Download Solving Equations

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?
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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?
_________________________________________________________________________________________
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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?
_________________________________________________________________________________________
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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
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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?
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
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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?
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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
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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? ______________________
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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.
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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.
____________________________________
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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.
_________________________________________________________________________________________
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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.
____________________________________________________________
____________________________________________________________
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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
________________________
_______________________
________________________
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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: ________________
________________________
_______________________
________________________
________________________
_______________________
________________________
________________________
_______________________
________________________
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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.
_________________________________________________________________________________________
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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.
________________________________________
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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: ________________________________
___________________________
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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.
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
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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: ___________________________
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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.
_____________________________________
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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: ________________________________________________________________________________
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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
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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
________________________________________
________________________________________
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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 ) =
_________________________________________________________________________________________
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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,
________________________________________
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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.
_________________________________________________________________________________________
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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?
_________________________________________________________________________________________
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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.
_________________________________________________
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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?
________________________________________
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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?
_________________________________________________________________________________________
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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.
________________________________________
________________________________________
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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. ________________________________
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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.
_________________________________________________________________________________________
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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.
_________________________________________________________
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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?
_____________________________________________
_____________________________________________
_____________________________________________
_____________________________________________
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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?
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
_________________________________________________________________________________________
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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: _______
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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.
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
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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?
_________________________________________________________________________________________
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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.
_________________________________________________
_________________________________________________
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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.
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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).
___________________________________________________________
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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.
_________________________________________________________________________________________
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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
_________________________________________________________________________________________
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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.
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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.
______________________________________________________
______________________________________________________
______________________________________________________
______________________________________________________
______________________________________________________
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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.
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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?
_________________________________________________________________________________________
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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?
_________________________________________________________________________________________
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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?
_________________________________________________________________________________________
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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?
_________________
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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?
_______________________
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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? _______________________
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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?
_____________________
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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? ____________
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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
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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.
_______________________
________________________
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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?
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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
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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.
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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
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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.
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
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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.
____________________
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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.
_________________________________________________________________________________________
_________________________________________________________________________________________
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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?
_________________________________________________________________________________________
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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?
_________________________________________________________________________________________
_________________________________________________________________________________________
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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?
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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.
________________________________________
________________________________________
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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.
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
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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. ________________________________________________________
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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?
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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
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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.
_________________________________________________________________________________________
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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.
_______________________________________________________
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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?
_________________________________________________________________________________________
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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 =
________________________________________
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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?
________________________________________
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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?
_________________________________________________________________________________________
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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?
_________________________________________________________________________________________
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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?
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
_________________________________________________________________________________________
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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?
_________________________________________________________________________________________
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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?
_________________________________________________________________________________________
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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?
_________________________________________________________________________________________
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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?
_________________________________________________________________________________________
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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?
____________________________________________________
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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.
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
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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? ____________
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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. ______________________________________________________
_____________________________________________________________________________________
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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.
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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.
_________________________________________________________________________________________
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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: _____________________________________________
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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: __________________________________
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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: ________________________
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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 ______________
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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.
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
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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.
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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.
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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
________________________________________
________________________________________
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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. ______________________________________________________
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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.
_________________________________________________________________________________________
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
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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?
_________________________________________________________________________________________
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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239
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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.
_________________________________________________________________________________________
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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?
_________________________________________________________________________________________
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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
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245
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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?
_________________________________________________________________________________________
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246
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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
________________________________________
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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
________________________________________
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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
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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. __________
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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:
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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. _______________________________
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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.
_________________________________________________________________________________________
_________________________________________________________________________________________
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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.
________________________________________________________________________________________ .
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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 → +∞ = ____
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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.
_________________________________________________________________________________________
_________________________________________________________________________________________
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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 → +∞ = ____
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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).
_________________________________________________________________________________________
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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?
_________________________________________________________________________________________
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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?
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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.
________________________________________
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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.
_________________________________________________________________________________________
_________________________________________________________________________________________
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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
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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.
_________________________________________________________________________________________
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
_________________________________________________________________________________________
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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?
_________________________________________________________________
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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.
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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.
________________________________________
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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.
_________________________________________________________________________________________
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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?
_____________________________________________________________________________________
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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?
_________________________________________________________________________________________
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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.
_____________________________________
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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.
_________________________________________________________________________________________
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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.
__________________________
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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.
_________________________________________________________________________________________
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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).
_________________________________________________________________________________________
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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?
_____________________________________
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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.
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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? __________________
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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.
_________________________________________________________________________________________
_________________________________________________________________________________________
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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: _________________________
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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.
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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?
_______________________________________________________
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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
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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).
_________________________________________________________________________________________
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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).
_________________________________________________________________________________________
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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).
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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)
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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)
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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)
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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.
_________________________________________________________________________________________
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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?
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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?
_________________________________________________________________________________________
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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. ____________________________________
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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
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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
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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 = __________
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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?
______________________________________________________________
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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.
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
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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.
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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.
_________________________________________________________________________________________
_________________________________________________________________________________________
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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?
_________________________________________________________________________________________
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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
________________________________________
________________________________________
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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?
_________________________________________________________________________________________
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
_________________________________________________________________________________________
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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?
_________________________________________________________________________________________
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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.
_____________________________________________________________________________________
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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.
________________________________________________________________________
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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
_________________________________________________________________________________________
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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.
_____________________________________________________________________________________
_____________________________________________________________________________________
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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) =
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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?
_________________________________________________________________________________________
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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?
_________________________________________________________________________________________
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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).
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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: __________________
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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?
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
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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.
________________________________________
________________________________________
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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
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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.
_________________________________________________________________________________________
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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, _________________
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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.
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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)
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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.
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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.
_________________________________________________________________________________________
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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
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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
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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
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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
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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
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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
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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%
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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.
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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.
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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.
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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:
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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.
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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
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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}
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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)
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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
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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.
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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
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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
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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
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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.
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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.
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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.
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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
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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
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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
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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
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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
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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.
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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.
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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
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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
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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
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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
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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
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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)
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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
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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
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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
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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
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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
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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
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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%
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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
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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.
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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.
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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
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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.
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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)
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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)
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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)
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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.
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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
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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
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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
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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
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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.
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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
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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
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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 }
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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
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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
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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
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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.
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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
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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
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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
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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
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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.
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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
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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
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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
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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.
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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
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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
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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
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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.
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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
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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)
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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
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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
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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
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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
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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.
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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
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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.
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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
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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
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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)
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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).
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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
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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
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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
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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
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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
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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
⎠
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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
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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
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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
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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.
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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
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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}
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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}
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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
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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
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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
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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
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551