B 9. initial amount: $30; hourly rate: $60 10. $9 ⎛ 1 1⎞ 1. ⎜ , ⎟ ⎝3 4⎠ Practice and Problem Solving: C 2. (3, −4) 1. (5, 0.5) 3. (−9, 8) 2. (5, −8) 4. (13, −2) 3. (−1, 1) 5. (−2, −9) 4. (75, −25) 6. (−10, −1) 5. 12 adults 7. (3, 7) 6. Pearl solved an inconsistent system of equations. The system has no solution. The graphs of the two equations are parallel lines. 7. ax + by = c ⎛5 ⎞ 8. ⎜ , 2 ⎟ ⎝3 ⎠ 9. Bagel: $0.80; muffin: $1.25 10. 15 minutes dx − by = e Practice and Problem Solving: C ax + dx = c + e (a + d ) x = c + e 1. (−5, 6) 2. (42, −36) c+e x= a+d 3. (3, −1) 4. (−10, −3.25) Practice and Problem Solving: Modified 5. 300 dimes and 120 quarters 6. $5 1. substitution 2. addition/subtraction 7. 300 10-pound bags and 120 50-pound bags 3. substitution 8. 727 4. (12, 4) 5. (0, 1) Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 485 2. 4 x + y + 2z = 10 Practice and Problem Solving: Modified −( x + y − 6z = −23.5) 1. 2nd; 3 or −3 2 x + 3 y + 3z = 13 −3( x + y − 6z = −23.5) 3 x + 8z = 33.5 − x + 21z = 83.5 st 2. 1 ; 2 or −2 3 x + 8z = 33.5 3. 1st; 4 or −4 + 3( − x + 21z = 84.5) 4. 2nd; 4 or −4 6. (−10, −4) 7. (4, −1) x + y − 6z = −23.5 8. (2, 0) 0.5 + y − 6(4) = −23.5 y =0 9. One hot chocolate costs $2. 10. 9 two-point shots and 3 three-point shots MODULE 12 Modeling with Linear Systems 1. Multiply the first equation by 3 and the second equation by 5 to get a common coefficient of −15. LESSON 12-1 ⎧ 4(9x − 10y = 7) ⎧36x − 40y = 28 2. ⎨ ⇒⎨ 5(5x + 8y = 31) ⎩25x + 40y = 155 ⎩ Practice and Problem Solving: A/B 1. apple: $1.50, pear $1.25 4. (10, −10) 2. lemonade: $1.57, iced tea: $1.82 Success for English Learners 3. y = 8x + 50 1. The y-variable is eliminated. 4. y = 10x + 30 2. Multiply the equation by −3 so that the y-variable is eliminated. 5. The campgrounds will both charge $130 for 10 campers. 6. C1(n) = 2n + 4, C2(n) = 2.5n + 2 MODULE 11 Challenge x + y + z = 12 +(3 x + y − z = 32) 4 x + 2y = 44 4 x + 2y = 44 −(6 x + 2y = 56) 7. The functions represent the rates charged by 2 different dog walkers. The variable represents the number of dogs. 3 x + y − z = 32 +(3 x + y + z = 24) 6 x + 2y = 56 8. Yes Practice and Problem Solving: C 4(6) + 2y = 44 y = 10 1. y = − 2 x = −12 x=6 6 + 10 + z = 12 z = −4 Solution: (0.5, 0, 4) The solution checks in all three equations. Reading Strategies 1. x = 0.5 71z = 284 z=4 5. (2, 3) 3. (1, −3) 3 x + 8(4) = 33.5 1 x 2 2. Sample answer: The number of bales of hay needed to feed 3 elephants. 3. Sample answer: Let x = number of bales of hay; Let y = the number of elephants; 1 + 1, y = x − 3 (6, 3); y = 3x Solution: (6, 10, −4) The solution checks in all three equations. Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 486 4. Write a system of equations using the equations for each hotel and solve by substitution for x to find the number of nights for which the hotels will charge the same rate. Then substitute the value of x into one of the original equations to find the rate charged by both hotels for that number of nights. 4. Sample chart: Bales of Hay Fed Number of Elephants Circus A Circus B Circus C 2 4 3 5 4 8 9 7 5 10 12 8 LESSON 12-2 Practice and Problem Solving: A/B 5. Sample answer: 1. no The chart shows the number of bales of hay three circuses feed to their elephants for each meal. At what number of elephants do the circuses feed the same number of bales of hay? Students’ solutions should reflect the information from the chart in Exercise 4 and the information given in Exercise 5. 2. yes 3. no 4. Practice and Problem Solving: Modified 1. slope: 12; y-intercept: 25; equation: y = 12x + 25 2. slope: 10; y-intercept: 35; equation: y = 10x + 35 a. (0, 3) and (3, −2) 3. 10x + 35 b. (−2, 0) and (−4, 3) 4. 10x + 35 −25 5. 5. 10x + 10 6. 10x − 10x + 10 7. 10 10 2 9. 5 10. 5 11. $85 8. a. (0, 0) and (−2, 0) Reading Strategies b. (3, 0) and (2, 3) 1. brush = $4.50, paint = $2.00 Success for English Learners 1. The slopes represent the rate at which each person saves. 2. Write the system of equations represented by the graph and solve by substitution. 3. nights stayed Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 487 4. y ≥ x − 3 6. y ≥− 3 x+4 4 5. a. (0, 0) and (−2, −2) b. (−3, 3) and (4, 0) 7. a. x = babysitting hours, y = gardening hours, ⎧ x + y ≤ 12 ⎨ ⎩10 x + 5 y ≥ 80 6. The region is in the shape of a quadrilateral. Its vertices are (−5, −5), (−5, 6), (2, −1), and (0, −5). b. 7. The area is 48.5 square units. Possible explanation: Draw the rectangle with vertices at (−5, −5), (−5, 1), (0, 1), and (0, −5). Its area is (6)(5) = 30 square units. Above the rectangle is a right 1 triangle. Its area is (5)(5) = 12.5. To the 2 right of the rectangle is a second triangle. Its base has endpoints at (0, 1) and (0, −5), making b = 6. Its height extends from (2, −1) to (0, −1), making h = 2. So, 1 area = (6)(2) = 6. So, the region has 2 area of 30 + 12.5 + 6 = 48.5 square units. c. Any combination of hours represented by the ordered pairs in the solution region. d. 6 h babysitting, 4 h gardening; 8 h babysitting, 2 h gardening Practice and Problem Solving: Modified Practice and Problem Solving: C 1. y ≤ x − 3 3 y ≥− x+4 4 1. y = −2x + 4; below 2. y ≤ x − 3 3 y ≤− x+4 4 4. no 2. y = 3x − 6; above 3. y = 4x − 7; above 5. yes 6. no 3. y ≥ x − 3 3 y ≤− x+4 4 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 488 7. Success for English Learners 1. (−3, 1) is a solution because it lies in the shaded region for both inequalities. 2. (4, 2) is not a solution because it does not lie in the shaded region for both inequalities. LESSON 12-3 Practice and Problem Solving: A/B 1. s + c = 12; 12s + 10.5c = 138; 8 steak, 4 chicken a. (−1, 0) and (−3, 2) 2. c + l = 9; 3c + 2l = 23; 5 couches, 4 loveseats b. (0, −3) and (4, 0) 8. 3. a + s = 89; 5a + 3s = 371; 52 adults, 37 students 4. q + d = 110; 0.25q + 0.10d = 20.30; 62 quarters, 48 dimes 5. t + c ≥ 16; 25t + 15c ≤ 285; solution is all the points in the overlap region; 4 tables, 12 chairs a. (0, 0) and (1, 2) b. (1, 0) and (−4, 3) 9. w + c > 5 49w + 100c < 400 Reading Strategies ⎧2 x + 2y ≤ 30 1. ⎨ ⎩x > 8 2. Practice and Problem Solving: C 1. l + s = 41; 22l + 14s = 710; 17 large vases, 24 small vases 3. I = 10 ft, w = 5 ft and I = 11 ft, w = 4 ft Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 489 2. s + f = 27; 120s + 90f = 2880; 60 students, 24 faculty Reading Strategies 1. q + d = 85; 0.25q + 0.10d = 16.90; 56 quarters, 29 dimes 2. f + t = 25; 5f + 20t = 335; 11 $5 bills, 14 $20 bills Success for English Learners Possible answers are given. 1. Multiply both sides of x + y = 22 by −4. 2. Multiply both sides of x + y = 7 by −3.2. 3. Multiply both sides of x + y = 5 by 1.3. 4. d + s = 15; 7d + 4s = 75; 5 dress socks, 10 sports socks 3. t + c ≥ 8; 5t + 19c ≤ 100; 4 tetras, 4 cichlids MODULE 12 Challenge ⎧y ⎪y ⎪ 1. ⎨ ⎪y ⎪⎩ y ≥x−4 ≤x+4 ≥ −x − 4 ≤ −x + 4 ⎧⎪ y ≥ x − 4 2. ⎨ ⎪⎩ y ≤ − x + 4 3. a. Practice and Problem Solving: Modified 1. markers; pads of paper; 2m + 4p = 26; m + p = 8; −2m − 2p = −16; 0 + 2p; 5; 5; 3; 3 2. c + d = 7; 9c + 17d = 87; 4 CDs, 3 DVDs ⎪⎧ x < 3 b. ⎨ ⎪⎩ y < 5 3. p + s ≥ 12; 4p + 7s ≤ 57; 9 plain candles, 3 scented candles ⎧⎪ x > 3 c. ⎨ ⎪⎩ y > 5 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 490 MODULE 13 Piecewise-Defined Functions 2. Look for a pattern in the values of f for even and odd multiples of 2. f (0) = f (2 × 0) = −2 f (2) = f (2 × 1) = 2 f (4) = f (2 × 2) = −2 f (6) = f (2 × 3) = 2 LESSON 13-1 Practice and Problem Solving: A/B 1. Since 48 = 2 × 24, f(48) = −2. Since 30 = 2 × 15, f(30) = 2. Therefore f(30) > f(48). 3. Complete the diagram to show the four parts of the letter, and help determine the four equations that define f. 2. Use two points to write equations for line segments. For I, use (−5, 3) and (−3, −4). ⎧ ⎪⎪ − x − 8 3. f ( x ) = ⎨0.5 x + 0.5 ⎪ ⎪⎩ 2 x − 3 ⎧ ⎪⎪0.5 x − 2.5 4. f ( x ) = ⎨0.5 x − 0.5 ⎪ ⎪⎩ 0.5 x + 1.5 −4 − 3 ( x − ( −5)) −3 − ( −5) 7 29 y =− x− 2 2 y −3= x < −3 −3 ≤ x ≤ 3 x >3 For II, use (−3, −4) and (−1, −2). −2 − ( −4) ( x − ( −3)) −1 − ( −3) y = x −1 x < −3 −3 ≤ x < 1 y − ( −4) = x ≥1 For III, use (−1, −2) and (1, −4). 5. $2.00; $4.00; $0.00; $0.00 −4 − ( −2) ( x − ( −1)) 1 − ( −1) y = −x − 3 y − ( −2) = Practice and Problem Solving: C 1. The endpoints of the curved portion of the graph modeled by f(x) = ax2 + c are (−2, 2) and (1, −1). For IV, use (1, −4) and (3, 3). 3 − ( −4) ( x − 1) 3 −1 7 15 y= x− 2 2 By substitution: y − ( −4) = 4a + c = 2 and a + c = −1. So, 4a + (−1 + (−a)) = 2 and a = 1. Therefore, c = −2. Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 491 10. . ⎧ ⎪ − 7 x − 29 − 5 ≤ x < −3 ⎪ 2 2 ⎪⎪ − 3 ≤ x < −1 Therefore, f ( x ) = ⎨ x − 1 ⎪ −x − 3 − 1≤ x ≤ 3 ⎪ ⎪ 7 x − 15 3<x≤3 ⎪⎩ 2 2 Other representations come from variations of the inequality symbols used for the parts. ⎧ 11. f ( x ) = ⎨⎪ −0.5 x − 1.5 x < 1 ⎪⎩ −5 x ≥1 Practice and Problem Solving: Modified 1. (1, 5) and (0, 3) Reading Strategies 2. 1. a function that has different rules for different parts of its domain 2. Find the part of the domain that contains x. Use the rule associated with that part of the domain. 3. You use dots when the graph makes a transition from one rule to another. You use a closed dot at the point (x, y) if x is included in the domain for the rule. You use an open dot at the point (x, y) if x is not included in the domain for the rule. 3. (1, 2) and (2, 0) 4. You usually must write one equation for each distinct piece of the graph. If the greatest integer function is involved, you might be able to use the greatest integer notation [ ] to write just one equation. 4. 5. Sample answer: The situation involves different intervals, and there is a different way to calculate a result over each interval. Success for English Learners 1. For f(6.5), use the rule f(x) = x + 2; f(6.5) = 6.5 + 2 = 8.5. For f(−6.5), use the rule f(x) = −2x; f(−6.5) = −2(−6.5) = 13. 5. (−1, 1) and (−3, 2) 2. The open dot means that the point with coordinates (2, 5) is not part of the graph. 6. −0.5 7. f(x) = −0.5x + 0.5 3. Sample answer: The greatest integer function is a piecewise-defined function because it has different rules for different parts of its domain. For example, when 0 ≤ x < 1, the rule is y = 0; when 1 ≤ x < 2, the rule changes to y = 1; when 2 ≤ x < 3, the rule changes to y = 2; and so on. 8. (−1, −3) and (1, −3) 9. f(x) = −3 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 492 LESSON 13-2 Practice and Problem Solving: C 1. Sample table Practice and Problem Solving: A/B 1. Sample table x f(x) x f(x) −2 −4 0 5 −1 −2 1 4 0 0 2 3 1 2 3 2 2 0 4 3 3 −2 5 4 Domain = {Real Numbers}, Range = { y ≤ 2 } Domain = {Real Numbers}, Range = { y ≥ 2 } 2. Sample table 2. Sample table x f(x) x f(x) −5 1 −4 4 −3 2 −3 2 −1 3 −2 0 1 2 −1 −2 3 1 0 0 5 0 1 2 Domain = {Real Numbers}, Range = { y ≤ 3 } Domain = {Real Numbers}, Range = { y ≥ −2 } 3. f ( x ) = − 3. f ( x ) = x + 1 + 1 1 x +1 +1 2 1 x −2 −2 3 4. f ( x ) = − x − 2 + 2 4. f ( x ) = 5. f ( x ) = x − 22.3 5. d ( x ) = 175 − 25 x − 7 , 14 days Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 493 4. Sample table Practice and Problem Solving: Modified 1. Sample table x f(x) −2 3 −1 2 0 1 1 2 2 3 3 4 x f(x) −4 −1 −3 0 −2 1 −1 2 0 1 1 0 Domain = {Real Numbers}, Range = { y ≤ 2 } Domain = {Real Numbers}, Range = { y ≥ 1 } 5. f ( x ) = x + 1 + 1 6. f ( x ) = x − 2 + 2 2. Sample table Reading Strategies x f(x) −3 3 −2 2 3. ( −3, − 4) −1 1 4. f(x) is stretched vertically by a factor of 2. 0 2 1 3 2 4 1. 3 units left 2. 4 units down Success for English Learners 1. Domain = {Real Numbers}, Range = { y ≥ 1 } 3. Sample table x f(x) −4 5 −3 3 −2 1 −1 −1 0 1 1 3 Domain = {Real Numbers}, Range = { y ≥ −1 } Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 494 10. 2. Sample table x f (x ) = x − 3 − 2 −1 2 0 1 1 0 2 −1 3 −2 4 −1 5 0 x = −1 or x = 3 11. 3. x = 2 or x = 8 4. Domain = {x, all real numbers} 12. x − 68 = 3.5 5. Range = { y ≥ −2 } LESSON 13-3 13. 64.5°; 71.5° Practice and Problem Solving: A/B Practice and Problem Solving: C 1. two 1. No solution 2. one 2. x = −7 or x = 7 3. none 3. x = −11 4. x = −12 or x = 12 5. x = − 4. x = − 1 1 or x = 2 2 3 5 or x = 2 2 5. no solution 6. x = −10 or x = 10 6. x = −6.6 or x = 8.6 7. x = −9 or x = 9 7. 8. x = −8 or x = 8 9. x = −13 or x = 7 x = −0.25 or x = 0.75 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 495 8. 10. x − 24 = 2 11. min: 22 mpg; max: 26 mpg Reading Strategies 1. one 2. two 3. none 4. one 5. none 6. two Success for English Learners x = −0.4 or x = 1.2 1. 4 and −4 9. x − 3 = 0.005; 2.995 m; 3.005 m 2. 4 and −4 10. x + 5 = 1.5; −6.5°C; −3.5°C 3. Subtract 8 from both sides. Practice and Problem Solving: Modified 4. There is no solution because the absolute value expression equals a negative number. 1. −3; −3; 2; −2; 2 LESSON 13-4 2. −7; 7; 4; 4; 4; 4; −11; 3 Practice and Problem Solving: A/B 3. 6; −6; 6; −5; 7 1. x ≥ −5 and x ≤ 5 4. x = −8 or x = 8 5. x = −14 or x = 14 2. x > −3 and x < 1 6. x = −17 or x =17 7. x = −7 or x = 3 3. x ≥ 3 and x ≤ 9 8. 4. x > −7 and x < 1 5. x < −3 or x > 3 6. x < 2 or x > 10 x = −2 or x = 2 7. x ≤ −9 or x ≥ −1 9. 8. x ≤ 0.5 or x ≥ 3.5 9. x − 350 ≤ 35; 315 ≤ x ≤ 385 10. x − 88 ≤ 7.5; 80.5 ≤ x ≤ 95.5 x = −6 or x = 2 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 496 7. x ≤ −2 or x ≥ 4 Practice and Problem Solving: C 1. x > −3 and x < 3 8. x < −5 or x > −1 2. x > 1 and x < 5 9. x − 85 ≤ 4 3. x ≥ −5 and x ≤ 1 10. 81 ≤ x ≤ 89; To get a B grade, the score must be greater than or equal to 81 and less than or equal to 89. 4. x < 1 or x > 9 Reading Strategies 5. x ≤ −3 or x ≥ 3 1. 5 2. 1 and 9 1 1 6. x ≤ −4 or x ≥ 3 2 2 3. 4. −1 5. −3 and 1 7. all real numbers 6. 7. 8. all real numbers Success For English Learners 9. x − 36.5 ≤ 1.5; 35 ≤ x ≤ 38 1. The solutions to x < 5 are between 5 and −5. The solutions to x > 5 are less 10. x − 23.5 ≤ 2.1; 21.4 ≤ x ≤ 25.6 than −5 and greater than 5. 11. Possible answer: Ben is correct. There is no solution. When the inequality is simplified, the result is an inequality that sets the absolute value of the expression to a negative number. Since absolute values are always positive, the inequality will have no solution. 2. −2 ≤ x + 4 and x + 4 ≤ 2 MODULE 13 Challenge 1. a. Practice and Problem Solving: Modified 1. 7; 7; 2; −2; 2 2. −3; 3; 1; 1; 1; 1; −2; 4 3. x > −4 and x < 4 b. Domain is all real numbers; range is. {y | y ≥ 0} . 4. x ≥ −4 and x ≤ 0 c. The part of the graph where the y-coordinates were negative has been reflected across the x-axis so that all y-coordinates are positive. 5. x ≥ −5 and x ≤ 5 6. x < −2 or x > 2 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 497 2. a. b. The part of the graph where the y-coordinates were negative has been reflected across the x-axis so that all y-coordinates are positive. c. y = 5 if x ≤ −2 or x ≥ 3, y = 2x − 1 if 1 < x < 3, and y = −2x + 1 if 2 1 −2 < x < 2 b. y = 5 if x ≥ 3 and y = 2x − 1 if −2 < x < 3 3. a. Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 498 UNIT 6 Exponential Relationships 10. 4 MODULE 14 Rational Exponents and Radicals 11. 3 12. 81 LESSON 14-1 13. 7 Practice and Problem Solving: A/B 14. Sample answer: By the Quotient of Power am Property, n = a m − n Suppose m = n = 2 . a 2 a a2 Then 2 = a 2− 2 = a 0 . And 2 must also a a equal 1. So, a0 must equal 1. 1. Power of a Product Property 2. Power of a Power Property 3. 4 4. 1 5. 3 15. Sample answer: Call the number x. The 6. 125 1 7. 32 cube root of x can be written as x 3 . Then the square root of the cube root of x can 8. 3 1 ⎛ 1 ⎞2 be written as ⎜ x 3 ⎟ . Finally, by the Power ⎝ ⎠ 9. 5 10. 196 11. 0.1 1 1 ⎛ 1 ⎞2 of a Power Property, ⎜ x 3 ⎟ = x 6 , and this ⎝ ⎠ is the sixth root of x. 12. 48 13. 12 14. 1 Practice and Problem Solving: Modified 15. 16 16. 2 seconds ( ) = ( 16 ) 1 4 17. 16 (16 ) 3 3 1 4 4 3 1. 4 = 2 = 8 and 3 2. 3 3. 100 1 4 = 4096 = 4 4096 = 8 4. 2 18. 150 square inches 1 5. 125 3 Practice and Problem Solving: C 3 1. 4 6. 5 4 12 2. a 3. 5 7. 64 6 1 , or 0.008 125 1 8. 10 2 4. 8b3 9 9. a 5. 0 6. n 6 10. m 7. 6400 42 11. c 1 12. 5 8. k 4 13. 2 9. w 2 14. 3 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 499 15. 8 14. 51.3 mph 16. 125 15. 4 cm 17. 16 Practice and Problem Solving: C 18. 2.5 seconds 19. 6 inches 9 20. 814 Reading Strategies 1. 3rd or cube 2. 5th 3. 5th; 4th 4. 2 5. 9 6. 2 7. 20 8. 4 9. 27 10. 8 11. 1024 12. 64 Success for English Learners Practice and Problem Solving: Modified 1. 3 2. 1 1. B 3. The exponent in the exponential expression is the quotient of the exponent in the radical expression and the root index. 2. D 3. C 4. A LESSON 14-2 1 5. x 5 Practice and Problem Solving: A/B 5 1. y 6. x 4 2. x2y6 2 7. 18 3 2 3. a b 6 4. 5y2 8. 10 2 5. x2y3 9. 7 6. 81y4 10. 3 7. 8y3 11. 1 8. x2y4 12. 12 9. 729y6 13. 32 10. (xy)2 14. x 8 11. x5 15. 14 cm 12. x 16. 6 s 13. 20 m Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 500 Reading Strategies 1. 125x MODULE 15 Geometric Sequences and Exponential Functions 12 2. 4 x 4 LESSON 15-1 Success for English Learners 3 1. They both have 2. Problem 2 has 3 3. 16 4 = = ( 4 16 ) (2) 4 4 = ( 2) 3 Practice and Problem Solving: A/B 8. 1. r = 3; 243, 729, 2187 1 4 2. r = ; 12, 4, 3 3 3. 192; 12 8 squared. 3 3 4. 1575 5. d = 5; 26, 31, 36 3 6. d = −3; −5, −8, −11 =8 2 3 4. 27 = = ( 3 27 ) ( ) 3 33 = (3) 7. 1.5 ft 2 8. $1871.77 9. B, C, D 64 10. 3 2 2 Practice and Problem Solving: C =9 1. r = 1.25; 7.8125, 9.7656, 12.2070 1 32 32 , − 2. r = − ; −32, − 3 3 9 3. −38.4, −60. MODULE 14 Challenge 1. Possible answers: 2, 15, π , 3 + 1, 4π , and the square root of any number that is not a perfect square. 4. 16.66 2. 5. d = 9.6; 36.8, 46.4, 56 6. d = −6.5; −15.5, −22, −28.5 7. 11.39 ft 8. $1766.10 9. A 10. 0.4219 3. No. The ratio of C to d for any circle is equal to π, which is not a whole number, c so if d were a whole number then d would be a rational number, and could not be π which is irrational. Practice and Problem Solving: Modified 1. r = 5; 1250, 6250, 31250 2. r = 6; 5184, 31,104, 186,624 3. 48, 12 4. The name of the point is π. One revolution is equal to the circumference of a circle c c with a diameter of 1. Since = = π , d 1 then C = π. 4. 96 5. d = 3; 18, 21, 24 6. d = −3; −7, −10, −13 7. Common difference: 5, 5, 5 8. Common ratio: 6, 6, 6 9. 15.2, 11.4 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 501 Reading Strategies Practice and Problem Solving: C 1. 2 1. f (n ) = 2. 3 2 n −1 (8) 3 2. f(1) = −10; f(n) = f(n − 1) i (0.4) for n ≥ 2. 3. Divide term 2 by term 1, or term 3 by term 2, etc. 3. r = 3, f(n) = 6(3)n − 1 4. 162 4. r = 0.5, a1 = 18; f(n) = 18(0.5)n − 1 5. 354,294 5. r = 0.01, a1 = 10000; f(n) = 10000 (0.01)n − 1 1 1 n −1 6. r = 0.25, a1 = ; f (n ) = ( 0.25 ) 3 3 7. r = 0.4, a1 = 200; f(n) = 200(0.4)n − 1 8 8 n −1 8. r = 1.5, a1 = − ; f (n ) = − (1.5 ) 3 3 9. r = 4, a1 = 2.5; f(n) = 2.5(4)n − 1 9 12 . No common ratio. 6. Not geometric. ≠ 6 9 7. −640 8. 18144 Success for English Learners 1. Each term is the product of r and the term before it. r = −4, a1 = 2.5; f(n) = 2.5(−4)n − 1 2. Multiply the last term by the ratio, and repeat. 10. 88,573 3. 1.5 11. 5% 4. No, the terms just keep getting smaller. 12. Tn = 64(0.5)n − 1 LESSON 15-2 Practice and Problem Solving: Modified Practice and Problem Solving: A/B 1. 3(4)0, 3(4)1, 3(4)2, 3(4)3, 3(4)4; f(n) = 3(4)n − 1 1. 3(4)0, 3(4)n − 1 2. f(1) = 11; f(n) = f(n − 1) i 2 for n ≥ 2. 3. 270 3. f(n) = 2.5(3.5)n − 1 4. 250 ⎛ 1⎞ 4. f(n) = 27 ⎜ ⎟ ⎝3⎠ 2. 6(2)n − 1 5. 189 n −1 6. 32 7. f(n) = 9(2)n − 1 5. f(1) = −4; f(n) = f(n − 1) i 0.5 for n ≥ 2 8. f(n) = 2(10)n − 1 6. r = 4; f(n) = 90(4)n − 1 7. r = 9. R: P(1) = 20,000; P(n) = p(n − 1) i 1.04 for 1 ; f(n) = 16(0.5)n − 1 2 n≥2 E: f(n) = 20,000(1.04)n − 1 8. r = 3; f(n) = 2(3)n − 1 9. r = 1 ⎛ 1⎞ , a1 = 90; f (n ) = 90 ⎜ ⎟ 3 ⎝3⎠ 10. P(5) = 23,397; P(10) = 28,466 n −1 Reading Strategies 1. a. a2, a5 n−1 10. r = 1.0025; f(n) = 18000(1.0025) b. r = 1.13, a1 = 708 11. f(5) = $18,180.68 c. 708(1.13)n − 1 12. f(60) = $20,856.96 Success for English Learners 1. f(n) = 3(2)n − 1; 3, 6, 12, 24 2. f(n) = 1 n−1 1 (4) ; , 2, 8, 32 2 2 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 502 6. 3. f(1) = 4, f(n) = f(n − 1) i 3 for n ≥ 2; 4, 12, 36, 108 1 1 , f(n) = f(n − 1) i 4 for n ≥ 2; , 4 4 1, 4, 16, 4. f(1) = LESSON 15-3 Practice and Problem Solving: A/B 1. y = 6(3)x 2. y = 84(0.25)x 3. 3 3 , , 3, 6, 12 4 2 7. 4. 16, 8, 4, 2, 1 8. 3.6 ft Practice and Problem Solving: C 1. 5. Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 503 2. Reading Strategies 1. y = 372,000(1.05)t; 549,613 2. y = 4200(1.03)t; 5165 3. y = 350,000(0.97)t; 291,540 4. y = 1200(0.98)t; 1085 Success for English Learners 1. a. (2, 96) and (3, 384) b. 4 c. 6 d. y = 6(4)x 3. y = 1000(.6)x e. 6144 4. 9 min 2. a. 3, 1500 5. $7401.22 b. y = 1500(3)x 6. No. The function cannot equal zero. Range is (0, ∞). LESSON 15-4 Practice and Problem Solving: Modified Practice and Problem Solving: A/B 1. 1, 2, 4, 8, 16; a = 4, b = 2, y-intercept = 4; end behavior = 0, ∞ 1. 18 2. 1 3. 0.5 4. 1 16 5. 21.6 6. 1 7. 0.1875 8. 18 9. 5x ⎛ 1⎞ 10. y = 81⎜ ⎟ ⎝3⎠ 11. 2. x x −2 −1 0 1 2 y 24 12 6 3 1.5 1 1 1 1 , , ,1, 3; a = , b = 3, y-intercept = 27 9 3 3 1 ; end behavior = 0, ∞ 3 12. 3125 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 504 3 3 3. − , − , − 3, − 6, −12; a = −3, b = 2, 4 2 y-intercept = −3; end behavior = 0, −∞ 2. 3 3 1 4. 12, 6, 3, , ; a = 3, b = , y-intercept = 3; 2 4 2 end behavior = 0, ∞ 1 1 1 3 9 1 , , , , ; a = , b = 3, 18 6 2 2 2 2 1 y-intercept = ; end behavior = 0, ∞ 2 3. 4. Practice and Problem Solving: C 1. 0.875, 1.75, 3.5, 7, 14; a = 3.5, b = 2, y-intercept = 3.5; end behavior = 0, ∞ 5. 9 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 505 Practice and Problem Solving: Modified 1. 4. 3 3 , , 3, 6, 12 4 2 2. 24, 12, 6, 3, 1 1 1 1 , , , 1, 3; a = , b = 3, 27 9 3 3 1 y-intercept = ; end behavior = 0, ∞ 3 Reading Strategies 3 2 1. Always 2. Sometimes 3. Always 4. Never 5. x −1 3. 1, 2, 4, 8, 16; a = 4, b = 2, y-intercept = 4; end behavior = 0, ∞ y − 3 2 0 −3 1 −6 2 −12 Success for English Learners 1. 54 2. I and II; III and IV; the y value has the same sign as a. 3. As x increases, y gets closer to zero; As x increases, y approaches positive infinity if a > 0 or negative infinity if a < 0. LESSON 15-5 Practice and Problem Solving: A/B 1. Y2 = 6(0.25)x 2. Y2 = 0.5(0.25)x 3. Y2 = (0.25)x − 4 4. Y2 = (0.25)x + 11 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 506 Reading Strategies 5. f ( x ) = 5 x ; g ( x ) = 0.4(5)x 1. Possible answer: The shapes of the curves are the same, but the curve for g(x) is shifted 4 units down from the curve for f(x). 6. It is a vertical compression. You can tell because multiplying f ( x ) by 0.4 brings the value in closer to 0. 7. They do not meet. For every value of x, g(x) = 0.4f(x). The only way that they could be equal would be if both equaled 0. But neither function ever equals 0. 2. Possible answer: The shapes of the curves are the same, but the curve for g(x) is shifted 5 units right of the curve for f(x). 8. Each value of h(x) is the opposite of f(x). For example, (0, 1) becomes (0, −1), and (1, 5) becomes (1, −5). So, the graph of h(x) is a reflection of f(x) across the x-axis. 3. Possible answer: The curve for g(x) is a vertical stretch of the parent function by a factor of 8. 4. Possible answer: The curve for g(x) is the curve for f(x) reflected across the x-axis. Practice and Problem Solving: C 5. g ( x ) = e x + 7 1. Y2 = 2(0.8)x + 8 6. g ( x ) = e − x 1 2. Y2 = (0.8)x − 12 3 7. g ( x ) = e 6 x 3. Y2 = −(0.8)x Success for English Learners −x 4. Y2 = (0.8) , or Y2 = (1.25) x 1. It is of the form f ( x ) = b x + k , where k is the vertical translation. 5. Y2 = −(0.8)x − 3 2. It is of the form f ( x ) = b x − h , where h is the horizontal translation. 6. Y2 = −(0.8)x + 3 7. Y2 = (0.8)− x − 10 , or Y2 = (1.25)x − 10 8. Y2 = (0.8) −x 3. It is the number that multiplies the exponential. For example, in Problem 2, 1 − represents a vertical compression and 3 a reflection. − 10 , or Y2 = (1.25) − 10 x 9. Y2 = −(0.8)− x , or Y2 = −(1.25)x 10. Y2 = (0.8)x 4. The exponential is multiplied by a negative number. Practice and Problem Solving: Modified MODULE 15 Challenge 1. Y1 : 1, Y2 : 3 1. 2. The graph of Y2 is three times farther away from the x-axis. 3. Possible answer: y = 0.5(0.4)x 4. y = (0.4)x + 5 5. f ( x ) = 2 x 6. It is 4 times greater. 7. g ( x ) = 4 i (2)x The bases are reciprocals. The graphs are reflections across the y-axis. 8. It is a vertical stretch of f ( x ) . You can tell because multiplying f ( x ) by 4 moves the graph away out from the x-axis. Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 507 2. 10. f(x) = 11; g ( x ) = 9 x ; The bases are reciprocals. The graphs are reflections across the y-axis. x ≈ 1.1 11. f(x) = 120; g ( x ) = 12x ; 3. The bases are reciprocals of one another and the graphs are reflections of one another across the y-axis. 4. Because 10 and 1 are reciprocals 10 1 ⎛ 1⎞ 5. Because 0.2 = ⎜ ⎟ and 5 and 5 ⎝5⎠ are reciprocals x 1 ⎛ 1⎞ are 6. Because 4−x = ⎜ ⎟ and 4 and 4 ⎝4⎠ reciprocals −x ⎛5⎞ 7. Because 1.25 = ⎜ ⎟ ⎝4⎠ 4 are reciprocals and 5 −x x ≈ 1.9 12. 600(1.05)x = 900; 8.3 years 13. 20,000(1.035)x = 40,000; 20.1 years x 5 ⎛4⎞ = ⎜ ⎟ and 4 ⎝5⎠ Practice and Problem Solving: C 1. x = 5 2. x = 9 3. x = 4 MODULE 16 Exponential Equations and Models 4. x = 0 5. x = −1 LESSON 16-1 6. x = −2 Practice and Problem Solving: A/B 7. x ≈ 1.1 1. x = 4 2. x = 5 3. x = 4 4. x = 4 5. x = 3 6. x = 2 7. x = 2 8. x = 5 9. x = 3 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 508 8. x ≈ 0.8 8. f(x) = 32; g ( x ) = 3 x ; x ≈ 3.2 9. x ≈ 6.5 9. 400(1 + 0.08)t = 700, about 6.5 years 10. Not exactly. It takes about 23.4 years at 3% and about 11.9 years at 6%. The time at 6% is slightly more than half the time at 3%. 10. 10,000(1 + 0.04)t = 20,000; t ≈ 17.7 years Reading Strategies 1. Possible answer: 240 < y < 250. 11. There were 225 years from 1789 to 2014. So, the penny would be worth 0.01(1.05)225 ≈ $585.59 2. Possible answer: 4000 < y < 4200 3. Possible answer: −2 < y < 5 4. Possible answer: 250 < y < 260 Practice and Problem Solving: Modified 5. Possible answer: 25 < y < 35 1. x = 5 6. Possible answer: 2 < y < 10 2. x = 3 7. 3. x = 4 4. x = 3 5. x = 2 6. x = 4 x = −2 7. 8. x=2 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 509 9. 5. y = 20,000(1.05)t x=3 10. 6. y = 45,000(0.8)t x=4 Success for English Learners 1. Instead of 34, it would be 33, which would make the final answer x = −5. Practice and Problem Solving: C 2. Set exponents equal. 1. v1(t ) = 10,000(1.04)t; 3. Yes, because an exponent can be a fraction. v 2 (t ) = 8000(1.06)t 4. Enter the value for the variable back into the original equation. Evaluate to see if the equation is true. 2. v1(5) = $12,166.53 ; v 2 (5) = $10,705.80; the difference is less because the smaller investment is growing at a greater interest rate. LESSON 16-2 3. Yes, the value of Investment 2 will exceed the value of Investment 1 by the end of Year 12. At that point, v 2 (12) = $16,097.57 and v1(12) = $16,010.32. Practice and Problem Solving: A/B 1. y = 650,000(1.04)t ; sales ≈ $790,824.39 D = set of real numbers t ≥ 0 R = set of real numbers y ≥ 650,000 4. Odette would earn more. For example, at the end of 1 year, Investment 1 is worth $10,400 using annual compounding and $10,408.08 using daily compounding. 2. y = 800(1.02)x ; population ≈ 901 students D = set of real numbers t ≥ 0 5. Graham is incorrect. Even though the car loses 20% each year, that 20% is taken from the original amount only in the first year. The correct way to figure the value after 5 years is with the expression (0.8)5 =0.32768. So, a car has 32.768% of its original value after 5 years. R = set of real numbers y ≥ 800 3. y = 2500(0.97)t ; population ≈ 2147 people D = set of real numbers t ≥ 0 R = set of real numbers 0 ≤ y ≤ 2500 6. The workers are making less now than they did before the pay cut. Specifically, let S be a salary. After the pay cut, the salary was 0.9S. Then, after the 10% raise, the salary became (1.1)(0.9S) = 0.99S. So, the final salary is 1% less than the original salary. 4. y = 25,000(0.85) ; value ≈ $6,812.26 t D = set of real numbers t ≥ 0 R = set of real numbers 0 ≤ y ≤ 25,000 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 510 Practice and Problem Solving: Modified Reading Strategies 1. a. exponential decay 3 1. y = 270,000(1 + 0.07) = $330,761.61 b. P (6) = 1250(0.97)6, 1041 birds 2. 2200; 0.02; 6; 2478 people 2. a. exponential growth 3. 200(1 + 0.08) ; $503.63 12 b. P (20) = 3800(1.02)20 , 5647 people 4. y = (1 − 0.02)4 ≈ 738 3. a. exponential growth 5. 2300; 0.04; 10; 1529 birds 0.04 ⎞ ⎛ b. P (12) = 800 ⎜ 1 + 12 ⎟⎠ ⎝ 6. y = 30,000(0.82)t x y 0 2 4 6 8 (12×10) , $1192.67 Success for English Learners 30,000 20,1722 13,564 9,120 6,132 1. The amount of interest earned each period is paid on the original amount and accumulated interest. 2. It gets compounded three more times. LESSON 16-3 Practice and Problem Solving: A/B 1. y = 9.186(1.029)x 2. 2.9% 3. and 4. Major League Baseball Total Attendance (yd), in millions, vs. Years Since 1930 (x) x 0 10 20 30 40 50 60 70 80 yd 10.1 9.8 17.5 19.9 28.7 43.0 54.8 72.6 73.1 ym 9.2 12.2 16.3 21.7 28.8 38.4 51.1 68.0 90.4 residual 0.9 −2.4 1.2 −1.8 −1.0 4.6 3.7 4.6 −17.3 7. The prediction for 2020 is 9.186(1.029)90 ≈ 120.4 million. This does not seem reasonable. The fact that attendance only increased from 72.6 million to 73.1 million between 2000 and 2010 makes it very unlikely that it would jump so much by 2020. 5. Possible answer: The residuals behave well until the last column in the table. It seems like the equation is a good fit through the year 2000 but then does not work very well. 6. The correlation coefficient is approximately 0.986. This is very close to 1, and so it suggests that the equation is a good fit for the data. Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 511 Practice and Problem Solving: C 1. y = 91.213(0.954)x 2. Temperature of Water (yd), in degrees Celsius, after cooling for x minutes x 0 5 10 15 20 25 30 35 40 45 50 yd 100 75 57 44 34 26 21 17 14 11 10 ym 91.2 72.1 57.0 45.0 35.6 28.1 22.2 17.5 13.9 11.0 8.7 0 −1.0 −1.6 −2.1 −1.2 −0.5 0.1 0 1.3 residual 8.8 2.9 3. y = −1.64x ++ 78.182 4. Temperature of Water (yd), in degrees Celsius, after cooling for x minutes x 0 5 10 15 20 25 30 35 40 45 50 yd 100 75 57 44 34 26 21 17 14 11 10 ym 78.2 70.0 61.8 53.6 45.4 37.2 29.0 20.8 12.6 4.4 −3.8 residual 21.8 5.0 −4.8 −9.6 −11.4 −11.2 −8.0 −3.8 1.4 6.6 13.8 5. Possible answer: Both sets of residuals show a pattern where they run from positive to negative and back to positive, which is not good. But the exponential model shows much smaller residuals overall. So, it seems to be the better model. 6. For the exponential equation r = −0.996. For the linear equation r = −0.928. The exponential equation is the better model since its r value is closer to −1. Practice and Problem Solving: Modified 1. y = 2.80(1.05)x 2. and 3. U.S. First-Class Postage Rate (yd) vs. Years Since 1950 (x) x 0 10 20 30 40 50 60 yd 3 4 6 15 25 33 44 ym 2.80 4.56 7.43 12.10 19.71 32.11 52.30 residual 0.20 −0.56 −1.43 2.90 6.29 0.89 −8.30 4. Possible answer: The residuals are getting pretty big, relative to the data. So, it seems like the equation might not be such a good fit. Reading Strategies 1. y = 15000(0.75)x 2. y = 5.06(1.30)x 5. The correlation coefficient is approximately 0.985. This is very close to 1, and so it suggests that the equation is a good fit for the data 6. 2.80(1.05)63 ≈ 61 cents. 3. y = 15.02(1.25)x 4. y = 100.07(0.92)x Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 512 6. f(20) = $22,000 and g(20) = $26,532.98; The compound interest account is the better choice. Success for English Learners 1. 1.36; 1.36; 1.35 2. Yes. It has a constant ratio of y-values for equally spaced x-values. 7. 8.2 years 8. Possible answer: If you want to invest for 8 years or less, in this case (for these interest rates) choose simple interest. Otherwise, choose compound interest. 3. V(w) = 814.96 (1.38)w. 2,325. LESSON 16-4 Practice and Problem Solving: A/B Practice and Problem Solving: C 1. Neither (or both). The account balance initially changes at a constant amount per month and then that changes to a constant percent per year. 1. Constant percent per unit interval. The amount given to Josh is doubling with each new book. So, the amount is increasing at a rate of 100%. 2. Constant amount per unit interval. Jin Lu’s bonus is increasing by $50 for each new sale. 2. Constant amount per unit interval. The amount of increase is $15 per year. 3. f ( x ) = A + 0.04 Ax, or f ( x ) = A(1 + 0.04 x ); 3. f ( x ) = 10,000 + 600 x; g ( x ) = 10,000(1.05) g ( x ) = A(1.035)x x 4. f (3) = 1.12 A and g (3) ≈ 1.109 A. So, the simple interest rate would be better. 4. f(x) is a linear function and g(x) is an exponential function. You can tell from their equations or from their original descriptions. Simple interest makes a function grow along a straight line. Compound interest grows exponentially. 5. f (15) = 1.6 A and g (15) ≈ 1.675 A. So, the compound interest rate would be better. 6. 8.6 years 7. No, the amount deposited does not matter. For example, in Problem 4, the fact that 1.12 > 1.109 makes you decide to choose the simple interest rate. It has nothing to do with the amount deposited. 5. f(3) = $11,800 and g(3) = $11,576.25; The simple interest account is the better choice. Practice and Problem Solving: Modified 1. Constant amount per unit of time. The change is $0.10 per month. 2. Neither (or both). It starts with a constant amount of change and switches to a constant percent of change. 3. Constant percent per unit of time. The change is 3% per year. 4. Day 0 1 2 3 4 5 6 7 8 f(x) $1.00 $1.05 $1.10 $1.15 $1.20 $1.25 $1.30 $1.35 $1.40 g(x) $1.00 $1.04 $1.08 $1.12 $1.17 $1.22 $1.27 $1.32 $1.37 8. Possible answer: Dana made a bad decision. At first, the 5 percent plan works out better than the 4% plan. But, eventually, the 4% plan works out better for Dana. For example, by the 20th day, the 4% plan is already $0.19 better. 5. f(x) is a linear function and g(x) is an exponential function. 6. f ( x ) = 1 + 0.05 x; g ( x ) = (1.04)x 7. f (20) = 2; g (20) = 2.19. Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 513 Reading Strategies MODULE 16 Challenge 1. 1. a. Differences are all 2. Square n Grains of Wheat on Square n Total Grains of Wheat on Board 1 1 1 2 2 3 3 4 7 3. Possible answer: The graph of f(x) is a straight line, and the graph of g(x) is a curve; the range of f(x) is all real numbers, but the range of g(x) is y > 0; g(x) has an asymptote at y = 0. 4 8 15 5 16 31 6 32 63 4. Exponential function; ratios of the 3 differences are the same, 4 7 64 127 8 128 255 9 256 511 10 512 1,023 b. Possible answer: because all the differences are the same 2. a. 0.5, 1, 2, 4, 8 b. All ratios are 2. c. Possible answer: because the ratios of the differences are all the same Success for English Learners 1. It is an example of an exponential model because the number of teams in the tournament is decreased by a multiple of 2 each round. 2. 2n − 1 3. 263 = 9,223,372,036,854,775,808 2. No. An exponential model needs to have constant ratios, so differences won’t help. 4. 2n − 1 5. 264 − 1 = 18,446,744,073,709,551,615 or about 1.845 × 1019 6. 147, 573, 952, 589, 676 kilograms or about 1.475 × 1014 7. about 254.4 years Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 514 UNIT 7 Polynomial Operations MODULE 17 Adding and Subtracting Polynomials 12. 14 1 3 11 p − 1 pq 4 24 13. 5x2 − 2x − 4; 4.75 LESSON 17-1 14. 6x3 − 4x2 + 7; 48 Practice and Problem Solving: A/B 15. 192 ft; 192 ft; they are the same because at 2 seconds the rocket is ascending and at 3.5 seconds the rocket is descending. 1. binomial; degree 2 2. trinomial; degree 6 Practice and Problem Solving: Modified 3. monomial; degree 4 4. none of the above 5. trinomial; degree 7 1. monomial; degree 2 6. none of the above 2. binomial; degree 3 7. 3n4 + 6n3 + 4n2 3. monomial; degree 6 8. −2c3 − 2c 4. trinomial; degree 5 5. trinomial; degree 4 9. 9b2 + b − 9 6. binomial; degree 1 10. −2a4b3 + 5a3b4 7. 5n2 + 3n 11. 5x2 + 15x − xy 8. 5c3 − 2c 12. p2q + 13p3 + 2p 9. b − 9 13. 5x2− 2x − 4 10. 4a4 − 9a3 − 4a 14. 7x3− 6x2 + 4 11. −4x2 + 5x 15. 192 ft 12. 13p2 + p − 6 16. 33b − 8 13. 19; 35; 5x2− 2x − 4 Practice and Problem Solving: C 2 14. 39; 36; 6x3− 4x2 + 7 2 1. 3ab − 3a b; binomial; degree 3 15. 296 ft 2. 8xy − 9x + 2y2; trinomial; degree 2 16. 10x3 − 14x2 3. −4 y y ; none of the above Reading Strategies 4. 3n2 + 11n; binomial; degree 2 1. 3; 1 5. b2 − 3b6; binomial; degree 6 6. 2. The exponents on the variables have a sum of 4. 3 x ; none of the above 5 3 7. 9mn + 14mn 3. cubic; binomial 2 4. −4g2 + 8g + 1 8. −2.1c3 + 12.9c2 − 2.4c 5. −4 1 7 11 9. 9 b 2 + 7 b − 4 6 12 12 6. 2 4 3 3 4 7. trinomial 2 5 10. −5a b − a b − 2a b 8. quadratic 2 11. 11.6x + 2.7x − 5.5xy Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 515 6. 21a4 + 4a2 + 2a Success for English Learners 7. 4x3 − 6 1. 3xy; Possible answers: 3x + y; 3 + xy; 3y + x 8. 12g2 + 4g − 1 2. A number is not a variable so it does not have a degree. 9. 20p5 + 14 10. 13b2 + 5b + 7 LESSON 17-2 11. a. 12n + 28 b. 8n + 20 Practice and Problem Solving: A/B 2 1. 12g + 4g − 1 3 Reading Strategies 2 2. 7x + 2x + 6x 1. Possible answer: Grouping like terms is similar to aligning like terms because you have to match the like terms. 3. 13b2 + 5b + 7 4. −2c3 + 3c2 − 2c 2. 9x2 + 16x + 2 5. 4ab2 + 20b − 3a 3. 2x3 − 2x2 + 3 6. −13r2 + 6pr + 7p Success for English Learners 7. 5y2 + y + 12 1. Represent m2 with a square with side length equal to m. 8. 6z3 + 4z2 + 5 9. 9s3 + 13s 2. 15, 2, 6 10. 21a4 + 4a2 + 2a 3. Like terms need to be grouped together before adding. 11. −3a2b3 − 2a3b − 8ab 12. 10p4q2 + 2p3q − 3pq LESSON 17-3 13. 16x − 2 Practice and Problem Solving: A/B Practice and Problem Solving: C 1. 3g2 + 4g − 19 2 1. 4ab − 3a + 20b − 3 3 2. 6x3 + 3x2 2 2. 10x − x − x − 4 3. 8b2 + 4b − 3 3. −10r2 + 6pr + 7p 4. 10c3 − 7c2 + 4c 2 4. 7rs − 3s − 5 5. 10ab2 + 4b − 4a 5. 4x2 − 16 6. 3x3 + 4x2+ 6x 6. Possible answer: (6n + 2) + (3n − 2) = 9n 7. 3y2 − 9y + 4 2 7. Possible answer: (n + 6n + 2) + (n2 + 3n − 2) = 2n2 + 9n 8. 2z3 + 4z2 + 11 9. 7s3 + s + 19 n 4 5 n 3 3n 2 n 8. + + + ; 3410 4 6 4 6 10. a4 + 14a2 11. −2(a2)(b3) + 2(a3)b − 2ab 9. 20x + 30 12. −2p4q2 + 10p3q + 6 Practice and Problem Solving: Modified 13. c2 − 15c − 100 14. 2x3 − 22x2 1. 3m + 6 Practice and Problem Solving: C 2. 5y2 + y + 12 3. 6z + 4z + 5 1. −2ab2 + 5a + 20b − 15 4. 16k + 5 2. 10x3 − 3x2 + 5x − 4 5. 9s3 + 13s 3. 14r 2 + 10pr + 7p 3 2 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 516 4. −rs2 − 5s − 5 MODULE 17 Challenge 2 5. 2y − 4y + 5 1. 2t 2 + 55t 6. Possible answer: (6n + 2) − (3n − 4) = 3n + 6 2. 55t 7. Possible answer: (n2 + 6n) − (3n − 4) = n2 + 3n + 4 4. 2t 2 + 175t ; 450 mi 3. 65t 5. Anusha drove 2(2.5)2 + 55(2.5) = 150 mi. Bella drove 55(2.5) = 137.5. Celia drove 65(2.5) = 162.5 miles. Celia drove the farthest, and Bella drove the least distance. 8. 10p + 16 9. a. c3 − 2c2 + 6c + 200 b. difference is $590 Practice and Problem Solving: Modified MODULE 18 Multiplying Polynomials 1. 4p + 4 2. 4y2 − 3y + 1 3. 3z3 + 5z2 + 7 LESSON 18-1 4. 10k + 4 Practice and Problem Solving: A/B 5. 2s + 2s + 40 1. 10x5y3 6. 15a4 + 11a2 + 5a 2. −15p4r2 7. 3x3 + 15 3. 22 a6b6 8. 5g2 + 4g − 5 4. 18c5d6 9. 4p5 + 2 5. 12a2 + 8a − 28 3 10. 2b2 + 5b − 6 6. 9x5 − 36x4 − 27x3 11. w + 8 7. −12s5 + 24s4 − 60s3 12. 10p + 200 8. 30a8 − 10a6 − 5a5 9. −56pr3 − 16p2r2 + 64p2r Reading Strategies 10. 6m2n6 + 2mn5 + 8m2n4 1. 4 11. −6x6y2 − 15x5y3 − 27x4y4 2. −1 12. 9v5w3 + 12v4w4 − 6v2w5 3. 4x and −2x; 8 and 12 13. −28a4b6 − 7a3b4 + 35a5b4 4. x2 and −3x2; 8x and 3x; −4 and −2 14. 16p8q4 − 6p7q3 + 10p6q3 5. 2x3 + 2x + 7 15. a. w(w + 3) or w2 + 3w 6. −8x5 + 2x4 b. 28 in.2 7. 9x2 + 16x + 2 16. a. w(3w − 8) or 3w2 − 8w 8. 2x3 − 2x2 + 3 b. 220 cm2 9. 4x4 + 2x − 2 Practice and Problem Solving: C 10. −2x3 + 3x2 − 10x + 4 1. 4m6 Success for English Learners 2. −27x9 1. 1; −3 3. 2. You must first distribute the negative sign. 3. The Commutative Property of Addition 1 3 3 xy 3 4. −36c6d7 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 517 4. polynomial 1 5. 3 x 3 + 5 x 2 + 2 x 2 4 3 6. 2x − 3.2x − 0.56x 5. monomial 2 6. polynomial 7. −6v + 6v 7. monomial 8. 28a8 − 20a6 − 5a5 + 6a 8. polynomial 9. −45s3 9. polynomial 3 10. 6j 2k6 − 7j 3k3 + 7j 2k4 10. monomial 11. −6x6y2 − 14x5 y3 − 29x4y4 11. polynomial 12. 41pr3 + 2p2r2 − 16p2r 12. polynomial 13. a. 3x3 − 27x2 + 15x Success for English Learners 1. Possible answer: Subtracting is the same as adding a negative, which would give a −4mn. Multiplying that by −5 is multiplying two negatives, which makes a positive. So it would change to addition. b. 5x3 − 30x2 − 45x c. 2x3 − 3x2 − 60x 14. Possible answer: 8a(3a2b2 − 2ab); 4a(6a2b2 − 4ab); 8ab(3a2b − 2a); 8a2(3ab2 − 2b) 2. Possible answer: Because the monomial you are multiplying by does not have a variable or power. Practice and Problem Solving: Modified 3. Possible answer: All of the positive numbers in the answer would be negative, and all the negatives would be positive. 1. 32x6 2. 15p4 3. 22a7b4 LESSON 18-2 4. 18c5d Practice and Problem Solving: A/B 5. 45r 4s3 1. x2 + 11x + 30 6. −16x7y5 2. a2 − 10a + 21 7. 21a2 + 14a − 49 3. d 2 + 4d − 32 8. 27x2 − 36x − 27 4. 2x2 + 5x − 12 9. −12s5 + 24s4 − 60s3 5. 5b2 − 9b − 2 10. 30a6 − 10a4 − 5a2 6. 6p2 + 5p − 6 11. −56r3 − 16pr2 + 64rp 7. 10k2 − 38k + 36 12. 6n6 + 2m2n5 − 8n4 8. 6m2 + m − 40 13. −24x6y2 + 15x5y3 − 27x4y4 9. 20 + 3g − 56g2 14. 10v5w3 + 20v4w4 − 5v2w5 10. r 2 − 4rs − 12s2 15. a. w b. w + 5 11. 6 − 19v + 10v2 c. w2 + 5w 12. 25 − h2 d. 24 in.2 13. y2 − 9 e. 126 in.2 14. z2 − 10z + 25 15. 9q2 − 49 Reading Strategies 16. 16w2 + 72w + 81 1. monomial 17. 9a2− 24a + 16 2. polynomial 18. 25q2− 64r2 3. polynomial Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 518 19. x3 + 7x2 + 17x + 20 3 Practice and Problem Solving: Modified 2 20. 3m − 5m + 3m + 20 21. 8x3 − 26x2 + 17x − 5 1. x2 + 7x + 10 22. 5x2 + 6x + 1 2. x2, −3x, 4x, −12; x2 + x − 12 23. 105 in.2 3. x2 + 11x + 30 24. 3x2 − 12; $351 4. a2 − 10a + 21 5. d2 + 4d − 32 Practice and Problem Solving: C 6. x2 + 10x + 25 1. 2x2 + 22x + 60 7. x2 − 20x + 100 2. 3a2 − 30a + 63 8. x2 − 49 3. −160 + 20d + 5d 2 9. x2 + 8x + 16 4. 8x2 + 20x − 48 10. b2 − 4b + 4 5. 30b2 − 54b − 12 11. p2 − 81 6. −12p2 − 10p + 12 12. x3, 4x2, 7x, 3x2, 12x, 21; x3 + 7x2 + 19x + 21 7. 20(k3) − 76(k2) + 72k 8. 6m4 + m3 − 40m2 2 3 9. −160g − 24g + 448g 3 2 2 10. r s − 4r s − 12rs 2 11. 24v − 76v + 40v 2 12. 150h − 486h 13. y3 + 8y2 + 17y + 10 4 14. p3 + p2 − 14p − 8 3 15. n3− 6n2 + 9n − 2 3 16. x2 + x − 6 4 Reading Strategies 13. 4y5 − 9y 1. 4 14. 108z2 − 180z + 75 2. They have the same exponent on the same variable. 15. 36c3 − 196cd2 16. −48w3 − 216w2 − 243w 3. There are no like terms. 17. 18a3 − 48a2 + 32a 5 18. 25q r − 64qr 4. −6x5 + 12x4 − 3x3 5 5. 18x3 + 57x2 + 30x 19. 6x3 − 11x2 − 18x + 7 6. 7x2 − 19x − 6 20. 20z3 + 24z2 − 5z − 6 7. 2x5 − 10x4 + 8x3 − 22x2 − 34x + 8 3 2 21. 25x + 20x − 51x + 18 Success for English Learners 22. 8x3 + 36x2 + 54x + 27 1. The final product is x2 − 3x − 10. 23. Substitute 4 for x and evaluate the polynomial; 1,331 in.3; substitute 4 in 2x + 3 and cube it. The results should be the same. 2. There is 1 x2-tile, 10 x-tiles, and 25 1-tiles. 3. 24. n2 − 1; n3 − 1; n4 − 1; the greatest power of n in the polynomial is increased by 1 and the product is the difference between that power of n and 1; n5 − 1 x2 x x x x x 1 1 1 1 x 1 1 1 1 x 1 1 1 1 x 1 1 1 1 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 519 LESSON 18-3 14. 0.64x4 − y4 Practice and Problem Solving: A/B 15. 16x6y2 − 25x2 1. x2 + 4x + 4 2. m2 + 8m +16 3. 9 + 6a + a2 b. 10 9 − x2 16 c. 19 11 16 4. 4x2 + 20x + 25 5. 64 − 16y + y2 6. a2− 20a + 100 17. x − 3.5 7. b2 − 6b + 9 Practice and Problem Solving: Modified 8. 9x2 − 42x + 49 9. 36 − 36n + 9n2 1. x; x; 5; 5; x2 + 10x + 25 2 10. x − 9 2. m; m; 3; 3; m2 + 6m + 9 11. 64 − y2 3. 2; 2; a; a; 4 + 4a + a2 2 12. x − 36 4. x2 + 8x + 16 13. 25x2 − 4 14. 16 − 4y 5. a2 + 14a + 49 2 2 15. 100x − 49y 6. 64 + 16b + b2 2 7. y2 − 8y + 16 16. a. 36 − x2 b. 4 − x 1 − x2 4 16. a. 30 8. y; y; 6; 6; y2 − 12y + 36 2 9. 9; 9; x; x; 81 − 18x + x2 c. 32 10. x2 − 20x + 100 17. a. 16 − x2 11. b2 − 22b + 121 b. 20 12. 9 − 6x + x2 Practice and Problem Solving: C 13. x; 7; x2 − 49 2 1. 9x + 6x + 1 14. 4; y; 16 − y2 2 2. 25m + 5m + 0.25 3. 49 + 28a + 4a 2 4. 4x + 12xy + 9y 4 2 16. x2 − 64 2 5. 4a + 36a b + 81b 4 15. x; 2; x2 − 4 2 17. 9 − y2 2 2 2 6. 25a + 40a b + 16b 18. x2 − 1 4 19. x2 − 16 ⎛ 1⎞ ⎛ 1 ⎞ 7. (y4) − ⎜ ⎟ (y2) + ⎜ ⎟ ⎝2⎠ ⎝ 16 ⎠ Reading Strategies 1. difference of squares 1 8. − y 2 4 2. perfect-square trinomial 1 9. a 6 − 3a 3 + 9 4 4. c4 + 20 c2d + 100d 2; perfect-square trinomial 3. It will have 3 terms. 10. x2 − 0.36 6 5. 4s2 − 9; difference of squares 3 11. 9(x ) − 42(x ) + 49 12. x2 − 0.0625 13. a4b2 − a2b2 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 520 3. The coefficients of (x + y)n are in row (n + 1) of the triangle. Success for English Learners 1. The middle term in a square of a sum is positive and in a square of a difference it is negative. 4. (x + y)8 = x8 + 8x7y + 28x6y2 + 56x5y3 + 70x4y4 + 56x3y5 + 28 x2y6 + 8xy7 + y8 2. There are 3 terms. 5. 112 = 121 3. There are 2 terms. 113 = 1,331 114 = 14,641 MODULE 18 Challenge 115 = 161,051 1. Row 7: 1, 6, 15, 20, 15, 6, 1 6. 116 = 1 (6 + 1) (5 + 2) (0 + 1) 5, 6, 1 = 1,771,561 Row 8: 1, 7, 21, 35, 35, 21, 7, 1 Row 9: 1, 8, 28, 56, 70, 56, 28, 8, 1 2. (x + y)2 = x2 + 2xy + y2 (x + y)3 = x3 + 3x2y + 3xy2 + y3 (x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4 (x + y)5 = x5 + 5x4y + 10x3y2 + 10 x2y3 + 5xy4 + y5 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 521 UNIT 8 Quadratic Functions 6. Since (1, 5) is on the graph of f, an equation for f is y = 5x2. Since (1, 0.2) is on the graph of g, an equation for g is y = 0.2x2. The graph of g is wider (vertical compression) than the graph of y = x2. This follows because 0.2 < 1 and 5 > 1. MODULE 19 Graphing Quadratic Functions LESSON 19-1 Practice and Problem Solving: A/B 1. a. upward Practice and Problem Solving: Modified b. minimum 0 c. no 1. (0, 0), 0, none, y = x2, compression d. stretch 2. (0, 0), none, 0, y = x2, compression 2. a. downward 3. a = 5 b. maximum 0 4. a = −3 c. yes 5. 0, 0 d. stretch Reading Strategies 3. a. downward b. maximum 0 1. B, C c. yes 2. A, C, D d. stretch 3. a. Possible answer: The graph is a parabola that opens downward. Its vertex is at the origin. It is a vertical stretch of the graph of the parent function, and so it is narrower than the graph of the parent function. 4. a. upward b. minimum c. no d. compression b. maximum value 5. (0, 0), 0, none, no, stretch 4. Possible answer: The graphs are alike in that they are both parabolas with vertex at the origin. Neither is a vertical stretch or compression of the other, and so they both have the same width. They are different in that the graph of f(x) = x2 opens upward, while the graph of g(x) = −x2 opens downward. This means that zero is the minimum value of f(x) = x2, but it is the maximum value of g(x) = −x2. 6. (0, 0), none, 0, yes, stretch 7. 3 Practice and Problem Solving: C 1. a. y = −4x 2 b. stretch 2. a. y = 0.5x 2 b. compression (shrink) 3. y = −2.5x 2 Success for English Learners 4. y = −3.5x 2 1. When the value of a is negative, the graph of f(x) = x2 is reflected across the x-axis and opens downward. 5. If (m, n) is on the graph of y = ax2, then, by substitution, n = a(m2). By division by n m2, a = 2 . m 2. The value of a is −2. When the value of a is less than −1, the graph of f(x) = x2 is stretched vertically. Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 522 3. The graph of g(x) is a parabola that opens downward and has the same width as the graph of f(x) = x2. Possible explanation: The expression −x2 is equivalent to −1x2, and so the value of a is −1. Since the value of a is negative, the graph is the reflection of f(x) = x2 when it is reflected across the x-axis. That is the reason the graph of g(x) opens downward. For every x, the value of g(x) is the opposite of the value of f(x). That is the reason the graph of g(x) has the same width as the graph of f(x). Practice and Problem Solving: C 1. (3, 4) 2. down 3. 4 4. −3 5. 2 6. positive 7. y = (x + 3)2 + 2 8. LESSON 19-2 Practice and Problem Solving: A/B 1. (3, −4) 2. up 3. −4 4. 2 9. 5. −4 6. y =(x − 2)2 − 4 7. 10. (4, 8) 11. 8. 9. (5, 9) 10. x = 5 11. (4, 7) and (6, 7) 12. 12. At x = 2 and x = 6 the ball is at y = 0 or ground level. Practice and Problem Solving: Modified 1. 3 to the right 2. down 4 3. (3, −4) Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 523 4. −4 2. 5. −2 6. y = (x + 4)2 − 2 7. 2 8. −2 3. ⎛ 1⎞ 9. ⎜ ⎟ ( x − 2)2 − 2 ⎝2⎠ 10. 4. 11. 5. Reading Strategies 1. (−8, −10); left 8; down 10; a = 3; up; stretch; x = −8; maximum value none; minimum value y = −10 1 2. (5, 7); right 5; up 7; a = − ; down; 2 compression; x = 5; maximum value y = 7; minimum value none 6. Success for English Learners 1. LESSON 19-3 Practice and Problem Solving: A/B 1. Quadratic 2. Not quadratic 3. Not quadratic 4. y = x 2 + x + 2, x = − 1 2 5. y = − x 2 + 2 x − 1, x = 1 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 524 6. y = −5 x 2 + 2 x − 2, x = 1 5 5. y = 2 x 2 − x − 1, x = 7. y = 2 x 2 + 12 x + 12 6. y = 5 x 2 + 2 x − 2, x = − 8. y = 3 x 2 − 30 x + 79 1 5 7. y = x 2 + 2 x + 3 9. y = 5( x − 1) − 3 2 ⎛ 1⎞ 8. y = ⎜ ⎟ x 2 + 1 ⎝9⎠ y = 5 x 2 − 10 x + 2 10. y = 4( x + 3)2 − 2 9. y = 1( x − 0)2 + 0 y = 4 x + 24 x + 34 2 y = x2 11. y = −( x − 3)2 + 4 10. y = 1( x − 0)2 + 2 Practice and Problem Solving: C y = x2 + 2 1. Quadratic 11. y = 2. Quadratic 3. Not quadratic 4. y = 2 x 2 + 3 x + 2, x = − 1 4 Reading Strategies 3 4 5. y = −2 x 2 + 1.5 x − 0.5, x = 1 ( x − 0)2 + 1 9 1. Possible answer: y = 2( x + 2)2 + 3 2. Possible answer: x = −2 3 8 Possible answer: ( −2, 3) 6. y = −5 x − 2, x = 0 3. Possible answer: y = 2 x 2 + 8 x + 11 7. y = 3 x 2 + 3 x − 1.65 4. Possible answer: x = −2 2 Possible answer: ( −2, 3) 2 3⎛ 7⎞ 5 x− ⎟ + 2 ⎜⎝ 2⎠ 2 2 y = −6 x + 30 x − 34 8. y = 5. y = 2(x − 5)2 − 2 y = 2(x2 − 10x + 25) − 2 y = 2x2 − 20x + 50 −2 9. y = 1.5( x − 1)2 + 2 y = 2x2 − 20x + 48 y = 1.5 x 2 − 3 x + 3.5 Success for English Learners 10. y = −7( x − 3)2 + 5 1. Compare the equation to the vertex form y = a( x − h )2 + k . The coordinates of the vertex are the ordered pair (h, k ) . y = −7 x 2 + 42 x − 58 2 11. y = 3⎛ 7⎞ 5 x− ⎟ + ⎜ 2⎝ 2⎠ 2 2. Compare the equation to the standard form f ( x ) = ax 2 + bx + c. The coordinates of the vertex are the ordered pair ⎛ −(b ) ⎛ −(b ) ⎞ ⎞ , f⎜ ⎜ ⎟⎟ . ⎝ 2a ⎠ ⎠ ⎝ 2a Practice and Problem Solving: Modified 1. Quadratic 2. Not quadratic 3. Quadratic 4. y = x 2 + x + 2, x = −0.5 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 525 MODULE 19 Challenge Practice and Problem Solving: C 1. y = x2 − 2x + 1 1. (−1, 0), (3, 0) 2. (−1, 0), (3, 0) 3. (−1, 0) 4. no solution 5. (1, −2), (−4, −2) x −1 0 1 2 3 y 4 1 0 1 4 x=1 2. y = −2x2 + 4x − 2 6. (0, 0) 7. 0; 1; 2 MODULE 20 Connecting Intercepts, Zeros, and Factors x −1 0 1 2 3 y −8 −2 0 −2 −8 x=1 LESSON 20-1 Practice and Problem Solving: A/B 1. y = x2 − 2x + 1 x −1 0 1 2 3 y 4 1 0 1 4 x=1 3. Yes. The two quadratic functions above are different (one parabola opens up and the other opens down), but they have the same zeros. 4. t = 2.5 s 5. about 8.9 ft Practice and Problem Solving: Modified 2. y = 2x2 + 4x 1. y = x2 − 4 x −3 −2 −1 0 1 y 6 0 −2 0 6 x = −2 and x = 0 x −2 −1 0 1 2 y 0 −3 −4 −3 0 x = −2 and x = 2 3. t = 11 sec 4. t = 2.2 sec Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 526 2. y = x2 − x − 6 2. x = −1 and x = 4 x −2 −1 0 1 2 y 0 −4 −6 −6 −4 x = −2 and x = 3 LESSON 20-2 Practice and Problem Solving: A/B 1. 3. t = 1.4 sec 4. t = 1.5 sec Reading Strategies 1. x = −3 2. (−3, −4) x - intercepts 1 and 5 3. axis of symmetry Axis of symmetry: x = 3 4. 2 2. 5. x = −1 and x = −5 6. y = 0 7. x - intercepts − 2 and 3 Axis of symmetry: x = 1 2 3. y = 5 x 2 + 5 x − 30 8. x = 2 4. y = −2 x 2 + 8 x − 6 9. Yes. f(2) = −22 + 4(2) = −4 + 8 = 4; f(0) = −02 + 4(0) = 0 + 0 = 0; f(4) = −42 + 4(4) = −16 + 16 = 0 5. Success for English Learners 1. f(x) = x2 − 3x −4 x −1 0 1 2 3 y 0 −4 −6 −6 −4 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 527 6. 6. Practice and Problem Solving: Modified Practice and Problem Solving: C 1. 1. x - intercepts 1 and 5 Axis of symmetry: x = 3 x - intercepts − 2 and 2 2. Axis of symmetry: x = 0 2. x - intercepts − 2 and 3 Axis of symmetry: x = 1 2 3. y = −6 x 2 − 5 x + 4 x - intercepts − 5 and − 1 4. y = 6 x 2 + 8 x − 8 Axis of symmetry: x = −3 3. y = x 2 + 5 x + 6 5. 4. y = x 2 − 4 x + 3 5. Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 528 Practice and Problem Solving: C 6. 1. x = 1 3 , x= − 3 2 2. x = −12, x = 8 3. x = 12, x = 9 4. x = 0, x = 1 5. x = −6, x = 5 6. x = 3, x = −2 Reading Strategies 7. x = −9, x = 2 1. 3, − 5 8. x = 1 2. 3, − 5 3 + −5 = −1 3. x = 2 9. x = 1, x = −1, x = 2, x = −2 10. x = −7, x = 1 11. x = 2, x = −3 4. ( −1, − 16) 12. x = −7, x = 5. (0, − 15) Success for English Learners 2 3 13. x = −4, x = 3 1. The x-intercepts of the two linear factors are the same as the zeros of the parabola. 14. x = −4, x = − 2. Multiply the two linear factors. 15. 3 s Combine like terms. 3 2 16. 2 s Multiply the resulting trinomial by 2. Practice and Problem Solving: Modified LESSON 20-3 1. a = 0 or b = 0 Practice and Problem Solving: A/B 1. x = 3, x = −5 2. x = 7, x = −2 2. x = 0, x = 1 3. x = 5, x = 1 3. x = −1 4. x = 0, x = 5 4. x = 5, x = −1 5. x = −2, x = −1 5. x = 0, x = 3 6. x = 9, x = −3 6. x = 6, x = −1 7. x = −5, x = −3 7. x = 11, x = 1 8. i = −2, x = −6 8. x = −13, x = −5 9. x = 9. x = −5, x = 8 4 , x=3 3 10. x = 7, x = −2 10. x = 5, x = 1 11. x = −7, x = 2 11. x = 6, x = −2 12. x = 2, x = 4 12. 8 and 7 13. x = −5, x = 3 7 14. x = − , x = 7 3 15. 4 s 13. 10 and 11 14. Ari is 8 years old. Jan is 5 years old. Bea is 11 years old. 16. 6 s Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 529 Reading Strategies MODULE 20 Challenge 1. yes, because 0 × 0 = 0 1. (−6, 0); (−3, −9); (0, 0) 2. Because you need two or more quantities being multiplied 2. 0 = 36a − 6b + c; −9 = 9a − 3b + c; 0 = c 3. 1; 6; 0 3. −2 because −2 + 2 = 0 4. y = x2 + 6x 4. If abc = 0 then a = 0 or b = 0 or c = 0. ⎧0 = 9a − 3b + c ⎪ ; y = −2 x 2 + 18 5. ⎨18 = c ⎪0 = 9a + 3b + c ⎩ 5. x = −5, x = 2 6. x = 5, x = −2 7. x = −4, x = 3 ⎧0 = 25a − 5b + c ⎪ ; 6. ⎨0 = 4a + 2b + c ⎪−12.25 = 2.25a − 1.5b + c ⎩ Success for English Learners 1. zero 2. No, because one factor must be zero. y = x 2 + 3 x − 10 3. The value for a, (x + 5), is a binomial. The values for b, x, and c, 2, are monomials. Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 530 UNIT 9 Quadratic Equations and Modeling 10. x = −2, x = 3 MODULE 21 Using Factors to Solve Quadratic Equations 11. x = −9, x = 6 12. x = 8, x = −5 LESSON 21-1 13. x = −7, x = 4 Practice and Problem Solving: A/B 14. x = −7, x = −9 1. (x + 2)(x + 3) 15. x = 5, x = −4 2. (x − 3)(x + 1) 16. 9 and 10, or 0 and 1 3. (x + 1)(x − 4) 17. n = 22 4. (x + 1)(x + 3) Practice and Problem Solving: Modified 5. (x − 9)(x − 5) 6. (x + 3)(x + 8) 1. 3, 5 7. (x − 8)(x − 4) 2. Answer should include a table: 8. (x − 3)(x − 12) Factors of 6 Sum of factors 10. (x − 9)(x − 9) 2, 3 5 11. (x + 4)(x − 11) −2, −3 −5 12. x = 0, x = 5 1, 6 7 13. x = 6, x = 3 −1, −6 −7 9. (x + 3)(x − 14) 14. x = 5, x = 10 2, 3 15. x = −7, x = 3 3. (x + 2)(x − 1) 16. x = −8, x = 1 4. (x + 3)(x − 2) 17. x = − 5, x = 3 5. (x + 1)(x + 1) 18. 9 and 8 6. (x − 4)(x + 3) 19. 14 and 6 7. (x − 5)(x − 1) Practice and Problem Solving: C 8. (x + 3)(x + 3) 2 9. (x − 3)(x + 2) 2 2. x − 9 10. (x − 5)(x − 3) 3. Both factors must be negative. 11. (x + 3)(x + 4) 4. x = 5, x = −5 12. x = 1, x = 2 5. x = 1 13. x = −3, x = 1 6. x = 4, x = 1 14. x = −2, x = −4 7. x = −4, x = 2 15. x = −5 8. x = −6, x = 5 16. x = −7, x = −3 9. x = 6, x = −6 17. x = 8, x = 3 1. x − 2x − 8 18. 5 and 6 19. 10 and 11 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 531 Reading Strategies 10. x = − 1. Both the product and sum are positive, so the factors are both positive. 11. x = 2. Both the sum and product are negative, so one factor is positive and the other is negative. 5 ,3 2 5 2 12. x = 2 2 3 3. The sum is negative and the product is positive, so both factors are negative. 13. x = −3, 4. (x + 5)(x + 6) 14. no solution 5. (x − 7)(x + 4) 15. x = 0, 3 6. (x + 6)(x − 4) 16. x = 2 5 , 3 3 17. x = 1 49 , 2 2 7. (x + 9)(x − 8) 8. (x + 3)(x + 12) 9. (x − 7)(x − 7) 10. (x − 3)(x + 8); x = 3, (x = −8) 18. x = − 11. (x + 4)(x + 11); x = −4, x = − 11 7 , −2 4 1 1 , 8 3 12. (x − 6)(x − 7); x = 6, x = 7 19. x = Success for English Learners 20. x = −5, 1 1. positive; negative 21. 7 s 2. positive; negative Practice and Problem Solving: C 3. negative 4. sum; product LESSON 21-2 Practice and Problem Solving: A/B 1. x = 1 ,2 2 2. x = 1 ,3 3 2 ,2 3 1. x = 1 ,2 2 3. x = 2. x = 1 ,3 3 4. x = −2, − 3. x = 2 ,2 3 5. x = 2 or −6 4. x = −2, − 6. x = −4, 4 1 5 7. x = −7, 5. x = −6, 2 8. x = − 6. x = −4, 4 7. x = −7, 8. x = − 1 5 3 2 9 ,4 7 9. x = −3, 3 3 2 10. x = − 9 ,4 7 11. x = 9. x = −3, 3 5 ,3 2 5 2 12. x = 2 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 532 13. x = −3, 14. x = − 15. a. −16t2 + 1600 = 0 2 3 b. t = −10, 10 3 ,2 2 c. 10 s Reading Strategies 15. x = 0, −3 1. 1 i 3 2 5 16. x = , 3 3 17. x = 2. 1 i 12; 2 i 6; 3 i 4 3. minus, minus 1 49 , 2 2 4. (x−)(x−) 5. Possible answer: (↓ x − ↓) (↓ x − ↓) Outer + Inner 3 − 1 1 − 12 3 i − 12 + − 1 i 1 = −37 No 3 − 12 1 − 1 3 i − 1 + − 12 i 1 = −15 No 18. x = −5, 1 19. x = 1 1 , 8 3 20. x = − 1 ,1 3 3 −4 3 −3 3 −6 3 −2 21. 4 s and 6 s Practice and Problem Solving: Modified −4 −2 −6 1 − 3 + − 4 i 1 = −13 No − 4 + − 3 i 1 = −15 No − 2 + − 6 i 1 = −12 No − 6 + − 2 i 1 = −20 Yes 1. rectangle 2. Use FOIL to multiply and check your answer. 2 ,2 3 LESSON 21-3 1 5 Practice and Problem Solving: A/B 5. x = −6, 2 1. (x + 5y)2 6. x = −4, 4 7. x = 2, 1 1 3i 3i 3i 3i Success for English Learners 2. 4; 5; x + 5; x − 1; x + 5; x − 1; −5, 1 4. x = −2, − −3 6. (3x − 2)(x − 6) 1 1. x = , 2 2 3. x = 1 2. 2(4x + 5y)2 7 2 3. (9x + 11y)(9x − 11y) 4. 3x(5x + 4)(5x − 4) 9 8. x = − , 4 7 5. x = 9. x = −3, 3 6 6 ;x = − 5 5 5 10. x = − , 3 2 6. x = 0; x = − 5 11. x = 2 7. t = 4 3 1 s 4 8. A, C, E 12. x = 2 Practice and Problem Solving: C 2 13. x = −3, 3 1. 3(3x + 4y)2 2. x(5x − 6y)2 3 14. x = − , 2 2 3. (x + 3)(x − 3)(x2 + 9) 4. 4x2(3x + 2y)(3x − 2y) Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 533 5. x = −5, x = 5, x = 0 MODULE 22 Using Square Roots to Solve Quadratic Equations 6. x = 0, x = −2 7. t = 9 s 4 LESSON 22-1 8. A, C, D Practice and Problem Solving: A/B Practice and Problem Solving: Modified 1. x = −5 or x = 5 2. no solution 1. 2x; 5; negative; (2x −5)2 3. x = −1 or x = 1 2. 3x; 2; positive; (3x + 2)2 4. x = −3 or x = 3 3. 5x; 3; negative; (5x − 3)2 5. no solution 4. 6x; 2; positive; (6x + 2)2 6. x = 0 5. 7x; 4; (7x − 4)(7x + 4) 7. x = 11 or x = −11 6. 6; 5x; (6 − 5x)(6 + 5x) 8. x = 7 or x = −7 1 1 7. (7x − 1); (7x − 1); − ; 7 7 8. (6x − 11); (6x + 11); 9. x = 6 or x = −6 10. x = −12 or x = 2 11 11 ; − 6 6 11. x = 11 or x = −9 12. x = 15 or x = 13 Reading Strategies 13. x = −3 or x = 9 1. x = 3 3 ,− 2 2 14. no solution 2. x = 1 3 16. x = −1 ± 5 15. x = −6 or x = 4 17. x = 3 ± 6 Success for English Learners 18. x = 7 ± 3 3 3 1. x = − , 2 2 2. x = − 19. length = 200 ft and width = 100 ft 7 5 20. 2 s 21. 40 ft MODULE 21 Challenge Practice and Problem Solving: C 1. (a − b)(a2 + ab + b2) 2 1. Solve ax2 + b = c for x. 2 2. (a + b)(a − ab + b ) 4 4 2 2 2 2 2 2 2 ax 2 + b = c 2 3. a − b = (a ) − (b ) = (a + b )(a − b ) = (a2 + b2)(a − b)(a + b) ax 2 = c − b c −b x2 = a 4. a. If two polynomials are equal, their corresponding coefficients are equal. The coefficients of a2, ab, and b2 are equal. c −b a c−b Now examine , the expression a inside the square root symbol. c−b is negative, that is c − b and a If a x=± b. If t = 0 and u = 0, then r and s are given by undefined expressions. Thus, there are no numbers r, s, t, and u for which a2 + b2 can be factored as (ra + sb) (ta + bu). Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 534 3. no solution have opposite signs, then there are no real roots. This follows since the square root of a negative number is not a real number. 0 = 0. In this case, If c = b, then x = ± a there is exactly one real root, namely 0. c−b If is positive, that is c − b and a a have the same sign, then there are two real roots. This follows since a positive number has two square roots, opposites of one another. 4. x = −3 or x = 3 5. x = −6 or x = 6 6. x = −7 or x = 7 7. x = 3 or x = −7 8. x = −11 or x = −7 9. x = 10 or x = 2 10. (3w)(w) = 300; 3w2 = 300; w2 = 100; w = 10; the width is 10 ft and the length is 30 ft. 11. Let x represent the unknown number. x2 + 27 = 148; x2 = 121; x = 11 or x = −11 2. Look at the graph of y = 0.5(x − 1)2 + 3. Look for any x-intercepts. These provide information about the number of real roots. The graph shows that there are none. Therefore, 0.5(x − 1)2 + 3 = 0 has no real roots. What follows is an Algebraic argument: If 0.5(x − 1)2 + 3 = 0, then 0.5(x − 1)2 = −3. Since (x − 1)2 ≥ 0, then 0.5(x − 1)2 ≥ 3. Therefore, there is no real number such that 0.5(x − 1)2 + 3 = 0. Reading Strategies 2(m − 1)2 50 = ; 2 2 (m − 1) 2 = 25; m − 1 = ± 25; m − 1 = ±5; m = ±5 + 1; m = −5 + 1 or m = 5 + 1; m = −4 or m = 6 1. 2(m − 1)2 = 50; 2. Sample answer: Alike—Divide both sides of 2(m − 1) = 50 and both sides of 2(m − 1)2 = 50 by 2 and simplify. Add 1 to both sides of m − 1 = 25, to both sides of m − 1 = −5, and to both sides of m − 1 = 5 and simplify. Different—The quantity m − 1 is squared in the second equation, but not in the first. At m − 1 = 25 in solving the first equation, add 1 to both sides and obtain the only solution, which is 26. At (m − 1) 2 = 25 in solving the second equation, first use the definition of square roots and then add 1 to both sides of the resulting linear equations to find the two solutions, which are −4 and 6. 3. To show that a(x − h)2 = p, where a, h and p are positive real numbers has two real roots, solve the equation for x. a( x − h )2 = p p ( x − h )2 = a x −h =± x =h± p a p a p is positive. a Therefore, there are two real values of x. Since p and a are positive, Success for English Learners 1. the quantity x − 5 Add the roots. ⎛ ⎜⎜ h + ⎝ 2. To isolate the expression (x − 5)2 on one side of the equals sign. p⎞ ⎛ p⎞ ⎟⎟ + ⎜⎜ h − ⎟ = 2h a⎠ ⎝ a ⎟⎠ 3. In Problem 1, find the square roots of 16, which is a perfect square and has rational square roots. In Problem 2, find the square roots of 7, which is not a perfect square; its square roots are irrational numbers and must be rounded in order to record them as answers. The sum of the roots is 2h. Practice and Problem Solving: Modified 1. x = −10 or x = 10 2. x = −8 or x = 8 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 535 2. The expression on the right side of the equation involves a perfect square trinomial. Complete the square in the quadratic expression on the left side of the equation. LESSON 22-2 Practice and Problem Solving: A/B 1. x = −5 or x = 1 2. x = −2 or x = 4 x 2 + 4x + 9 = x 2 + 4x + 4 + 5 3. x = 5 ( ) = x 2 + 4x + 4 + 5 4. x = −5 or x = 3 = ( x + 2) + 5 2 5. x = 12 or x = −2 6. x = −8 or x = 4 Now the given equation becomes: 7. x = 1 + 2 or x = 1 − 2 ( x + 2)2 + 5 = ( x + b )2 + 5 8. x = 3 + 3 or x = 3 − 3 From this equation: 9. x = 2 + 3 or x = 2 − 3 ( x + 2)2 = ( x + b )2 Therefore, b = 2. 10. x = 1 + 5 or x = 1 − 5 3. If y = x2 − 6x + 14 and y = 5, then x2 − 6x + 14 = 5. Solve x2 − 6x + 14 = 5. 11. x = −2 + 3 or x = −2 − 3 12. x = 2 + 5 or x = 2 − 5 x2 − 6x + 14 = x2 − 6x + 9 + 5 13. x = 1 + 2 2 or x = 1 − 2 2 = (x − 3)2 + 5 14. x = 2 + 3 3 or x = 2 − 3 3 So, (x − 3)2 + 5 = 5 and x = 3. 15. x = 5 + 2 2 or x = 5 − 2 2 4. The x-intercepts of the graph of y = x2 + 4x − 21 are those values of x for which y = 0. Solve x2 + 4x − 21 = 0 by completing the square. 16. The width is 16 feet and the length is 20 feet. Practice and Problem Solving: C 1. In order for x2 + bx = −4 to have exactly one root, the equation x2 + bx + 4 = 0 must have exactly one root. This means that x2 + bx + 4 must be a perfect square. Thus, 4 must be the square of one half of b. ⎛1 ⎞ 4 = ⎜ b⎟ ⎝2 ⎠ x2 + 4x − 21 = x2 + 4x + 4 − 25 = (x + 2)2 − 25 Therefore, solve (x + 2)2 − 25 = 0. (x + 2)2 = 25 x + 2 = 5 or x + 2 = −5 2 x = 3 or x = −7 So, there are two x-intercepts and they are 3 and − 7. Solve for b. ⎛1 ⎞ 4 = ⎜ b⎟ ⎝2 ⎠ 2 ⎛1 ⎞ 4 = ⎜ b⎟ ⎝2 ⎠ 1 2= b 2 b=4 Practice and Problem Solving: Modified 1. 6; 3; 9; 9, 9; −2, −4 2 2. x = −2 or x = 10 3. x = −13 or x = 1 4. x = −5 or x = 7 5. x = 4 or x = 6 6. x = −1 or x = 17 7. x = −8 or x = −2 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 536 3. Sample answers: Multiply each side of the 1 equation by to obtain x2 − 3x = 4; you 2 can then make a perfect square on the left side of the equation by calculating 2 9 9 ⎛ −3 ⎞ ⎜ 2 ⎟ = 4 and adding 4 to each side. Or ⎝ ⎠ you can multiply each side of the equation by 2 to obtain 4x2 − 12x = 16; you can then make a perfect square on the left side of the equation by calculating ( −12)2 = 9 and adding 9 to each side. 4(4) 8. x = 1 + 5 or x = 1 − 5 9. x = −2 + 3 or x = −2 − 3 10. x = 2 + 5 or x = 2 − 5 11. a. w and w + 6; b. w2 + 6w = 91; c. 36; d. width 7 ft and length 13 ft Reading Strategies 1. a. Sample answer: Multiply each side of the equation by 2. b. Add 9 to each side of the equation. LESSON 22-3 9 1 , or , to each side of the 36 4 equation. c. Add Practice and Problem Solving: A/B 1. 3 and − 4 2. a. Sample answer: One side of the equation will be a perfect square trinomial. You factor it as two identical binomials. b. Sample answer: The other side of the equation will involve an addition of two numbers. You simplify by adding those numbers. 2. 5 and − 3 4 3. 3 and − 1 2 4. −11 + 61 −11 − 61 and 6 6 5. 7 and 4 3. One side of the equation will be a real number. If that number is positive, it has a positive square root and a negative square root. You must write an equation that involves each square root. 6. 7 and − 7 7. 1 1 and − 3 2 8. 2 and − 10 Success for English Learners 9. 02 − 4(1)(25) < 0, no real solution 1. To rewrite the expression on the left side of an equation either in the form (x ± n)2 or (mx ± n)2, the x2 term must be a perfect square. In Problem 1, the x2 term is already a perfect square because x2 = x i x, or 1x2 = 1x i 1x. In Problem 2, 3x2 is not a perfect square. So you multiply 3x2 by 3 to obtain 9x2, which is a perfect square (9x2 = 3x i 3x). 10. ( 7) 2 − 4(3)( −3) > 0, two real solutions 11. (8)2 − 4(1)(16) = 0 , one real solution 12. No; the discriminant is negative. There are no real solutions so the ball will not hit the roof. Practice and Problem Solving: C 2 2. The expression 9x − 6x is not a perfect square because it cannot be rewritten either in the form (x ± n)2 or (mx ± n)2. When you add 1 to each side of the equation, you create the expression 9x2 − 6x + 1, which is a perfect square because it can be rewritten in the second of these forms (9x2 − 6x + 1 = (3x − 1)2). 1. 2 real solutions −7 + 209 −7 − 209 and 8 8 2. 2 real solutions; 2 and − 2 3 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 537 3. 2 real solutions; 3 + 15 3 − 15 and 2 2 4. 2 real solutions; −1 + 43 −1 − 43 and 3 3 8. 10; 1; 25; 0 1 real solution 9. 36 − 4(1)(−7) = 8; two real solutions 10. 8, −8 11. no real solutions 5. no real solutions Reading Strategies 6. 2 real solutions; −3 + 105 −3 − 105 and 6 6 2. a = 2; b = −5; c = 3 3 7. 1 real solution; 2 4. 5 and 2 1. No real solutions 3. ( −5)2 − 4(2)(3) = 1; two real solutions 5. No real solutions 8. no real solutions 9. 2 real solutions; − Success for English Learners 4 and − 1 5 1. Rearrange the equation so that it is equal to zero. The coefficient of the x2 -term is the value of a, the coefficient of the x-term is the value of b, and the constant term is the value of c. 10. no real solutions 11. 1 real solution; 7 2 12. no real solutions 2. If the discriminant is equal to zero, then there is one solution. If the discriminant is greater than zero, then there are two solutions. If the discriminant is less than zero, then there is no solution. 2 13. The discriminant b − 4ac is negative when there are no real roots for a quadratic equation. 14. x = −8 ± 191 x ≈ 5.82 or − 21.82 LESSON 22-4 Length: 5.82 + 7 = 12.82 ft Practice and Problem Solving: A/B Width: 5.82 + 9 = 14.82 ft 1. x = 4 or x = −4; taking square roots because b = 0 −35 ± 284.2 −9.8 t ≈ 5.29 or 1.85 15. t = 11 1 or x = ; taking square roots 2 2 because equation is expressed as a squared binomial 2. x = The rocket will be at an altitude of 60 meters at about 5.29 seconds and 1.85 seconds. 3. x = 7 or x = −4; factoring because not too many factors to check Practice and Problem Solving: Modified 4. x = 3 or x = −2; factoring because not too many factors to check. 1. 1; 6; 5; 6; 6; 1; 5; 1; −1, −5 2 ± 10 , x = 2.58 or x = −0.58; 2 complete the square or use quadratic formula because trinomial doesn’t factor 5. x = 2. 1; −9; 20; −9; −9; 1; 20; 1; 5, 4 3. 2; 9; 4; − 1 , −4 2 6. x = −5 ± 2 7, x = 0.29 or x = −10.29; complete the square because a = 1 and b is an even number 4. 1; −3;−18; 6, −3 5. −8, 4 6. x = 1 or x = −5 2 7. 3; 1; 5; −11; no real solutions Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 538 Practice and Problem Solving: Modified 4.3 ± 11.29 , x = 2.55 or 0.31; 3 quadratic formula because trinomial doesn’t factor and the coefficients are not integers 7. x = 1. x = 4 or x = −4; taking the square roots because b = 0 2. x = 3 or x = −3; taking the square roots because b = 0 1 1 8. x = or − ; factor or taking square 2 2 roots because b = 0; difference of two square factors 3. x = 10 or x = −2; taking the square roots because binomial is squared 4. x = −1 or x = −6; factoring because not too many factors to check 9. 1.55 s and 2.83 s; quadratic formula because the trinomial doesn’t factor 5. x = 2 or x = −6; factoring because not too many factors to check 10. 4 s; taking square roots because b = 0 Practice and Problem Solving: C 1 or x = −2; factoring because not 2 too many factors to check or quadratic formula 6. x = − 1 1 1. x = or x = − , taking square roots 2 2 because b = 0 4 4 or x = − ; taking the square roots 3 3 because b = 0 1 2. x = 8 or x = − ; factoring because not too 2 many factors to check 7. x = 3. x = 0 or x = −1; taking square roots because binomial is squared 8. x = 0 or x = 2; factoring because c = 0 9. 2 s; factoring because c = 0 and the terms have a common factor 3 4. x = or x = −5; factoring because not too 2 many factors to check. 10. 4.5 s; taking square roots because b = 0 Reading Strategies −2 ± 7 , x = 0.22 or x = −1.55; 3 quadratic formula because the trinomial doesn’t factor 5. x = 1. x = 1 1 or x = − 2 2 2. x = 0 or x = − 6. x = 16 or x = 8, factoring because not too many factors to check. 2 3 3. x = 11 or x = −1 3 5 7. x = or x = − ; multiply by 100 and then 4 4 factoring because not too many factors to check, or quadratic formula 4. x = 3 5 or x = − 2 3 Success for English Learners 5 5 or x = − ; multiply by 100 and then 6 6 taking square roots because b = 0 8. x = 1. Take the square root of both sides of the equation, write two equations and solve for the variable. 9. 3 s; factoring because c = 0 2. x = 2 or x = −20 3. the quadratic equation must be written in standard form: ax 2 + bx + c = 0 10. No, the discriminant is negative so there is no solution; quadratic formula because discriminant is useful to answer the question 4. x = 1 or x = − 5 3 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 539 LESSON 22-5 ⎛ −5 + 21 7 − 21 ⎞ 4. ⎜ , ⎟⎟ ; ⎜ 2 2 ⎝ ⎠ ⎛ −5 − 21 7 + 21 ⎞ , ⎜ 2 2 ⎟⎠ ⎝ Practice and Problem Solving: A/B 1. (2, 2); (3, 7) ⎛ 3 49 ⎞ 5. ⎜ , ⎟ ; (−7, −11) ⎝4 4 ⎠ 6. (0, 7); (−3, 25) ( ) ( 7. 2 5, − 8 5 + 59 ; −2 5, 8 5 + 59 ) 8. (0, 0); (3, −27); (−4, 64) 9. (3, 34) ⎛3 ⎞ ⎛5 ⎞ 10. ⎜ , 24 ⎟ ; ⎜ , 40 ⎟ 5 3 ⎝ ⎠ ⎝ ⎠ 2. (1, 3); (2, 2) 11. about 3.3 seconds 9 seconds and when 16 t = 2 seconds 12. a. When t = b. The balloon starts out higher than the ball. At first, the ball is traveling faster than the balloon and they attain the 9 ⎞ ⎛ same height ⎜ at t = . The ball then 10 ⎟⎠ ⎝ passes the balloon but is slowing down. Eventually, the balloon catches up to the ball again (at t = 2) and passes it. By this point, the ball is heading back toward the ground. 3. (−3, 6), (2, 1) 4. no real solutions Practice and Problem Solving: Modified 5. (−1, −2), (2, 7) 6. (−5, 0); (6, 11) 1. (−1, 1); (2, 4) 7. (−1, 0); (3, 8) 8. (−2, −1), (−1, 0) 9. 1.875 seconds Practice and Problem Solving: C 1. (3, 2); (−11, −68) 2. no real solutions ⎛1 ⎞ 3. ⎜ , 16 ⎟ ; (−3, 9) ⎝2 ⎠ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 540 Reading Strategies 2. 1. axis; symmetry; slope; y-intercept 2. (−1, −4), (3, 0) Success for English Learners 1. Yes; one equation is linear and the other equation is nonlinear. 2. You can check your answer by graphing the two equations and finding the points of intersection. (1, 2); (−2, −1) 3. MODULE 22 Challenge 1. Divide both sides by a to get x2 = 0, so x = 0; no solution method necessary. 2. Add c to both sides to obtain ax 2 = c. Divide both sides by a to obtain x 2 = c . a Take the square root of both sides to c solve: x = ± . a Method: Taking square roots. (−2, 4); (1, −2) 4. 3. Factor an x from the binomial to get x(ax + b). Set each factor equal to 0, to b get x = 0 and x = − . a Method: Factoring. Completing the square is another option. 4. Using the quadratic formula; (−3, −1); (3, 5) x= 5. x 2 = 2 x + 8 −b ± b 2 − 4ac . 2a 5. Answers may vary: x 2 − 2x − 8 = 0 ( x − 4)( x + 2) = 0 x = 4 or x = −2 (4, 16); (−2, 4) 6. x 2 + 9 x + 12 = 6 x + 30 x + 3 x − 18 = 0 ( x + 6)( x − 3) = 0 2 x = −6 or x = 3 (−6, −6); (3, 48) Example Method if b = 0 and c = 0 5x2 = 0 none; x = 0 if b = 0 and c ≠ 0 x2 − 9 = 0 taking square roots if b ≠ 0 and c = 0 x2 + 6x = 0 factoring or completing the square if b ≠ 0 x2 + 5x + 1 = 0 and c ≠ 0 7. 5 seconds quadratic formula Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 541 MODULE 23 Linear, Exponential, and Quadratic Models Practice and Problem Solving: Modified 1. first differences: 9, 15, 21, 27, 33 second differences: 6, 6, 6, 6 3x2 − 5 LESSON 23-1 2. first differences: 12, 20, 28, 36, 44 second differences: 8, 8, 8, 8 4x2 − 1 Practice and Problem Solving: A/B 1. is first differences: 9, 15, 21, 27, 33 second differences: 6, 6, 6, 6 3. first differences: 5, 9, 13, 17, 21 second differences: 4, 4, 4, 4 2x2 − x 2. is not first differences: 16, 20, 30, 38, 46 second differences: 4, 10, 8, 8 The second differences are not the same. 4. first differences: 4, 6, 10, 16, 20 second differences: 2, 4, 6, 4 2x2 − 4x + 1 3. a = 4, b = 0, c = −7; g(x) = 4x2 − 7 4. a = 3, b = 1, c = 0; g(x) = 3x2 + x 5. a = 1.9, b = 0.4, c = 4.3; g(x) = 1.9x2 + 0.4x + 4.3 Success for English Learners 1. For the first differences, subtract the temperature for each hour from the temperature for the hour before (38 − 23 = 15, 49 − 38 = 11, and so on). For the second differences, subtract the first difference for each hour from the first difference for the hour before (11 − 15 = −4, 8 − 11 = −3, and so on). 6. g(x) = 0.2x2 + 0.4x + 0.9 Practice and Problem Solving: C 1. a. first differences: 12.8, 23.2, 33.4, 42.6, 52.9 second differences: 10.4, 10.2, 9.2, 10.3 These differences are about equal. b. g(x) = 5.0x2 − 1.7x + 0.7 2. f(x) = −0.375x2 + 9.393x + 6 c. The values of x in the second table are all 2 more than the corresponding values of x in the first table. But the values of f(x) in the second table are the same as those in the first table. 3. Sample answer: Solve the equation −0.375x2 + 9.393x + 6 = 32. This can be done by graphing the equation y1 = 0.375x2 − 9.393x + 6 and the equation y2 = 32 on the same graphing calculator screen and identifying the x-coordinate of the point where the graphs intersect. (The x-coordinate is approximately 3.2. This means the temperature first rose above freezing about 3.2 hours after midnight, or about 3:12 a.m.) Let g’(x) = g(x − 2). Then g’(x) = 5.0(x − 2)2 − 1.7(x − 2) + 0.7, or, after multiplication, g’(x) = 5.0x2 − 21.7x + 24.1. This function will fit the second table just as well as g(x) fits the data in the first table. 2. a. first differences: 0.2, 0.5, 1.1, 1.5, 1.7, 2.1 second differences: 0.3, 0.6, 0.4, 0.2, 0.4 These differences are close to one another. Reading Strategies 1. There are only two sets of data, which are the x-values and the corresponding y-values. It is not necessary to enter anything into the L3 column. b. The set of coordinates can be used to make a table whose entries can be put into a graphing calculator. The equation output is y = 0.2x2 + 0.1x + 2.9. Let x = 2.5. then 0.2(2.5)2 + 0.1(2.5) + 2.9 = 4.4 If x = 2.5, then y is about 4.4. 2. y = 1.708x2 + 2.994x + 2.089 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 542 6. quadratic 3. Yes. Sample explanation: The value of the correlation coefficient, R2, is 0.9999411374. This is very close to 1. When the correlation coefficient is very close to 1, the equation is a good fit for the data. Also, the fourth screen provides visual confirmation that the graph of the regression equation passes very close to all the data points. 7. f(x) = 1.1, 2, 11, 101, 1001, 10001; 1st differences = −, 0.9, 9, 90, 900, 9000; 2nd differences = −, −, 8.1, 81, 810, 8100; ratios = −, 1.82, 5.50, 9.18, 9.91, 9.99 8. approaches one 9. exponential 10. h(x) = x2 − 2x. Approaches − ∞ LESSON 23-2 11. Sample answer: f(x) = 0.5x + 10 Practice and Problem Solving: A/B 12. quadratic 1. f(x) = −3, −1, 1, 3, 5, 7; 1st differences = −, 2, 2, 2, 2, 2; 2nd differences = −, −, 0, 0, 0, 0; ratios = −, 0.33, −1, 3, 1.67, 1.40 Practice and Problem Solving: Modified 1. quadratic 2. linear 2. increases without bound 3. exponential 3. linear 4. f(x) = 4, 1, −2, −5, −8, −11; 1st differences = −, −3, −3, −3, −3, −3; 2nd differences = −, −, 0, 0, 0, 0; ratios = 0.25, −2, 2.5, 1.6, 1.4 4. f(x) = −2, −3, −2, 1, 6, 13; 1st differences = −, −1, 1, 3, 5, 7; 2nd differences = −, −, 2, 2, 2, 2; ratios = −, 1.50, 0.67, −0.50, 6, 2.17 5. decreases without bound 5. increases without bound 6. linear 6. quadratic 7. f(x) = −1, −2, −1, 2, 7, 14; 1st differences = −, −1, 1, 3, 5, 7; 2nd differences = −, −, 2, 2, 2, 2; ratios = −, 2, 0.5, −2, 3.5, 2 1 1 7. f(x) = , , 1, 3, 9, 27; 9 3 1st differences = −, 0.22, 0.67, 2, 6, 18; 2nd differences = −, −, 0.45, 1.33, 4, 12; ratios = −, 3, 3, 3, 3, 3 8. increases without bound 9. quadratic 8. approaches zero 1 , 1, 2, 4, 8, 16; 2 1st differences = −, 0.5, 1, 2, 4, 8; 2nd differences = −, −, 0.5, 1, 2, 4; ratios = −, 2, 2, 2, 2, 2 10. f(x) = 9. exponential 10. Exponential. Common ratio is 0.5 11. $12 Practice and Problem Solving: C 11. increases without bound 1. f(x) = 11, 1, −9, −19, −29, −39; 1st differences = −, −10, −10, −10, −10, −10; 2nd differences = −, −, 0, 0, 0, 0; ratios = −, 0.09, −9, 2.1, 1.5, 1.3 12. exponential 13. linear 14. f(x) = 125 + 15x 2. decreases without bound Reading Strategies 3. linear 1. exponential 4. f(x) = −2, 4, 8, 10, 10, 8; 1st differences = −, 6, 4, 2, 0, −2; 2nd differences = −, −, −2, −2, −2, −2; ratios = −, −2, 1.25, 1, 0.8 3. quadratic 5. decreases without bound 5. exponential 2. linear 4. quadratic Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 543 6. linear 2. On the table the average rate of change over the interval [0, 15] is 2550 − 0 = 170 ft/s . On the graph the 15 − 0 average rate of change over the interval 3600 − 0 [0, 15] is = 240 ft/s . The graph is 15 − 0 not a good model (because it does not account for wind resistance). 7. linear 8. quadratic 9. exponential Success for English Learners 1. f(x) = 9, 13, 17, 21, 25; 1st differences = −, 4, 4, 4, 4 2. f(x) = 5, 4, 5, 8, 13; 1st differences = −, −1, 1, 3, 5; 2nd differences = −, −, 2, 2, 2 3. f(x) = 5, 25, 125, 625, 3125; ratios = −, 5, 5, 5, 5 MODULE 23 Challenge 1. On the table the average rate of change 174 − 14 over the interval [0, 8] is = 20. 8−0 The average rate of change over the 128 − 14 interval [0, 15] is = 7.6. On the 15 − 0 graph, the average rate of change over 58 − 20 the interval [0, 8] is = 4.75. The 8−0 average rate of change on the interval 140 − 20 = 8. Over [0, 8], the [0, 15] is 15 − 0 graph is not a good model, but is better over the interval [0, 18]. However, since the wolf population peaked in year 8 and then declined in the following years, the graph does not model the population well. Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 544 UNIT 10 Inverse Relationships MODULE 24 Functions and Inverses LESSON 24-1 Practice and Problem Solving: A/B 1. 4; negative 2. 3; positive 3. odd; positive If a turning point is in the second quadrant, the leading coefficient is positive. If the turning point is in the third quadrant, the leading coefficient is negative. 4. even; negative 5. 4; even; negative 6. 3; neither; positive Practice and Problem Solving: Modified Practice and Problem Solving: C 1. Graph A represents the function. The pattern down/up/down/up indicates that the leading coefficient is positive and that the polynomial has degree 4. Since the graph is symmetric about the vertical axis, the graph represents and even function. 1. Graph D is the reflection of graph A in the horizontal axis. It represents a polynomial of degree 4 that is even but has negative leading coefficient. Graph C does not represent a polynomial of degree 4 but rather degree 3. Graph B is a translation of graph A. It too has positive leading coefficient and degree 4, but is not an even function. 4 2. 2 3. 4 2. f(x) = 2x − 3x and f(−x) = 2(−x) − 3(−x), or 2x4 + 3x. The equation f(x) = f(−x) becomes: 2x 4 − 3 x = 2x 4 + 3 x This equation is true only if x = 0. It is not true for all real numbers. Therefore, the definition of even function is not satisfied. Thus, the function is not even. 3. The sketch below shows exactly one turning point left of the vertical axis, here in the second quadrant. Because f is odd, there is a companion turning point right of the vertical axis, here in the fourth quadrant. The sketch shows an up/down/up pattern. This is characteristic of a polynomial function with degree 3. 4. positive 5. 3 6. No 7. No Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 545 Range: { y | −1 ≤ y ≤ 7} 8. No 9. No 10. Neither 11. degree: 3; function type: odd; leading coefficient: positive 12. degree: 3; function type: neither; leading coefficient: negative 13. degree: 3; function type: odd; leading coefficient: positive 14. degree: 4; function type: even; leading coefficient: positive Inverse relation: Reading Strategies 1. Sample answer: As x approaches zero, the value of f at x approaches positive infinity. x −1 1 3 5 7 y −3 −2 −1 0 1 Domain: { x | −1 ≤ x ≤ 7} Range: { y | −3 ≤ y ≤ 1} 2. Sample answer: The value of f at the opposite of x equals the value of f at x. 2. Original relation: 3. Sample answer: The value of f at the opposite of x sub one equals the value of f at x sub one, and the value of f at the opposite of x sub two equals the opposite of the value of f at x sub two. Domain: { x | −2 ≤ x ≤ 2} Range: { y | 0 ≤ y ≤ 7} 4. As x → −∞, f(x) → 0. 5. −f(x) ≠ f(x) Success for English Learners 1. The terms odd degree, odd function, and negative leading coefficient should be circled. 2. No. Sample explanation: When the degree of the polynomial is even, the end behavior of the function as x approaches negative infinity is the same as its end behavior as x approaches positive infinity. However, if the function is an even function, it also must be true that the value of the opposite of x is the same as the value of x for all x. This is not always the case. For example, the equation f(x) = x2 + 2x defines a function for which the degree of the polynomial is even, but which is not an even function. Inverse relation: x 0 1 4 5 7 y −2 −1 0 1 2 Domain: { x | 0 ≤ x ≤ 7} Range: { y | −2 ≤ y ≤ 2} 3. f −1( x ) = x−2 3 Sample check for x = 10 f (10) = 3(10) + 2 = 32 LESSON 24-2 f −1(32) = Practice and Problem Solving: A/B 32 − 2 = 10 3 1. Original relation: Domain: { x | −3 ≤ x ≤ 1} Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 546 4. f −1( x ) = 5( x + 3) 2 3. Sample check for x = 10 f (10) = 2(10) −3 =1 5 f −1(1) = 5(1 + 3) = 10 2 5. f −1( x ) = 5x −5 2 f −1( x ) = −4 x −4 3 4. f −1( x ) = 3 x − 9 6. 5. c = 4 + 0.13m , c−4 = m, 240 miles 0.13 6. d = 1.3858e + 5, 176.79 euros f −1( x ) = −2 x − 4 Practice and Problem Solving: Modified Practice and Problem Solving: C 1. f −1( x ) = d −5 = e, 1.3858 1. Domain { x | −2 ≤ x ≤ 2} , Range { y | 1 ≤ y ≤ 5} 3x − 3 4 Sample check for x = 3 f (3) = 4(3) + 3 =5 3 3(5) − 3 =3 4 5x + 10 2. f −1( x ) = 3 f −1(5) = Sample check for x = 5 f (5) = 3(5) − 6 = −3 5 f −1( −3) = 5( −3) + 10 = 5 3 x 1 2 3 4 5 y −2 −1 0 1 2 Domain { x | 1 ≤ x ≤ 5}, Range { y | −2 ≤ y ≤ 2} Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 547 2. x + 3 2. x+3 3. 2 4. f −1( x ) = x −3 −1 0 y 4. f −1( x ) = 5. Sample check: Let x = 4 , 6. f −1( x ) = y = 3 5 2 3 4 3. Exchange x and y and solve for y. x+3 2 y = 2(4) − 3 = 5 f −1(5) = y = 1 1 5+3 =4 2 x −9 3 LESSON 24-3 Practice and Problem Solving: A/B x −3 2 1. Translation 6 units right; domain x ≥ 6, range: y ≥ 0 2. Translation 9 units right; vertical stretch, factor 10; domain x ≥ 9, range: y ≥ 0 3. Translation 1 unit right; domain: x ≤ 1, range: y ≥ 0 7. f −1( x ) = 4. Translation 2 units right; vertical 1 2 compression, factor ; domain: x ≥ , 2 3 range: y ≥ 0 x+4 3 5. 6. Reading Strategies 1. f −1( x ) = 5( x − 4) 2. f −1( x ) = x+3 6 3. f −1( x ) = 5( x + 2) 3 7. t = 2 x+ −1 3 4. f ( x ) = 3 d 4.9 8. 4.5 s Practice and Problem Solving: C Success for English Learners 1. x ≥ −3 1. Switch the x and y values of the function. 2. x ≤ 3 3. x ≥ 9 4. x ≥ 1 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 548 8. Domain: x ≥ 8, Range: y ≥ 3 9. x (x, y) y = x −2 5. Reflection across the x-axis followed by a translation 1 unit to the right and 10 units up; vertical stretch by a factor of 4; Domain: x ≥ 1; Range: y ≤ 10. 2 2−2=0 (2, 0) 6 2 (6, 2) 11 3 (11, 3) 6. Translation 4.5 units to the left and 1 unit up; vertical stretch by a factor of 2; Domain: x ≥ −4.5; Range: y ≥ 1. 10. 7. y 2 = x is not a function because each x does not have a unique corresponding y-value. For example, if x = 4, y could equal 2 or −2. To graph y 2 = x, you can graph y = x and y = − x . x x +1 (x, y) 0 1 (0, 1) 4 3 (4, 3) 9 4 (9, 4) 8. The function has domain of {x| x ≥ 0} with f(0) = 2. For x > 0, the function is decreasing. However, the function never reaches a value of 0 because x + 4 > x . By choosing larger and larger x, you can get f(x) as close as you want to 0. So, the range is all positive numbers less than or equal to 2. Practice and Problem Solving: Modified Reading Strategies 1. 3 4. 8 1. a. yes b. no c. no d. yes 5. Domain: x ≥ −2, Range: y ≥ 0 2. The graph is shifted up 2 units. 2. 4 3. 1 6. Domain: x ≥ 0, Range: y ≥ −10 7. Domain: x ≥ 0, Range: y ≥ 0 Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 549 8. 3. all real numbers greater than or equal to −5 4. 80 ft s 5. 9. d = 3 4V π 10. 7.3 in. Practice and Problem Solving: C Success for English Learners 1. 40 1. f −1( x ) = ft s 2. f −1( x ) = 3 x − 2 − 3 2. Set the radicand greater than or equal to zero and solve for x. 3. f −1( x ) = −10 3 x − 27 LESSON 24-4 4. f −1( x ) = 1 + 3 Practice and Problem Solving: A/B 3 −1 6. f ( x ) = −0.6 3 x 3 2. f ( x ) = 2 x , or f ( x ) = 8 x 7. T (a ) = a 3 ; a(T ) = 3 T 2 1 x 3. f −1( x ) = − 3 x , or f −1( x ) = 3 − 3 27 4. f −1( x ) = 3 6−x 5 5. f ( x ) = 10 3 x 1. f −1( x ) = 3 x −1 13 x +1 2 8. T = a 3 = (0.387)3 ≈ 0.05796 ≈ 0.24. Mercury’s orbital period is approximately 0.24 years. x 5 9. a = 3 T 2 = 3 (11.9)2 = 3 141.61 ≈ 5.21. Jupiter’s mean distance from the Sun is approximately 5.21 astronomical units. 1 x+7 5. f −1( x ) = 3 x + 7 , or f −1( x ) = 3 5 125 6. f −1( x ) = 3 x − 8 Practice and Problem Solving: Modified 7. 1. 2 2. 1 3. 3 4. 1 5. y = 3 0.25 x Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 550 6. 7. x 0 0.5 −0.5 1 −1 f(x) 0 0.5 −0.5 4 −4 x 0 0.5 −0.5 4 −4 f 1(x) 0 0.5 −0.5 1 −1 Success for English Learners 1. Switch the x and y values. 2. The inverse would be exactly the same as the original function because the x and y values are the same in all the points, so switching them won't change it. 3. The inverse of an inverse would just be the original function again. 8. MODULE 24 Challenge 1. (4, −2), (0, −2), (−1, −6), (3, −6) 9. e = 3 V 2. (x, y) → (y, −x) 10. e = 3 V = 3 216 = 6 mm 3. (x, y) → (−y, x) 11. e = 3 V = 3 100 ≈ 4.64 mm 4. (0, 4), (0, 0), (4, −1), (4, 3) Reading Strategies 5. (−4, 0), (0, 0), (1, 4), (−3, 4) 1. Accept any answers that have an a value with an absolute value less than 1. 6. (−2, 0), (2, 0), (3, 4), (−1, 4) 2. Accept any answers that have an a value with an absolute value greater than 1. 3. shrinking 4. stretching 5. stretching 6. shrinking 7. shrinking 8. no change 9. shrinking 10. stretching 11. shrinking Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 551