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SKILLS REVIEW HANDBOOK Logical Argument A logical argument has two given statements, called premises, and a statement, called a conclusion, that follows from the premises. Below is an example. Premise 1 Premise 2 Conclusion If a triangle has a right angle, then it is a right triangle. In nABC, ∠B is a right angle. nABC is a right triangle. Letters are often used to represent the statements of a logical argument and to write a pattern for the argument. The table below gives five types of logical arguments. In the examples, p, q, and r represent the following statements. p: a figure is a square Type of Argument q: a figure is a rectangle Pattern r: a figure is a parallelogram Example Direct Argument If p is true, then q is true. p is true. Therefore, q is true. If ABCD is a square, then it is a rectangle. ABCD is a square. Therefore, ABCD is a rectangle. Indirect Argument If p is true, then q is true. q is not true. Therefore, p is not true. If ABCD is a square, then it is a rectangle. ABCD is not a rectangle. Therefore, ABCD is not a square. Chain Rule If p is true, then q is true. If q is true, then r is true. Therefore, if p, then r. If ABCD is a square, then it is a rectangle. If ABCD is a rectangle, then it is a parallelogram. Therefore, if ABCD is a square, then it is a parallelogram. Or Rule p is true or q is true. p is not true. Therefore, q is true. ABCD is a square or a rectangle. ABCD is not a square. Therefore, ABCD is a rectangle. And Rule p and q are not both true. q is true. Therefore, p is not true. ABCD is not both a square and a rectangle. ABCD is a rectangle. Therefore, ABCD is not a square. An argument that follows one of these patterns correctly has a valid conclusion. EXAMPLE State whether the conclusion is valid or invalid. If the conclusion is valid, name the type of logical argument used. a. If it is raining at noon, Peter’s family will not have a picnic lunch. Peter’s family had a picnic lunch. Therefore, it was not raining at noon. c The conclusion is valid. This is an example of indirect argument. b. If a triangle is equilateral, then it is an acute triangle. Triangle XYZ is an acute triangle. Therefore, triangle XYZ is equilateral. c The conclusion is invalid. c. If x 5 4, then 2x 2 7 5 1. If 2x 2 7 5 1, then 2x 5 8. x 5 4. Therefore, if x 5 4, then 2x 5 8. c The conclusion is valid. This is an example of the chain rule. d. If it is at least 808F outside today, you will go swimming. It is 858F outside today. Therefore, you will go swimming. c The conclusion is valid. This is an example of direct argument. 1000 Student Resources n2pe-9020.indd 1000 11/21/05 10:27:14 AM A compound statement has two or more parts joined by or or and. • For an or compound statement to be true, at least one part must be true. EXAMPLE State whether the compound statement is true or false. a. 12 < 20 and 212 > 220 True b. 2 < 4 and 4 < 3 True True c True, because each part is true. c. 10 > 0 or 210 > 0 True False c False, because one part is false. d. 28 > 27 or 27 > 26 or 26 > 25 False False c True, because at least one part is true. False SKILLS REVIEW HANDBOOK • For an and compound statement to be true, each part must be true. False c False, because every part is false. PRACTICE State whether the conclusion is valid or invalid. If the conclusion is valid, name the type of logical argument used. 1. If Scott goes to the store, then he will buy sugar. If he buys sugar, then he will bake cookies. Scott goes to the store. Therefore, he will bake cookies. 2. If a triangle has at least two congruent sides, then it is isosceles. Triangle MNP has sides 5 in., 6 in., and 5 in. long. Therefore, triangle MNP is isosceles. 3. If a horse is an Arabian, then it is less than 16 hands tall. Andrea’s horse is 13 hands tall. Therefore, Andrea’s horse is an Arabian. 4. If a figure is a rhombus, then it has four sides. Figure WXYZ has four sides. Therefore, WXYZ is a rhombus. 5. Jeff cannot buy both a new coat and new boots. Jeff decides to buy new boots. Therefore, Jeff cannot buy a new coat. 6. If x 5 0, then y 5 4. If y 5 4, then z 5 7. Therefore, if z 5 7, then x 5 0. 7. Kate will order either tacos or burritos for lunch. Kate does not order tacos for lunch. Therefore, Kate orders burritos for lunch. 8. If a triangle is equilateral, then it is equiangular. Triangle ABC is not equiangular. Therefore, triangle ABC is not equilateral. 9. An animal cannot be both a fish and a bird. Courtney’s pet is not a fish. Therefore, Courtney’s pet must be a bird. State whether the compound statement is true or false. 10. 27 < 25 and 25 < 26 11. 6 > 2 or 8 < 4 12. 0 ≤ 21 or 5 ≥ 5 13. 4 ≤ 3 or 12 ≥ 13 14. 3 < 5 and 23 < 25 15. 1 5 21 or 1 5 1 or 1 5 0 16. 7 < 8 and 8 < 12 17. 22 < 2 and 3 ≥ 2 18. 3(24) 5 12 or 23(4) 5 12 19. 28 > 8 or 28 5 8 or 28 ≥ 0 20. 140 Þ 145 or 140 > 2145 or 2140 < 2145 21. 28(9) 5 272 and 8(29) 5 272 22. 22 ≤ 23 and 222 < 223 and 23 > 22 Skills Review Handbook n2pe-9020.indd 1001 1001 11/21/05 10:27:15 AM SKILLS REVIEW HANDBOOK Conditional Statements and Counterexamples A conditional statement has two parts, a hypothesis and a conclusion. When a conditional statement is written in if-then form, the “if” part contains the hypothesis and the “then” part contains the conclusion. An example of a conditional statement is shown below. If a triangle is equiangular, then each angle of the triangle measures 608. Hypothesis Conclusion The converse of a conditional statement is formed by switching the hypothesis and the conclusion. The converse of the statement above is as follows: If each angle of a triangle measures 608, then the triangle is equiangular. EXAMPLE Rewrite the conditional statement in if-then form. Then write its converse and tell whether the converse is true or false. a. Bob will earn $20 by mowing the lawn. If-then form: If Bob mows the lawn, then he will earn $20. Converse: If Bob earns $20, then he mowed the lawn. False b. x 5 8 when 5x 1 1 5 41. If-then form: If 5x 1 1 5 41, then x 5 8. Converse: If x 5 8, then 5x 1 1 5 41. True A biconditional statement is a statement that has the words “if and only if.” You can write a conditional statement and its converse together as a biconditional statement. A triangle is equiangular if and only if each angle of the triangle measures 608. A biconditional statement is true only when the conditional statement and its converse are both true. EXAMPLE Tell whether the biconditional statement is true or false. Explain. a. An angle measures 90° if and only if it is a right angle. Conditional: If an angle is a right angle, then it measures 908. True Converse: If an angle measures 908, then it is a right angle. True c The biconditional statement is true because the conditional and its converse are both true. b. Bonnie has $.50 if and only if she has two quarters. Conditional: If Bonnie has two quarters, then she has $.50. True Converse: If Bonnie has $.50, then she has two quarters. False c The biconditional statement is false because the converse is not true. 1002 Student Resources n2pe-9020.indd 1002 11/21/05 10:27:17 AM A counterexample is an example that shows that a statement is false. SKILLS REVIEW HANDBOOK EXAMPLE Tell whether the statement is true or false. If false, give a counterexample. a. If a polygon has four sides and opposite sides are parallel, then it is a rectangle. c False. A counterexample is the parallelogram shown. b. If x2 5 49, then x 5 7. c False. A counterexample is x 5 27, because (27)2 5 49. PRACTICE Rewrite the conditional statement in if-then form. Then write its converse and tell whether the converse is true or false. 1. The graph of the equation y 5 mx 1 b is a line. 2. You will earn $35 for working 5 hours. 3. Abby can go swimming if she finishes her homework. 4. In a right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse. 5. x 5 5 when 4x 1 8 5 28. 6. The sum of two even numbers is an even number. Tell whether the biconditional statement is true or false. Explain. 7. Two lines are perpendicular if and only if they intersect to form a right angle. 8. x 3 5 27 if and only if x 5 3. 9. A vegetable is a carrot if and only if it is orange. 10. A rhombus is a square if and only if it has four right angles. 11. The graph of a function is a parabola if and only if the function is y 5 x2. 12. An integer is odd if and only if it is not even. Tell whether the statement is true or false. If false, give a counterexample. 13. If an integer is not negative, then it is positive. 14. If you were born in the summer, then you were born in July. 15. If a polygon has exactly 5 congruent sides, then the polygon is a pentagon. 16. If x 5 26, then x2 5 36. 17. If B is 6 inches from A and 8 inches from C, then A is 14 inches from C. 18. If a triangle is isosceles, then it is obtuse. 19. If Charlie has $1.00 in coins, then he has four quarters. 20. If you are in Montana, then you are in the United States. Skills Review Handbook n2pe-9020.indd 1003 1003 11/21/05 10:27:18 AM SKILLS REVIEW HANDBOOK Venn Diagrams A Venn diagram uses shapes to show how sets are related. EXAMPLE Draw a Venn diagram of the positive integers less than 13 where set A consists of factors of 12 and set B consists of even numbers. Positive integers less than 13: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 Positive integers less than 13 A Set A (factors of 12): 1, 2, 3, 4, 6, 12 1 Set B (even numbers): 2, 4, 6, 8, 10, 12 Both set A and set B: 2, 4, 6, 12 B 2 4 3 6 8 12 10 5 11 7 9 Neither set A nor set B: 5, 7, 9, 11 EXAMPLE Use the Venn diagram above to decide if the statement is true or false. Explain your reasoning. a. If a positive integer less than 13 is not even, then it is not a factor of 12. c False. 1 and 3 are not even, but they are factors of 12. b. All positive integers less than 13 that are even are factors of 12. c False. 8 and 10 are even, but they are not factors of 12. PRACTICE Draw a Venn diagram of the sets described. 1. Of the positive integers less than 11, set A consists of factors of 10 and set B consists of odd numbers. 2. Of the positive integers less than 10, set A consists of prime numbers and set B consists of even numbers. 3. Of the positive integers less than 25, set A consists of multiples of 3 and set B consists of multiples of 4. Use the Venn diagrams you drew in Exercises 1–3 to decide if the statement is true or false. Explain your reasoning. 4. The only factors of 10 less than 11 that are not odd are 2 and 10. 5. If a number is neither a multiple of 3 nor a multiple of 4, then it is odd. 6. All prime numbers less than 10 are not even. 7. If a positive odd integer less than 11 is a factor of 10, then it is 5. 8. There are 2 positive integers less than 25 that are both a multiple of 3 and a multiple of 4. 9. If a positive even integer less than 10 is prime, then it is 2. 1004 Student Resources n2pe-9020.indd 1004 11/21/05 10:27:19 AM Mean, Median, Mode, and Range The mean of a data set is the sum of the values divided by the number of values. The mean is also called the average. EXAMPLE The median of a data set is the middle value when the values are written in numerical order. If a data set has an even number of values, the median is the mean of the two middle values. The mode of a data set is the value that occurs most often. A data set can have no mode, one mode, or more than one mode. The range of a data set is the difference between the greatest value and the least value. SKILLS REVIEW HANDBOOK Mean, median, and mode are measures of central tendency; they measure the center of data. Range is a measure of dispersion; it measures the spread of data. Find the mean, median, mode(s), and range of the data. Daily High Temperatures, Week of June 21–27 Day Sunday Monday Tuesday Wednesday Thursday Friday Saturday 76 74 70 69 70 75 78 Temperature (8F) Mean Add the values. Then divide by the number of values. 76 1 74 1 70 1 69 1 70 1 75 1 78 5 512 mean 5 512 4 7 ø 73 The mean of the data is about 738F. Median Write the values in order from least to greatest. Find the middle value(s). 69, 70, 70, 74, 75, 76, 78 median 5 74 Mode Find the value that occurs most often. mode 5 70 Range The median of the data is 748F. The mode of the data is 708F. Subtract the least value from the greatest value. range 5 78 2 69 5 9 The range of the data is 98F. PRACTICE Find the mean, median, mode(s), and range of the data. 1. Apartment rents: $650, $800, $700, $525, $675, $750, $500, $650, $725 2. Ages of new drivers: 15, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17, 18, 18 3. Monthly cell-phone minutes: 581, 713, 423, 852, 948, 337, 810, 604, 897 4. Prices of a CD: $12.98, $14.99, $13.49, $12.98, $13.89, $16.98, $11.98 5. Cookies in a batch: 36, 60, 52, 44, 48, 45, 48, 41, 60, 45, 38, 55, 60, 48, 40 6. Ages of family members: 41, 45, 8, 10, 40, 44, 3, 5, 42, 42, 13, 14, 67, 70 7. Hourly rates of pay: $8.80, $6.50, $10.85, $7.90, $9.50, $9, $8.70, $12.35 8. Weekly quiz scores: 8, 9, 8, 10, 10, 7, 9, 8, 9, 9, 10, 7, 8, 6, 10, 9, 9, 8, 8, 10 9. People on a bus: 9, 14, 5, 22, 18, 30, 6, 25, 18, 12, 15, 10, 8, 22, 10, 11, 20 Skills Review Handbook n2pe-9020.indd 1005 1005 11/21/05 10:27:20 AM SKILLS REVIEW HANDBOOK Graphing Statistical Data There are many ways to display data. An appropriate graph can help you analyze data. The table at the right summarizes how data are shown in some statistical graphs. EXAMPLE Bar Graph Compares data in categories. Circle Graph Compares data as parts of a whole. Line Graph Shows data change over time. Use the bar graph to answer the questions. a. On which day of the week were the greatest Cars Parked in Student Lot number of cars parked in the student lot? 120 b. How many cars were parked in the student lot on Monday? 80 Cars c The tallest bar on the graph is for Friday. So, the answer is Friday. 40 c The bar for Monday shows that about 70 cars were parked in the student lot. EXAMPLE 0 Tu W Th F Use the circle graph to answer the questions. a. Which type of transportation is used almost half the Transportation to School time? Car 45% Bus 20% c Almost half of the total area of the circle is labeled “Car 45%.” So, a car is used almost half the time. Walk or bike 35% b. Which type of transportation is used the least often? c The smallest part of the circle is labeled “Bus 20%.” So, a bus is used the least often. EXAMPLE M Use the line graph to answer the questions. a. In which month(s) was Jamie’s balance Jamie’s Savings Account Balance $250? b. Between which two consecutive months did Jamie’s balance increase the most? c Of the graph’s line segments that have positive slope, the graph is steepest from June to July. So, Jamie’s balance increased the most between June and July. 400 300 Dollars c The points on the graph to the right of $250 show that Jamie’s balance was $250 in May and December. 200 100 0 J F M A M J J A S O N D Month 1006 Student Resources n2pe-9020.indd 1006 11/21/05 10:27:21 AM PRACTICE Friday at Ferraro’s Restaurant 1. At which hour did Ferraro’s have 22 diners? 3. How many diners were at Ferraro’s at 11 P.M.? Were they gone by midnight? Diners 30 2. At which hour did Ferraro’s have the most diners? 20 10 0 4. Between which two consecutive hours did the 5 number of diners at Ferraro’s change the most? 6 7 8 9 10 11 Time (hours since noon) 12 5. How many fewer diners were at Ferraro’s at 10 P.M. than at 6 P.M.? Use the bar graph to answer Exercises 6–8. SKILLS REVIEW HANDBOOK Use the line graph to answer Exercises 1–5. Seasons of Students’ Birthdays 6. In which season were the fewest students born? 12 Students 7. In which season(s) were 7 students born? 8. How many more students were born in spring than in summer? 8 4 0 Use the circle graph to answer Exercises 9–11. Fall Winter Spring Summer Heat Sources for U.S. Homes 9. What is the heat source of more than half the Natural gas 52% Electricity 22% homes in the United States? 10. What percent of homes in the United States are Fuel oil 10% heated with electricity? Other 16% 11. If you randomly selected 500 U.S. homes, about how many would be heated with fuel oil? 12. The table below shows the high temperatures in degrees Fahrenheit for one week. Display the data in a line graph. Mon. Tues. Wed. Thurs. Fri. Sat. Sun. 83 89 79 73 69 67 71 13. A high school conducted a survey to determine the numbers of students involved in various school activities. Display the survey results in a bar graph. Computer club Music club Yearbook club Drama club Student council Chess club 34 75 16 57 28 12 14. The table below shows the items sold at a café in one day. Display the data in a circle graph. Juice Soda Water Muffin Cookie 95 180 100 55 40 Skills Review Handbook n2pe-9020.indd 1007 1007 11/21/05 10:27:22 AM SKILLS REVIEW HANDBOOK Organizing Statistical Data Because it is difficult to analyze unorganized data, it is helpful to organize data using a line plot, stem-and-leaf plot, histogram, or box-and-whisker plot. EXAMPLE Sydney’s math test scores are 90, 85, 88, 95, 100, 77, 85, 100, 80, 77, and 90. a. Draw a line plot to display the data. Make a number line from 75 to 100. Each time a value is listed in the data set, draw an X above the value on the number line. 75 3 3 3 3 3 3 3 3 3 3 3 77 80 85 88 90 95 100 b. Draw a stem-and-leaf plot to display the data. First write the leaves next to their stems. 7 7 7 8 5 8 5 9 0 5 0 10 0 Then order the leaves from least to greatest. 7 7 8 0 5 5 9 0 0 5 10 0 0 0 Key: 7 | 7 5 77 0 7 8 Key: 7 | 7 5 77 c. Draw a histogram to display the data. First make a frequency table. Use equal intervals. Then make a histogram. Sydney’s Math Test Scores Tally Frequency 71–80 3 3 81–90 5 5 91–100 3 3 6 Frequency Score 4 2 0 71–80 81– 90 Score 91–100 d. Draw a box-and-whisker plot to display the data. Write the data in order from least to greatest. Ordered data are divided into a lower half and an upper half by the median. The median of the lower half is the lower quartile, and the median of the upper half is the upper quartile. 77 77 Low value 80 Lower quartile 85 85 88 90 90 Median Plot the median, quartiles, and low and high values below a number line. Draw a box between quartiles with a vertical line through the median as shown. Draw whiskers to the low and high values. 95 100 100 Upper quartile High value Sydney’s Math Test Scores 75 80 77 80 85 90 88 95 100 95 100 1008 Student Resources n2pe-9020.indd 1008 11/21/05 10:27:23 AM PRACTICE SKILLS REVIEW HANDBOOK Use the following list of ticket prices to answer Exercises 1–4: $50, $42, $65, $54, $70, $65, $59, $30, $67, $49, $54, $30, $73, $47, and $54. 1. Draw a line plot to display the data. 2. How many ticket prices are $50 or less? 3. Draw a stem-and-leaf plot to display the data. 4. What is the range of ticket prices costs? Use the following list of hourly wages of employees to answer Exercises 5–8: $8.50, $6, $10, $14.25, $5.75, $7, $6.50, $14, $10, $9, $6.50, $8.25, $8.50, $11.25, $7, $16, $12, $6, $6.75. 5. Draw a histogram to display the data. Begin with the interval $5.00 to $6.99. 6. Copy and complete: The greatest number of employees earn from ? to ? per hour. 7. Draw a box-and-whisker plot to display the data. 8. Copy and complete: About half of the employees have an hourly wage of ? or less. Use the line plot, which shows the results of a survey asking people the average number of e-mails they receive daily, to answer Exercises 9 and 10. 9. Copy and complete: Most people surveyed receive an average of ? e-mails per day. 3 3 3 3 3 4 5 3 3 3 3 3 3 3 7 10 12 15 17 10. How many people receive an average of more than 10 e-mails per day? Use the stem-and-leaf plot, which shows the weights (in pounds) of dogs at an animal shelter, to answer Exercises 11–13. 11. How many dogs were at the shelter? 12. Find the median of the data. 2 2 5 5 9 3 1 3 5 8 4 0 0 1 2 2 5 6 7 5 0 3 5 8 9 6 4 5 Key: 2 | 2 5 22 13. Find the range of the data. Use the histogram to answer Exercises 14–16. Baseball Game Attendance 9 9 –6 60 –5 9 50 –4 40 –3 9 30 20 oldest group? –2 9 0 –1 9 16. Which age group had the same attendance as the 20 9 the baseball game? 40 10 15. How many children up to the age of 9 years attended 0– baseball game? Which had the least? People 14. Which age group had the greatest attendance at the Age (years) Use the box-and-whisker plot to answer Exercises 17–19. 17. What is the median number of songs on Sam’s CDs? Number of Songs on Sam’s CDs 10 12 14 10 11 12 14 16 18 18. What is the upper quartile of songs on Sam’s CDs? 19. What is the least number of songs on one of Sam’s CDs? What is the greatest number? 18 Skills Review Handbook n2pe-9020.indd 1009 1009 11/21/05 10:27:24 AM E xt xtrra P ra racc tice Chapter 1 1.1 Graph the numbers on a number line. 5 , 0.2, 2Ï} 5 1. 22, } 2 , 2} 4 3 4 , 1, 21.2, Ï 3 , 1.9 2. 2} 3 } 1 , 4, Ï} 3. 3.7, 2Ï 7 , 2} 15 2 } 1.1 Perform the indicated conversion. EXTRA PRACTICE 4. 18 feet to inches 5. 20 ounces to pounds 6. 3 years to hours 1.2 Evaluate the expression for the given value of the variable. 8. 3x 2 2 x 1 7 when x 5 21 7. 22p 1 5 when p 5 25 9. 8z3 2 6z when z 5 2 1.2 Simplify the expression. 10. 2y 2 2 3y 1 5y 2 11. 4r 2 2 5r 1 2r 2 1 12 12. 2w 3 1 w 2 2 7w 2 2 8w 3 13. 2(b 1 5) 1 3(2b 2 10) 14. 27(t 2 1 2) 1 9(t 2 2) 15. 4(m 2 3) 2 5(m2 2 m) 1.3 Solve the equation. Check your solution. 16. 3a 1 2 5 11 17. 29 5 b 2 14 18. 8 2 0.5c 5 1 19. 23n 2 7 5 2n 1 17 20. 12m 5 15m 2 7.5 21. 6p 1 1 5 21 2 4p 22. 6(x 1 1) 5 2x 2 10 23. 4(y 2 3) 5 2(y 1 8) 24. 11(z 2 5) 5 2(z 1 6) 2 13 1.4 Solve the equation for y. Then find the value of y for the given value of x. 25. 6y 2 x 5 18; x 5 2 26. 2x 1 3y 5 12; x 5 26 27. 4y 2 9x 5 230; x 5 6 28. 3x 2 xy 5 20; x 5 8 29. 4y 1 6xy 5 10; x 5 22 30. 5x 1 8y 1 4xy 5 0; x 5 21 1.5 Look for a pattern in the table. Then write an equation that represents the table. 31. 32. x 0 1 2 3 y 25 22 19 16 x 0 1 2 3 y 1.5 4 6.5 9 1.6 Solve the inequality. Then graph the solution. 33. x 1 2 > 9 34. 213 2 3x < 11 35. 4x 2 9 ≤ 2x 1 1 36. 23x 2 8 ≥ 29x 1 10 37. 27 < x 1 3 ≤ 1 38. 24 ≤ 3x 2 7 ≤ 4 39. 29 ≤ 5 2 2x < 7 40. x 1 3 < 22 or x 2 7 > 0 41. 2x 1 9 ≥ 3 or 25x 1 1 ≤ 0 1.7 Solve the equation. Check for extraneous solutions. 42. g 1 5 5 4 43. 1 2 }3 q 2 }3 5 1 44. 10 2 3t 5 t 1 4 45. 3z 1 1 5 26z 1.7 Solve the inequality. Then graph the solution. 46. a < 2 47. 2c > 14 48. g 1 11 ≥ 2 49. 4j 2 7 ≤ 9 50. 0.25m 1 3 ≥ 1 51. 10 2 2p > 9 52. 0.6r 1 8 ≤ 17 53. 5t 2 9 1 9 < 10 1010 Student Resources n2pe-9030.indd 1010 10/17/05 12:26:30 PM Chapter 2 2.1 Tell whether the relation is a function. Explain. 1. 2. Input Output 1 21 21 2 2 0 Input 3 3 1 4 5 2 Output 3. Input 4. Output 3 6 21 22 Input Output 27 14 8 4 28 9 6 12 4 2.2 Find the slope of the line passing through the given points. Then tell whether the line rises, falls, is horizontal, or is vertical. 6. (2, 21), (8, 21) 7. (3, 5), (3, 212) 8. (1, 8), (21, 24) 2.2 Tell whether the lines are parallel, perpendicular, or neither. 9. Line 1: through (5, 24) and (24, 2) 10. Line 1: through (0, 24) and (22, 2) Line 2: through (25, 24) and (22, 22) Line 2: through (4, 23) and (5, 26) 2.3 Graph the equation using any method. 11. y 5 2x 2 2 12. y 5 2x 1 2 2x 2 1 13. f(x) 5 } 3 14. x 1 2y 5 26 15. 24x 1 5y 5 10 16. y 2 2 5 0 17. 22x 5 6y 1 5 18. 2y 1 10 5 22.5x EXTRA PRACTICE 5. (23, 0), (5, 24) 2.4 Write an equation of the line that satisfies the given conditions. 19. m 5 7, b 5 23 1, b 5 4 20. m 5 } 3 21. m 5 0, passes through (7, 22) 1 , passes through (3, 6) 22. m 5 2} 4 23. passes through (21, 23) and (2, 7) 24. passes through (4, 22) and (0, 4) 2.5 The variables x and y vary directly. Write an equation that relates x and y. Then find y when x 5 22. 25. x 5 2, y 5 4 26. x 5 21, y 5 3 27. x 5 228, y 5 27 28. x 5 6, y 5 24 2.6 In Exercises 29 and 30, (a) draw a scatter plot of the data, (b) approximate the best-fitting line, and (c) estimate y when x 5 12. 29. x 1 2 3 4 5 y 8 11 13 16 18 30. x 1 2 3 4 5 y 50 41 37 22 20 2.7 Graph the function. Compare the graph with the graph of y 5 x. 31. y 5 x 1 3 32. y 5 22x 2 5 33. y 5 3x 1 1 2 2 1 x12 13 34. y 5 2} 2 2.8 Graph the inequality in a coordinate plane. 35. x < 4 36. y ≥ 22 37. y ≤ 2x 2 1 38. x 1 2y > 8 39. 2x 2 4y ≤ 6 40. 3x 1 4y > 12 41. y < x 1 1 42. y ≥ 3x 2 2 2 1 Extra Practice n2pe-9030.indd 1011 1011 10/17/05 12:26:34 PM Chapter 3 3.1 Graph the linear system and estimate the solution. Then check the solution algebraically. 1. y 5 2x 2 1 2. y 5 2x 1 3 y5x24 3. x 1 2y 5 6 y 5 24x 4. 22x 1 7y 5 27 25x 1 6y 5 22 4x 2 14y 5 14 3.2 Solve the system using any algebraic method. 5. 25x 2 y 5 23 6. 4x 2 2y 5 26 7. 4x 1 3y 5 25 23x 1 y 5 23 12x 1 4y 5 10 x 2 4y 5 9 8. 3x 1 2y 5 4 27x 2 5y 5 27 EXTRA PRACTICE 3.3 Graph the system of inequalities. 9. x > 4 10. x 1 y < 22 11. x ≤ 5 x 2 3y > 6 y>3 y>x y ≥ 21 12. x > 23 x≤2 2x 1 3y < 10 y > 24x 3.4 Solve the system using any algebraic method. 13. 3x 1 y 2 z 5 26 14. x 1 y 2 z 5 7 2x 1 2y 1 3z 5 21 5x 2 2y 1 6z 5 54 15. 2x 1 y 2 2z 5 1.5 16. 26x 1 y 1 9z 5 4 4x 2 y 1 5z 5 26 2x 1 y 2 2z 5 6 2x 2 3y 2 z 5 26 8x 1 5y 2 4z 5 10 2x 2 3y 1 z 5 2 4x 1 2y 2 2z 5 20 3.5 Perform the indicated operation. 17. F G F G 26 7 1 0 3 F 26 2 28 1 G 3 2 29 18. 2} 3 4 21 19. F 10 17 29 26 4 11 3.6 Find the product. If the product is not defined, state the reason. 20. F GF 4 1 23 0 G 27 5 7 23 21. F GF G 216 2 4 15 3.7 Evaluate the determinant of the matrix. 23. F G 5 8 22 10 24. F G 13 7 211 24 25. 22. F 1 23 22 7 4 0 27 2 3 F 5 21 0 4 22 9 2 GF 26 8 22 24 29 4 G G 12 27 3 G F 26. G F 6 0 5 24 2 1 1 0 0.5 G 3.7 Use Cramer’s rule to solve the linear system. 27. 2x 1 y 5 28 28. 8x 1 3y 5 1 25x 2 2y 5 13 29. 2x 2 2y 2 3z 5 9 7x 1 3y 5 21 30. 2x 1 y 1 3z 5 4 3x 1 z 5 10 x1y50 28x 1 4y 1 z 5 27 x 1 2y 1 3z 5 21 3.8 Find the inverse of the matrix. 31. F G 3 7 3 8 32. F G 1 4 0 5 33. F 22 25 3 8 G 34. F G 9 2 18 5 3.8 Use an inverse matrix to solve the linear system. 35. x 1 3y 5 24 22x 1 y 5 234 36. 2x 1 3y 5 6 2x 2 6y 5 29 37. 3x 2 8y 5 0 2x 1 y 5 219 38. x 1 y 5 7 25x 1 3y 5 23 1012 Student Resources n2pe-9030.indd 1012 10/17/05 12:26:35 PM Chapter 4 4.1 Graph the function. Label the vertex and axis of symmetry. 1. y 5 3x 2 1 5 2. y 5 2x2 2 4x 2 4 3. y 5 22x2 1 4x 1 1 4. y 5 2x 2 1 5x 1 6 4.2 Graph the function. Label the vertex and axis of symmetry. 5. y 5 4(x 2 2)2 1 1 6. y 5 2(x 1 3)2 2 2 7. y 5 3(x 2 1)(x 2 5) 1 (x 1 3)(x 1 2) 8. y 5 } 2 4.2 Write the quadratic function in standard form. 9. y 5 7(x 1 2)(x 1 4) 10. y 5 2(x 1 5)(x 2 3) 11. y 5 (x 2 7)2 1 7 12. y 5 2(x 1 1)2 2 4 13. x2 2 4x 1 4 14. t 2 2 11t 2 26 15. x2 1 21x 1 108 16. b2 2 400 18. x2 2 11x 1 24 5 0 19. c 2 1 6c 5 55 20. n2 5 5n 4.3 Solve the equation. 17. x2 1 5x 2 14 5 0 4.4 Factor the expression. If the expression cannot be factored, say so. 21. 2x2 1 x 2 15 22. 10a2 2 19a 1 7 23. 3r 2 1 9r 2 4 EXTRA PRACTICE 4.3 Factor the expression. If the expression cannot be factored, say so. 24. 4t 2 1 8t 1 3 4.4 Find the zeros of the function by rewriting the function in intercept form. 25. y 5 81x 2 2 16 26. y 5 2x 2 2 9x 2 5 27. y 5 4x 2 1 18x 1 18 28. y 5 23x 2 2 30x 2 27 4.5 Simplify the expression. } 29. Ï 56 } } Î 47 } 30. 3Ï 2 p Ï 50 31. 34. p2 1 6 5 127 35. (x 2 5)2 5 10 6 32. } } 1 1 Ï2 } 4.5 Solve the equation. 33. b2 5 8 36. 3(x 1 2)2 2 4 5 11 4.6 Write the expression as a complex number in standard form. 37. (5 1 2i) 1 (6 2 5i) 38. 23i(7 1 i) 1 1 2i 39. } 3 2 8i (3 2 2i) 1 2i 40. } (21 1 7i) 2 (2 1 3i) 43. 2c 2 2 12c 1 6 5 0 44. 3z2 2 3z 1 9 5 0 47. 4s 2 1 3s 5 12 48. 22r 2 5 r 1 17 51. 2x2 1 7x 1 6 > 1 52. 3x 2 1 16x 1 2 ≤ 3x 4.7 Solve the equation by completing the square. 41. x2 1 6x 5 10 42. x2 2 9x 2 2 5 0 4.8 Use the quadratic formula to solve the equation. 45. x2 1 10x 2 10 5 0 46. x2 2 x 2 1 5 0 4.9 Solve the inequality using any method. 49. x2 2 10x ≥ 0 50. x2 2 8x 1 12 < 0 4.10 Write a quadratic function in standard form for the parabola that passes through the given points. 53. (21, 26), (0, 27), (2, 9) 54. (22, 21), (1, 2), (3, 26) 55. (23, 36), (0, 36), (2, 16) Extra Practice n2pe-9030.indd 1013 1013 10/17/05 12:26:37 PM Chapter 5 5.1 Write the answer in scientific notation. 1. (3.4 3 103)(2.8 3 108) 4.6 3 1027 3. } 9.2 3 1029 2. (5.8 3 1026) 4 5.1 Simplify the expression. Tell which properties of exponents you used. 214x23y 5 4. } 35xy 3 5. (4a5b22)23 xy21 7x 3 7. } p} y24 x 2y 6. (2r 3s 3)(r27s5) EXTRA PRACTICE 5.2 Graph the polynomial function. 8. f(x) 5 x4 9. f(x) 5 x 3 1 x 1 4 10. f(x) 5 2x 3 1 3x 11. f(x) 5 x5 1 2x 3 5.3 Perform the indicated operation. 12. (4z3 1 9) 1 (3z2 2 4z 2 2) 13. (x2 1 3x 2 1) 2 (4x2 1 7) 14. (3x 2 4) 3 5.4 Factor the polynomial completely using any method. 15. 3x4 1 18x 3 1 27x2 16. 343x 3 1 1000 17. 2x 3 1 x2 2 8x 2 4 5.4 Find the real-number solutions of the equation. 18. 3x 3 1 18x2 5 48x 19. x4 1 32 5 14x2 20. 2x 3 1 48 5 3x2 1 32x 5.5 Divide using polynomial long division or synthetic division. 21. (2x 3 1 4x2 2 5x 1 16) 4 (x 2 3) 22. (x4 1 2x 3 2 7x2 2 14) 4 (x 1 2) 5.6 Find all real zeros of the function. 23. f(x) 5 2x 3 1 3x2 2 8x 1 3 24. f(x) 5 2x4 1 x 3 2 53x2 2 14x 1 20 5.7 Determine the possible numbers of positive real zeros, negative real zeros, and imaginary zeros of the function. 25. f(x) 5 2x 3 1 2x2 2 11x 2 1 26. f(x) 5 4x5 1 3x2 2 8x 2 10 27. f(x) 5 x4 2 3x 3 2 7x 2 13 5.8 Estimate the coordinates of each turning point and state whether each corresponds to a local maximum or a local minimum. Then estimate all real zeros and determine the least degree the function can have. 28. 1 29. y 1 x 1 30. y y x 1 1 3 x 5.9 Use finite differences and a system of equations to find a polynomial function that fits the data in the table. 31. x 1 2 3 4 5 6 y 2.5 11 27.5 55 96.5 155 32. x 1 2 3 4 5 6 y 27 26 39 188 525 1158 1014 Student Resources n2pe-9030.indd 1014 10/17/05 12:26:39 PM Chapter 6 6.1 Find the indicated real nth root(s) of a. 1. n 5 4, a 5 81 2. n 5 3, a 5 512 3. n 5 5, a 5 2243 6.1 Evaluate the expression without using a calculator. 5} 4 3 } 22 6. (Ï 216 ) 5. 645/6 4. 3621/2 7. (Ï 232 ) 6.1 Solve the equation. Round the result to two decimal places when appropriate. 8. x 3 5 28 9. x4 1 9 5 90 10. (x 2 3) 5 5 60 11. 24x6 5 2400 12. 45/2 p 421/2 5} 5} 16. 5Ï 7 2 7Ï 7 173/7 13. } 174/7 3} Ï135 15. } 3} Ï5 3241/4 18. } 421/4 19. 4Ï 108 p 2Ï 4 4} 3} 17. Ï 2 1 2Ï 128 3} 14. (Ï 5 p Ï 5 ) } 4 3} 3} 6.2 Write the expression in simplest form. Assume all variables are positive. } 20. Ï20x6y7 21. Î } 5} Ï18x3y14z20 22. 4 x5 y 23. } 16 3} EXTRA PRACTICE 6.2 Simplify the expression. 3} Ï16x7y 2 p Ï6xy 5 x . Perform the indicated operation and 6.3 Let f(x) 5 2x 1 4, g(x) 5 x3, and h(x) 5 } 4 state the domain. 24. f(x) 1 g(x) 25. g(x) 2 f(x) 26. g(x) p h(x) f (x) 27. } g(x) 28. f(g(x)) 29. g(h(x)) 30. h(f(x)) 31. f(f(x)) 6.4 Verify that f and g are inverse functions. x21 33. f(x) 5 3x2 1 1, x ≥ 0; g(x) 5 } 3 1x 1 2 32. f(x) 5 2x 2 4, g(x) 5 } 2 1 1/2 2 6.4 Find the inverse of the function. 34. f(x) 5 5x 2 3 4x 1 2 35. f(x) 5 } 3 1 x 2, x ≥ 0 36. f(x) 5 } 2 37. f(x) 5 2x6 1 2, x ≤ 0 4x4 2 1 , x ≥ 0 38. f(x) 5 } 18 39. f(x) 5 32x5 1 4 6.5 Graph the function. Then state the domain and range. 1 Ï} 40. y 5 2} x 3 3} 44. y 5 22Ï x 2 1 1 2 2 3} 41. y 5 } Ïx 5 3} 45. f(x) 5 3Ï x 5 Ï} 42. y 5 } x 6 43. y 5 Ï x 1 2 2 3 1 Ï} 46. g(x) 5 2} x22 2 47. h(x) 5 2Ï x 1 3 1 4 } } 6.6 Solve the equation. Check your solution. } 48. Ï 2x 1 3 5 7 3} 51. 2Ï 8x 1 9 5 5 } 54. x 2 8 5 Ï 18x } 3} 49. 25Ï x 1 1 1 12 5 2 50. Ï 5x 2 1 1 6 5 10 52. 7x4/3 5 175 53. (x 2 2) 3/4 5 1 } 55. x 5 Ï 4x 2 3 } } 56. Ï 2x 1 1 1 5 5 Ï x 1 12 2 8 Extra Practice n2pe-9030.indd 1015 1015 10/17/05 12:26:40 PM Chapter 7 7.1 Graph the function. State the domain and range. 4 1. y 5 } 3 1 2 x 2. y 5 22 p 2x 3. y 5 3x 2 3 2 2 1 p 3x 1 1 1 2 4. y 5 } 4 7. y 5 (0.8) x 2 3 2 2 2 8. y 5 2 } 3 7.2 Graph the function. State the domain and range. 3 5. y 5 } 5 1 2 x 1 6. y 5 22 } 4 1 2 x 1 2 x 11 EXTRA PRACTICE 7.3 Simplify the expression. 9. e23 p e28 10. 28e 3x 12. } 21e2x } (2e 2x)25 11. Ï81e 8x 7.3 Graph the function. State the domain and range. 13. y 5 0.5e 3x 15. y 5 1.5e x 1 1 1 3 14. y 5 2e2x 2 2 16. y 5 e 3(x 2 2) 1 1 7.4 Evaluate the logarithm without using a calculator. 1 17. log4 } 16 18. log 6 6 19. log5 125 64 20. log3/4 } 27 22. 10log 9 23. log4 16x 24. eln 5 26. y 5 log1/2 (x 2 4) 27. y 5 log5 x 1 3 28. y 5 log3 (x 2 2) 1 1 100x 2 30. log } y 31. ln 20x 3y 2 32. log 2 Ï 8x4 7.4 Simplify the expression. 21. 5log5 x 7.4 Graph the function. State the domain and range. 25. y 5 log 7 x 7.5 Expand the expression. 2x 29. log5 } 5 3} 7.5 Condense the expression. 33. log4 20 1 4 log4 x 34. log 7 1 2 log x 2 5 log y 35. 0.5 ln 100 2 2 ln x 1 8 ln y 7.5 Use the change-of-base formula to evaluate the logarithm. 36. log 2 5 37. log4 80 38. log5 100 39. log 7 27 7.6 Solve the equation. Check for extraneous solutions. x23 40. 24x 1 2 5 8x 1 2 1 41. } 9 43. ln (3x 1 7) 5 ln (x 2 1) 44. log5 (3x 1 2) 5 3 1 2 5 33x 1 1 42. 79x 5 18 45. log 6 (x 1 9) 1 log6 x 5 2 7.7 Write an exponential function y 5 ab x whose graph passes through the given points. 46. (1, 8), (2, 32) 47. (1, 3), (3, 12) 48. (2, 29), (5, 2243) 49. (1, 4), (2, 4) 7.7 Write a power function y 5 axb whose graph passes through the given points. 50. (2, 2), (5, 16) 51. (3, 27), (6, 432) 52. (1, 4), (8, 17) 53. (5, 36), (10, 220) 1016 Student Resources n2pe-9030.indd 1016 10/17/05 12:26:42 PM Chapter 8 8.1 The variables x and y vary inversely. Use the given values to write an equation relating x and y. Then find y when x 5 25. 1. x 5 2, y 5 210 1 , y 5 24 2. x 5 } 3 3. x 5 23, y 5 25 2 4. x 5 25, y 5 2} 5 8.1 Determine whether x and y show direct variation, inverse variation, or neither. 5. 6. 7. y 2.5 11 3.5 8.75 16 5 6.4 12.5 8 10 y 32 1 4 20 5 y 2.5 8. x y 30 1 12 14 61 3 4 12.5 16 85 8 1.5 8 20 24 92 12 1 9 22.5 27 105 15 0.8 EXTRA PRACTICE x x x 8.2 Graph the function. State the domain and range. 6 9. y 5 } x 22 1 3 10. y 5 } x 5 22 11. y 5 } x21 4x 1 19 12. y 5 } x13 x2 1 1 14. y 5 } 2 x 1 4x 1 3 2 1 2x 2 3 15. y 5 x} x12 2x 2 2 8 16. f(x) 5 } x 2 2 2x 2 2 5x 2 84 19. x} 2x 2 2 98 2 1 7x 1 10 20. x} x 2 2 7x 1 10 8.3 Graph the function. x 13. y 5 } x2 2 4 8.4 Simplify the rational expression, if possible. x2 1 x 2 6 17. } x 2 1 9x 1 18 x 3 2 100x 18. } 4 x 1 20x 3 1 100x 2 8.4 Multiply or divide the expressions. Simplify the result. 6x 2y 2y 21. } p} xy 2 9x 3 2x 2 2 x 2 6 p x 2 1 x 22. } } 2x 2 1 5x 1 3 x 2 2 4 3x 2 1 15x p (x 2 2 x 2 30) 23. } 2 x 2 12x 1 36 12x 8y 3y 2 24. } 4 } 5y 5 x2 6x 2 1 x 2 1 4 6x 2 2 2x 25. } } 4x 3 1 4x 2 x 2 2 4x 2 5 x 2 2 4x 2 32 4 x 26. } } 2x 2 2 13x 2 24 4x 2 2 9 8.5 Add or subtract the expressions. Simplify the result. x2 2 1 27. } } x11 x11 x15 1 1 28. } } x16 x22 5 1 35 29. } } x12 x 2 2 3x 2 10 x 3 31. } 1 }13 x 3 x 24 32. } x11 2 }2} x12 x2 2 x 2 6 8.5 Simplify the complex fraction. x 2x 1 1 30. } 3 51} x } } 2 }12 8.6 Solve the equation. Check for extraneous solutions. 7 14 33. } 5} 3x 2 7 x11 1 1 2 523 34. } } } 3 x x2 4 52 35. 2 2 } } x12 x 4 1 6x 2 5 3x 36. } } } x22 x12 x2 2 4 Extra Practice n2pe-9030.indd 1017 1017 10/17/05 12:26:43 PM Chapter 9 9.1 Find the distance between the two points. Then find the midpoint of the line segment joining the two points. 1. (25, 0), (5, 4) 2. (2, 1), (3, 7) 3. (212, 12), (14, 24) 4. (12, 21), (18, 29) 9.2 Graph the equation. Identify the focus, directrix, and axis of symmetry of the parabola. 5. y 2 5 2x 6. x2 5 24y 7. 14x 2 5 221y 8. 12y 2 1 3x 5 0 9.3 Graph the equation. Identify the radius of the circle. EXTRA PRACTICE 9. x2 1 y 2 5 4 10. x2 1 y 2 5 14 11. 3x 2 1 3y 2 5 75 12. 16x2 1 16y 2 5 4 9.3 Write the standard form of the equation of the circle that passes through the given point and whose center is at the origin. 13. (8, 0) 14. (0, 29) 15. (7, 21) 16. (25, 211) 9.4 Graph the equation. Identify the vertices, co-vertices, and foci of the ellipse. 2 x2 1 y 5 1 17. } } 81 16 y2 18. x2 1 } 5 1 9 19. 9x2 1 4y 2 5 576 20. 49x 2 1 64y 2 5 12,544 9.4 Write an equation of the ellipse with the given characteristics and center at (0, 0). 21. Vertex: (4, 0) 22. Vertex: (0, 25) Co-vertex: (0, 2) 23. Vertex: (9, 0) Co-vertex: (4, 0) 24. Co-vertex: (0, 10) Focus: (23, 0) Focus: (8, 0) 9.5 Graph the equation. Identify the vertices, foci, and asymptotes of the hyperbola. 2 x2 2 y 5 1 25. } } 36 16 26. x2 2 y 2 5 4 27. 49y 2 2 81x2 5 3969 9.5 Write an equation of the hyperbola with the given foci and vertices. 28. Foci: (0, 28), (0, 8) Vertices: (0, 26), (0, 6) 29. Foci: (22, 0), (2, 0) Vertices: (21, 0), (1, 0) 30. Foci: (0, 25), (0, 5) } } Vertices: (0, 23Ï 2 ), (0, 3Ï2 ) 9.6 Graph the equation. Identify the important characteristics of the graph. y2 (x 2 3)2 31. } 1 } 5 1 25 9 32. (x 1 2)2 1 (y 2 1)2 5 4 (x 1 1)2 33. (y 2 4)2 2 } 5 1 16 9.6 Classify the conic section and write its equation in standard form. Then graph the equation. 34. x2 1 y 2 1 2x 1 2y 2 7 5 0 35. 9x2 1 4y 2 2 72x 1 16y 1 16 5 0 36. 9x2 2 4y 2 1 16y 2 52 5 0 37. x2 2 6x 2 4y 1 17 5 0 9.7 Solve the system. 38. x2 1 y 2 5 4 2 2 9x 2 4y 5 36 39. y 5 x 2 2 2 2 x 1 y 2 6x 2 4y 2 12 5 0 40. y 2 5 x 2 5 9x2 2 25y 2 5 225 1018 Student Resources n2pe-9030.indd 1018 10/17/05 12:26:45 PM Chapter 10 10.1 For the given password configuration, determine how many passwords are possible if (a) digits and letters can be repeated, and (b) digits and letters cannot be repeated. 1. 8 digits 2. 8 letters 3. 5 letters followed by 1 digit 4. 2 digits followed by 2 letters 10.1 Find the number of permutations. 5. P 6. 5 2 P 7. 6 1 P 8. 9 9 P 12 4 10.1 Find the number of distinguishable permutations of the letters in the word. 10. CHOCOLATE 11. STRAWBERRY EXTRA PRACTICE 9. VANILLA 12. COFFEE 10.2 Find the number of combinations. 13. 7C3 14. C 15. 4 1 C 16. 10 9 C 15 6 10.2 Use the binomial theorem to write the binomial expansion. 17. (x 2 3) 3 18. (2x 1 3y)4 20. (x 3 1 y 2)6 19. (p2 1 4) 5 10.3 You have an equally likely chance of choosing any integer from 1 through 25. Find the probability of the given event. 21. An odd number is chosen. 22. A multiple of 3 is chosen. 10.3 Find the probability that a dart thrown at the given target will hit the shaded region. Assume the dart is equally likely to hit any point inside the target. 23. 24. 25. 8 10 4 4 8 20 10.4 Events A and B are disjoint. Find P(A or B). 26. P(A) 5 0.4, P(B) 5 0.15 27. P(A) 5 0.3, P(B) 5 0.5 28. P(A) 5 0.7, P(B) 5 0.21 10.4 Find the indicated probability. State whether A and B are disjoint events. 29. P(A) 5 0.25 P(B) 5 0.55 P(A or B) 5 ? P(A and B) 5 0.2 30. P(A) 5 0.52 P(B) 5 0.15 P(A or B) 5 0.67 P(A and B) 5 ? 31. P(A) 5 0.54 32. P(A) 5 0.5 P(B) 5 0.28 P(A or B) 5 0.65 P(A and B) 5 ? P(B) 5 0.4 P(A or B) 5 ? P(A and B) 5 0.3 10.5 Find the probability of drawing the given cards from a standard deck of 52 cards (a) with replacement and (b) without replacement. 33. A jack, then a 3 34. A club, then another club 35. A black ace, then a red card 10.6 Calculate the probability of tossing a coin 15 times and getting the given number of heads. 36. 1 37. 4 38. 7 39. 15 Extra Practice n2pe-9030.indd 1019 1019 10/17/05 12:26:46 PM Chapter 11 11.1 Find the mean, median, mode, range, and standard deviation of the data set. 1. 5, 5, 6, 9, 11, 12, 14, 16, 16, 16 2. 16, 18, 29, 30, 34, 35, 35, 38, 46 3. 24, 23, 23, 4, 1, 0, 0, 23, 22, 10, 11 4. 1.7, 2.2, 1.8, 3.0, 0.4, 1.2, 2.8, 2.9 5. 4.5, 5.7, 4.3, 6.9, 22.1, 5.7, 21.2, 3.8 6. 27.2, 3.9, 2.6, 29.1, 2.5, 27.2, 3.9, 27.2 11.2 Find the mean, median, mode, range, and standard deviation of the given data set and of the data set obtained by adding the given constant to each data value. EXTRA PRACTICE 7. 33, 36, 36, 39, 49, 56; constant: 2 8. 10, 12, 14, 16, 16, 18, 19; constant: 21 11.2 Find the mean, median, mode, range, and standard deviation of the given data set and of the data set obtained by multiplying each data value by the given constant. 9. 22, 22, 5, 4, 2, 22, 8, 3; constant: 1.5 10. 52, 52, 76, 56, 67, 89, 70; constant: 3 11.3 A normal distribution has a mean of 2.7 and a standard deviation of 0.3. Find the probability that a randomly selected x-value from the distribution is in the given interval. 11. Between 2.4 and 2.7 12. At least 3.0 13. At most 2.1 11.4 Identify the type of sample described. Then tell if the sample is biased. Explain your reasoning. 14. The owner of a movie rental store wants to know how often her customers rent movies. She asks every tenth customer how many movies the customer rents each month. 15. A school wants to consult parents about updating its attendance policy. Each student is sent home with a survey for a parent to complete. The school uses only surveys that are returned within one week. 11.4 Find the margin of error for a survey that has the given sample size. Round your answer to the nearest tenth of a percent. 16. 100 17. 600 18. 2900 19. 5000 11.4 Find the sample size required to achieve the given margin of error. Round your answer to the nearest whole number. 20. 61% 21. 62% 22. 65.5% 23. 66.2% 11.5 Use a graphing calculator to find a model for the data. Then graph the model and the data in the same coordinate plane. 24. 25. x 0 2 4 6 8 10 12 14 y 210 23 4 10 14 20 21 36 x 1 2 3 4 5 6 7 8 y 0.5 0.8 1.1 3 9 30 90 280 1020 Student Resources n2pe-9030.indd 1020 10/17/05 12:26:47 PM Chapter 12 12.1 For the sequence, describe the pattern, write the next term, and write a rule for the nth term. 1 , 2 , 1, 4 , . . . 2. } } } 3 3 3 1. 9, 16, 25, 36, . . . 3. 12.5, 7, 1.5, 24, . . . 12.1 Write the series using summation notation. 1 1 2 1 3 1 4 1 1 1... 5. } } } } } 7 6 8 9 2 4. 16 1 32 1 48 1 64 1 . . . 1 144 12.1 Find the sum of the series. 5 5 ∑ (3i 1 2) 7. i51 ∑ 4i2 6 8. i50 8 n ∑} n54 n 1 3 ∑ k3 9. k56 12.2 Write a rule for the nth term of the arithmetic sequence. Then graph the first six terms of the sequence. 10. a5 5 15, d 5 6 11 , d 5 2 2 12. a 6 5 2} } 5 5 11. a10 5 278, d 5 210 EXTRA PRACTICE 6. 12.2 Write a rule for the nth term of the arithmetic sequence. Then find a15. 13. 11, 20, 29, 38, . . . 7 , 5 , 1, . . . 15. 3, } } 3 3 14. 28, 215, 222, 229, . . . 12.2 Write a rule for the nth term of the arithmetic sequence that has the two given terms. 16. a2 5 9, a7 5 37 14 , a 5 2 42 18. a 3 5 2} } 10 5 5 17. a 8 5 10.5, a16 5 18.5 12.3 Write a rule for the nth term of the geometric sequence. Then find a10. 1 , 1 , 1 , 1, . . . 19. } } } 27 9 3 16 , 64 , 256 , . . . 21. 4, } } } 3 9 27 20. 5, 4, 3.2, 2.56, . . . 12.3 Find the sum of the geometric series. 4 22. ∑ 3(4)i 2 1 i51 7 23. ∑ 0.5(23)i 2 1 i51 5 24. ∑ 10 1 }35 2 7 i21 25. i51 ∑ 2(1.2)i 2 1 i51 12.4 Find the sum of the infinite geometric series, if it exists. 26. 8 1 4 1 2 1 1 1 . . . 27. 2 2 4 1 8 2 16 1 . . . 28. 26.75 1 4.5 2 3 1 2 2 . . . 12.4 Write the repeating decimal as a fraction in lowest terms. 29. 0.333. . . 30. 0.898989. . . 31. 0.212121. . . 32. 1.50150150. . . 12.5 Write a recursive rule for the sequence. The sequence may be arithmetic, geometric, or neither. 33. 2.5, 5, 10, 20, . . . 34. 2, 22, 26, 210, . . . 35. 1, 2, 2, 4, 8, 32, . . . 12.5 Find the first three iterates of the function for the given initial value. 36. f(x) 5 2x 2 5, x0 5 3 4 x 2 2, x 5 210 37. f(x) 5 } 0 5 38. f(x) 5 3x2 1 x, x0 5 21 Extra Practice n2pe-9030.indd 1021 1021 10/17/05 12:26:48 PM Chapter 13 13.1 Let u be an acute angle of a right triangle. Find the values of the other five trigonometric functions of u. 3 1. sin u 5 } 5 } 8 2. tan u 5 } 15 Ï7 4. cos u 5 } 4 3. sec u 5 2 EXTRA PRACTICE 13.1 Solve n ABC using the diagram and the given measurements. 5. A 5 218, c 5 8 6. B 5 668, a 5 14 7. B 5 608, c 5 20 8. A 5 298, b 5 6 9. A 5 188, c 5 18 10. B 5 568, c 5 7 B c A a b C 13.2 Convert the degree measure to radians or the radian measure to degrees. 11. 1008 3p 13. } 4 12. 268 p 14. 2} 6 13.2 Find the arc length and area of a sector with the given radius r and central angle u. 15. r 5 5 ft, u 5 908 16. r 5 2 in., u 5 3008 17. r 5 12 cm, u 5 π 13.3 Sketch the angle. Then find its reference angle. 18. 2508 19. 2308 8p 20. } 3 11p 21. 2} 6 7p 24. tan } 4 5p 25. cos 2} 4 13.3 Evaluate the function without using a calculator. 22. sin (2608) 23. csc 2408 1 2 13.4 Evaluate the expression without using a calculator. Give your answer in both radians and degrees. 26. sin21 0 } Ï3 27. cos21 2} 2 1 2 28. cos21 3 29. tan21 1 13.4 Solve the equation for u. 30. sin u 5 0.25; 908 < u < 1808 31. cos u 5 0.9; 2708 < u < 3608 32. tan u 5 2; 1808 < u < 2708 13.5 Solve n ABC. (Hint: Some of the “triangles” may have no solution and some may have two solutions.) 33. A 5 348, a 5 6, b 5 7 34. A 5 508, C 5 658, b 5 60 35. B 5 868, b 5 13, c 5 11 13.5 Find the area of n ABC with the given side lengths and included angle. 36. A 5 358, b 5 50, c 5 120 37. B 5 358, a 5 7, c 5 12 38. C 5 208, a 5 10, b 5 16 40. C 5 508, a 5 12, b 5 14 41. A 5 808, b 5 7, c 5 5 13.6 Solve n ABC. 39. a 5 16, b 5 23, c 5 17 13.6 Find the area of n ABC with the given side lengths. 42. a 5 6, b 5 3, c 5 4 43. a 5 14, b 5 30, c 5 27 44. a 5 16, b 5 16, c 5 20 1022 Student Resources n2pe-9030.indd 1022 10/17/05 12:26:50 PM Chapter 14 14.1 Graph the function. 1x 1. y 5 cos } 4 2. y 5 3 sin x 3. y 5 sin 2πx 4. y 5 2 tan 2x 14.2 Graph the sine or cosine function. p 11 5. y 5 sin 2 1 x 2 } 42 p 6. y 5 2sin 1 x 1 } 42 7. y 5 2 cos x 1 3 14.2 Graph the tangent function. p 21 10. y 5 tan 1 x 2 } 22 14.3 Simplify the expression. p 2 x 1 cos2 (2x) 11. cos2 1 } 2 2 (sec x 2 1)(sec x 1 1) 12. } tan x p 2 x cot x 2 csc2 x 13. tan 1 } 2 2 cos2 x 1 sin2 x 5 cos2 x 15. } tan2 x 1 1 16. 2 2 sec2 x 5 1 2 tan 2 x 14.3 Verify the identity. cos (2x) 14. } 5 sec x 1 tan x 1 1 sin (2x) EXTRA PRACTICE 1 tan 2x 9. y 5 2} 4 8. y 5 2 tan x 1 2 14.4 Find the general solution of the equation. 17. 12 tan 2 x 2 4 5 0 19. tan 2 x 2 2 tan x 5 21 18. 3 sin x 5 22 sin x 1 3 14.4 Solve the equation in the given interval. Check your solutions. 20. cos2 x sin x 5 5 sin x; 0 ≤ x < 2π 21. 2 2 2 cos2 x 5 3 1 5 sin x; 0 ≤ x < 2π 22. 8 cos x 5 4 sec x; 0 ≤ x < π 23. cos2 x 2 4 cos x 1 1 5 0; 0 ≤ x < π 14.5 Write a function for the sinusoid. 24. y 25. sπ4 , 3d 3 y (0, 3.5) (1, 2.5) π 4 π 1 x s 3π4 , 21d 1 x 14.6 Find the exact value of the expression. 26. sin (2158) 27. cos 1658 11p 28. tan } 12 p 29. cos } 12 14.7 Find the exact values of sin 2a, cos 2a, and tan 2a. 2 , π < a < 3p 30. tan a 5 } } 3 2 9 ,0<a< p 31. cos a 5 } } 10 2 3 , 3p < a < 2π 32. sin a 5 2} } 5 2 14.7 Find the general solution of the equation. 33. cos 2x 2 cos x 5 0 x 5 sin x 34. cos } 2 35. sin 2x 5 21 Extra Practice n2pe-9030.indd 1023 1023 10/17/05 12:26:52 PM Tables Symbols Symbol Meaning Page Symbol ... and so on 2 e ø is approximately equal to 2 p multiplication, times Page irrational number ø 2.718 492 log b y log base b of y 499 3 log x log base 10 of x 500 opposite of a 4 ln x log base e of x 500 } a 1 reciprocal of a, a Þ 0 4 n! n factorial; number of permutations of n objects 684 b1 b sub 1 P n r number of permutations of r objects from n distinct objects 685 C n r number of combinations of r objects from n distinct objects 690 2a 26 π pi; irrational number ø 3.14 26 < is less than 41 > is greater than 41 ≤ is less than or equal to 41 ≥ is greater than or equal to 41 P(A) probability of event A 698 absolute value of x 51 } P(A) probability of the complement of event A 709 is not equal to 52 < union of two sets 715 ordered pair 72 ù intersection of two sets 715 f of x, or the value of f at x 75 0⁄ empty set slope 82 715 m i # 84 is a subset of 716 is parallel to ⊥ is perpendicular to 84 P(BA) probability of event B given that event A has occurred 718 (x, y, z) ordered triple 178 }x x-bar; the mean of a data set 744 F G matrix 187 s sigma; the standard deviation of a data set 745 A determinant of matrix A 203 ∑ summation 796 A21 inverse of matrix A 210 u theta 852 Ïa the nonnegative square root of a 266 sin sine 852 cos cosine 852 i imaginary unit equal to Ï21 275 tan tangent 852 absolute value of complex number z 279 csc cosecant 852 x approaches positive infinity sec secant 852 339 cot cotangent 852 nth root of a 414 sin21 inverse sine 875 438 21 inverse cosine 875 21 inverse tangent 875 x Þ TABLES Meaning (x, y) f(x) 1 0 0 1 } z x → 1` n} Ïa f 21 } inverse of function f cos tan 1024 Student Resources n2pe-9040.indd 1024 10/14/05 11:05:58 AM Measures Time 60 seconds (sec) 5 1 minute (min) 60 minutes 5 1 hour (h) 24 hours 5 1 day 7 days 5 1 week 4 weeks (approx.) 5 1 month 365 days 52 weeks (approx.) 5 1 year 12 months 10 years 5 1 decade 100 years 5 1 century Metric United States Customary Length Length 10 millimeters (mm) 5 1 centimeter (cm) 12 inches (in.) 5 1 foot (ft) 100 cm 1000 mm 5 1 meter (m) 36 in. 5 1 yard (yd) 3 ft 1000 m 5 1 kilometer (km) 5280 ft 5 1 mile (mi) 1760 yd Area 100 square millimeters 5 1 square centimeter (mm2) (cm2) 2 10,000 cm 5 1 square meter (m2 ) 10,000 m2 5 1 hectare (ha) 144 square inches (in.2 ) 5 1 square foot (ft2 ) 9 ft2 5 1 square yard (yd2 ) Volume Volume 1000 cubic millimeters 5 1 cubic centimeter (mm3) (cm3) 3 1,000,000 cm 5 1 cubic meter (m3) 1728 cubic inches (in.3) 5 1 cubic foot (ft3) 27 ft3 5 1 cubic yard (yd3) Liquid Capacity 43,560 ft2 5 1 acre (A) 4840 yd2 Liquid Capacity 1000 milliliters (mL) 5 1 liter (L) 1000 cubic centimeters (cm3) 1000 L 5 1 kiloliter (kL) Mass 8 fluid ounces (fl oz) 5 1 cup (c) 2 c 5 1 pint (pt) 2 pt 5 1 quart (qt) 4 qt 5 1 gallon (gal) Weight 1000 milligrams (mg) 5 1 gram (g) 1000 g 5 1 kilogram (kg) 1000 kg 5 1 metric ton (t) Temperature Degrees Celsius (°C) 0°C 5 freezing point of water 37°C 5 normal body temperature 100°C 5 boiling point of water 16 ounces (oz) 5 1 pound (lb) 2000 lb 5 1 ton Temperature Degrees Fahrenheit (°F) 32°F 5 freezing point of water 98.6°F 5 normal body temperature 212°F 5 boiling point of water Tables n2pe-9040.indd 1025 TABLES Area 1025 10/14/05 11:06:01 AM Formulas Formulas from Coordinate Geometry y 2y 2 1 m5} x 2 x where m is the slope of the nonvertical line through Slope of a line (p. 82) 2 1 points (x1, y1) and (x 2, y 2) Parallel and perpendicular lines (p. 84) If line l1 has slope m1 and line l2 has slope m2, then: l1 i l2 if and only if m1 5 m2 1 l1 ⊥ l2 if and only if m1 5 2} m , or m1m2 5 21 2 }} d 5 (x2 2 x1)2 1 (y2 2 y1)2 where d is the distance between Ï Distance formula (p. 615) points (x1, y1) and (x 2, y 2) 1 x 1x y 1y 2 2 2 1 2 1 2 M } ,} is the midpoint of the line segment joining Midpoint formula (p. 615) points (x1, y1) and (x 2, y 2). TABLES Formulas from Matrix Algebra Determinant of a 2 3 2 matrix (p. 203) Determinant of a 3 3 3 matrix (p. 203) Area of a triangle (p. 204) det F G a b c d 5 a b ) c d ) 5 ad 2 cb F G) a b det d e g h c a b f 5 d e i g h c f 5 (aei 1 bfg 1 cdh) 2 (gec 1 hfa 1 idb) i ) The area of a triangle with vertices (x1, y1), (x 2, y 2), and (x 3, y 3) is given by 1 x1 y 1 1 ) Area 5 6} 2 x2 y 2 1 x3 y 3 1 ) where the appropriate sign (6) should be chosen to yield a positive value. Cramer’s rule (p. 205) Let A 5 F G a b c d be the coefficient matrix of this linear system: ax 1 by 5 e cx 1 dy 5 f If det A Þ 0, then the system has exactly one solution. e b a ) f d) e )c f) The solution is x 5 } and y 5 }. det A Inverse of a 2 3 2 matrix The inverse of the matrix A 5 (p. 210) F d 2b 2c a A 1 A21 5 } G det A F G F G 1 5} a b c d is d 2b a ad 2 cb 2c provided ad 2 cb Þ 0. 1026 Student Resources n2pe-9040.indd 1026 10/14/05 11:06:02 AM Formulas and Theorems from Algebra Quadratic formula (p. 292) The solutions of ax 2 1 bx 1 c 5 0 are } 2b 6 Ï b2 2 4ac x5 } 2a where a, b, and c are real numbers such that a Þ 0. Discriminant of a quadratic equation (p. 294) The expression b2 2 4ac is called the discriminant of the associated equation ax 2 1 bx 1 c 5 0. The value of the discriminant can be positive, zero, or negative, which corresponds to an equation having two real solutions, one real solution, or two imaginary solutions, respectively. Special product patterns Sum and difference: (a 1 b)(a 2 b) 5 a2 2 b2 (p. 347) Square of a binomial: (a 1 b)2 5 a2 1 2ab 1 b2 (a 2 b)2 5 a2 2 2ab 1 b2 Cube of a binomial: (a 1 b) 3 5 a 3 1 3a2b 1 3ab2 1 b3 (a 2 b) 3 5 a 3 2 3a2b 1 3ab2 2 b3 Sum of two cubes: a 3 1 b3 5 (a 1 b)(a2 2 ab 1 b2) (p. 354) Difference of two cubes: a 3 2 b3 5 (a 2 b)(a2 1 ab 1 b2) Remainder theorem (p. 363) If a polynomial f(x) is divided by x 2 k, then the remainder is r 5 f(k). Factor theorem (p. 364) A polynomial f(x) has a factor x 2 k if and only if f(k) 5 0. Rational zero theorem (p. 370) If f(x) 5 anx n 1 . . . 1 a1x 1 a 0 has integer coefficients, then every rational zero of f has this form: p TABLES Special factoring patterns factor of constant term a0 } q 5 }}} factor of leading coefficient an Fundamental theorem of algebra (p. 379) If f(x) is a polynomial of degree n where n > 0, then the equation f(x) 5 0 has at least one solution in the set of complex numbers. Corollary to the fundamental theorem of algebra (p. 379) If f(x) is a polynomial of degree n where n > 0, then the equation f(x) 5 0 has exactly n solutions provided each solution repeated twice is counted as 2 solutions, each solution repeated three times is counted as 3 solutions, and so on. Complex conjugates theorem If f is a polynomial function with real coefficients, and a 1 bi is an imaginary zero of f, then a 2 bi is also a zero of f. (p. 380) Irrational conjugates theorem (p. 380) Suppose f is a polynomial function with rational coefficients, and a and b are} } } rational numbers such that Ï b is irrational. If a 1 Ïb is a zero of f, then a 2 Ïb is also a zero of f. Descartes’ rule of signs Let f(x) 5 anx n 1 an 2 1x n 2 1 1 . . . 1 a2x 2 1 a1x 1 a 0 be a polynomial function with real coefficients. (p. 381) • The number of positive real zeros of f is equal to the number of changes in sign of the coefficients of f(x) or is less than this by an even number. • The number of negative real zeros of f is equal to the number of changes in sign of the coefficients of f(2x) or is less than this by an even number. Tables n2pe-9040.indd 1027 1027 10/14/05 11:06:04 AM Formulas and Theorems from Algebra (continued) Discriminant of a general second-degree equation (p. 653) Any conic can be described by a general second-degree equation in x and y: Ax 2 1 Bxy 1 Cy 2 1 Dx 1 Ey 1 F 5 0. The expression B2 2 4AC is the discriminant of the conic equation and can be used to identify it. Discriminant Type of Conic B2 2 4AC < 0, B 5 0, and A 5 C Circle 2 B 2 4AC < 0, and either B Þ 0 or A Þ C Ellipse B2 2 4AC 5 0 Parabola 2 B 2 4AC > 0 Hyperbola If B 5 0, each axis of the conic is horizontal or vertical. Formulas from Combinatorics Fundamental counting principle (p. 682) If one event can occur in m ways and another event can occur in n ways, then the number of ways that both events can occur is m p n. Permutations of n objects taken r at a time (p. 685) The number of permutations of r objects taken from a group of n distinct objects is denoted by nPr and is given by: n! P 5} TABLES n r Permutations with repetition (p. 685) (n 2 r)! The number of distinguishable permutations of n objects where one object is repeated s1 times, another is repeated s 2 times, and so on is: n! s1! p s2! p . . . p sk! }} Combinations of n objects taken r at a time (p. 690) The number of combinations of r objects taken from a group of n distinct objects is denoted by nCr and is given by: n! C 5} (n 2 r)! p r! n r Pascal’s triangle (p. 692) If you arrange the values of nCr in a triangular pattern in which each row corresponds to a value of n, you get what is called Pascal’s triangle. C 1 0 0 C C 1 0 C C 2 0 C C 4 0 C 2 1 C 3 0 C C 1 C 3 2 4 2 1 2 2 C 3 1 4 1 1 1 1 3 3 C 4 3 1 C 4 4 1 2 3 4 1 3 6 1 4 1 The first and last numbers in each row are 1. Every number other than 1 is the sum of the closest two numbers in the row directly above it. Binomial theorem (p. 693) The binomial expansion of (a 1 b)n for any positive integer n is: (a 1 b) n 5 C anb 0 1 C an 2 1b1 1 C an 2 2b2 1 . . . 1 C a 0bn n 0 n 5 ∑ r50 n 1 n 2 n n C a n 2 r br n r 1028 Student Resources n2pe-9040.indd 1028 10/14/05 11:06:05 AM Formulas from Probability Theoretical probability of an event (p. 698) When all outcomes are equally likely, the theoretical probability that an event A will occur is: Number of outcomes in A P(A) 5 }}} Total number of outcomes Odds in favor of an event When all outcomes are equally likely, the odds in favor of an event A are: (p. 699) Number of outcomes in A Number of outcomes not in A }}} Odds against an event When all outcomes are equally likely, the odds against an event A are: (p. 699) Number of outcomes not in A }}} Number of outcomes in A Experimental probability of an event (p. 700) When an experiment is performed that consists of a certain number of trials, the experimental probability of an event A is given by: of trials where A occurs P(A) 5 Number }}} Total number of trials Probability of compound events (p. 707) If A and B are any two events, then the probability of A or B is: P(A or B) 5 P(A) 1 P(B) 2 P(A and B) If A and B are disjoint events, then the probability of A or B is: P(A or B) 5 P(A) 1 P(B) The probability of the complement of event A, denoted } A, is: P(} A) 5 1 2 P(A) Probability of independent events (p. 717) If A and B are independent, the probability that both A and B occur is: P(A and B) 5 P(A) p P(B) Probability of dependent events (p. 718) If A and B are dependent, the probability that both A and B occur is: P(A and B) 5 P(A) p P(BA) Binomial probabilities For a binomial experiment consisting of n trials where the probability of success on each trial is p, the probability of exactly k successes is: P(k successes) 5 nCk p k (1 2 p) n 2 k (p. 725) TABLES Probability of the complement of an event (p. 709) Formulas from Statistics ... Mean of a data set (p. 744) Standard deviation of a data set (p. 745) 1 x2 1 1 xn 1 }x 5 x}} where } x (read “x-bar”) is the mean of the data x1, x 2, . . . , xn n s5 Î }}}} } )2 1 (x2 2 x} )2 1 . . . 1 (xn 2 x} )2 (x1 2 x }} where s (read “sigma”) is the standard n deviation of the data x1, x 2, . . . , xn Areas under a normal curve (p. 757) A normal distribution with mean } x and standard deviation s has these properties: z-score (p. 758) 2x } z 5 x} s where x is a data value, x is the mean, and s is the standard deviation • • • • The total area under the related normal curve is 1. About 68% of the area lies within 1 standard deviation of the mean. About 95% of the area lies within 2 standard deviations of the mean. About 99.7% of the area lies within 3 standard deviations of the mean. } Tables n2pe-9040.indd 1029 1029 10/14/05 11:06:06 AM Formulas for Sequences and Series n Formulas for sums of special series n ∑15n n 1 1) ∑ i 5 n(n } 2 ∑ i51 n(n 1 1)(2n 1 1) 6 i2 5 }} (p. 797) i51 Explicit rule for an arithmetic sequence (p. 802) The nth term of an arithmetic sequence with first term a1 and common difference d is: an 5 a1 1 (n 2 1)d Sum of a finite arithmetic series The sum of the first n terms of an arithmetic series is: (p. 804) i51 1 a 1a 1 n Sn 5 n } 2 2 Explicit rule for a geometric sequence (p. 810) The nth term of a geometric sequence with first term a1 and common ratio r is: an 5 a1r n 2 1 Sum of a finite geometric series The sum of the first n terms of a geometric series with common ratio r Þ 1 is: (p. 812) 1 2 rn Sn 5 a1 } 1 12r 2 Sum of an infinite geometric series (p. 821) The sum of an infinite geometric series with first term a1 and common ratio r is a TABLES 1 S5} 12r provided r < 1. If r ≥ 1, the series has no sum. Recursive equation for an arithmetic sequence (p. 827) an 5 an 2 1 1 d where d is the common difference Recursive equation for a geometric sequence (p. 827) an 5 r p an 2 1 where r is the common ratio Formulas and Identities from Trigonometry Conversion between degrees and radians (p. 860) p radians . To rewrite a degree measure in radians, multiply by } 180° 180° . To rewrite a radian measure in degrees, multiply by } p radians Definition of trigonometric functions (p. 866) Let u be an angle in standard position and (x, y) be any point } (except the origin) on the terminal side of u. Let r 5 Ïx 2 1 y 2 . y Law of sines (p. 882) y sin u 5 }r cos u 5 }xr tan u 5 }x , x Þ 0 csc u 5 }yr , y Þ 0 sec u 5 }xr , x Þ 0 cot u 5 }xy , y Þ 0 If n ABC has sides of length a, b, and c, then: sin A a sin B b sin C c }5}5} Area of a triangle (given two sides and the included angle) (p. 885) If n ABC has sides of length a, b, and c, then its area is: 1 bc sin A Area 5 } 2 1 Area 5 } ac sin B 2 1 Area 5 } ab sin C 2 1030 Student Resources n2pe-9040.indd 1030 10/14/05 11:06:07 AM Formulas and Identities from Trigonometry (continued) Law of cosines (p. 889) If n ABC has sides of length a, b, and c, then: a2 5 b2 1 c 2 2 2bc cos A b2 5 a2 1 c 2 2 2ac cos B c 2 5 a2 1 b2 2 2ab cos C Heron’s area formula (p. 891) The area of the triangle with sides of length a, b, and c is }} Area 5 Ïs(s 2 a)(s 2 b)(s 2 c) 1 (a 1 b 1 c). where s 5 } 2 Reciprocal identities (p. 924) 1 csc u 5 } sin u 1 sec u 5 } cos u Tangent and cotangent identities sin u tan u 5 } cos u cos u cot u 5 } sin u Pythagorean identities (p. 924) sin 2 u 1 cos2 u 5 1 1 1 tan 2 u 5 sec2 u 1 1 cot 2 u 5 csc2 u Cofunction identities (p. 924) π sin 1 } 2 u 2 5 cos u π cos 1 } 2 u 2 5 sin u π tan 1 } 2 u 2 5 cot u Negative angle identities (p. 924) sin (2u) 5 2sin u cos (2u) 5 cos u tan (2u) 5 2tan u (p. 924) 2 2 TABLES Sum formulas (p. 949) 2 1 cot u 5 } tan u sin (a 1 b) 5 sin a cos b 1 cos a sin b cos (a 1 b) 5 cos a cos b 2 sin a sin b tan a 1 tan b tan (a 1 b) 5 }} 1 2 tan a tan b Difference formulas (p. 949) sin (a 2 b) 5 sin a cos b 2 cos a sin b cos (a 2 b) 5 cos a cos b 1 sin a sin b tan a 2 tan b tan (a 2 b) 5 }} 1 1 tan a tan b Double-angle formulas (p. 955) cos 2a 5 cos2 a 2 sin 2 a sin 2a 5 2 sin a cos a cos 2a 5 2 cos2 a 2 1 2 tan a tan 2a 5 } 2 1 2 tan a cos 2a 5 1 2 2 sin 2 a } Half-angle formulas (p. 955) a 2 cos a sin } 5 6 1} 2 Ï 2 } a 1 cos a cos } 5 6 1} 2 Ï 2 a 2 cos a tan } 5 1} 2 sin a a sin a tan } 5} 2 1 1 cos a a a a and cos } depend on the quadrant in which } lies. The signs of sin } 2 2 2 Tables n2pe-9040.indd 1031 1031 10/14/05 11:06:08 AM Formulas from Geometry Basic geometric figures Area of an equilateral triangle See pages 991–993 for area formulas for basic two-dimensional geometric figures. } s Ï3 2 s where s is the length of a side Area 5 } s 4 s Arc length and area of a sector Arc length 5 ru where r is the radius and u is the radian measure of the central angle that intercepts the arc 1 2 Area 5 } r u 2 Area of an ellipse sector r arc length s central angle u Area 5 πab where a and b are half the lengths of the major and minor axes of the ellipse b a TABLES Volume and surface area of a right rectangular prism Volume 5 lwh where l is the length, w is the width, and h is the height h Surface area 5 2(lw 1 wh 1 lh) w l Volume and surface area of a right cylinder Volume 5 πr 2h where r is the base radius and h is the height Lateral surface area 5 2πrh h Surface area 5 2πr 2 1 2πrh Volume and surface area of a right regular pyramid 1 Bh where B is the area of the base and h is the height Volume 5 } 3 l 1 nsl where n is the number Lateral surface area 5 } 2 of sides of the base, s is the length of a side of the base, and l is the slant height 1 nsl Surface area 5 B 1 } 2 Volume and surface area of a right circular cone r h s B s 1 2 πr h where r is the base radius and h is the height Volume 5 } 3 Lateral surface area 5 πrl where l is the slant height 2 Surface area 5 πr 1 πrl l h r Volume and surface area of a sphere 4 3 Volume 5 } πr where r is the radius 3 Surface area 5 4πr 2 r 1032 Student Resources n2pe-9040.indd 1032 10/14/05 11:06:10 AM Properties Properties of Real Numbers Let a, b, and c be real numbers. Addition Multiplication Closure Property (p. 3) a 1 b is a real number. ab is a real number. Commutative Property (p. 3) a1b5b1a ab 5 ba Associative Property (p. 3) (a 1 b) 1 c 5 a 1 (b 1 c) (ab)c 5 a(bc) Identity Property (p. 3) a 1 0 5 a, 0 1 a 5 a a p 1 5 a, 1 p a 5 a Inverse Property (p. 3) a 1 (2a) 5 0 1 ap} a 5 1, a Þ 0 Distributive Property (p. 3) The distributive property involves both addition and multiplication: a(b 1 c) 5 ab 1 ac Zero Product Property (p. 253) Let A and B be real numbers or algebraic expressions. If AB 5 0, then A 5 0 or B 5 0. Properties of Matrices Let A, B, and C be matrices, and let k be a scalar. (A 1 B) 1 C 5 A 1 (B 1 C) Commutative Property of Addition (p. 188) A1B5B1A Distributive Property of Addition (p. 188) k(A 1 B) 5 kA 1 kB Distributive Property of Subtraction (p. 188) k(A 2 B) 5 kA 2 kB Associative Property of Matrix Multiplication (p. 197) (AB)C 5 A(BC) Left Distributive Property of Matrix Multiplication (p. 197) A(B 1 C) 5 AB 1 AC Right Distributive Property of Matrix Multiplication (p. 197) (A 1 B)C 5 AC 1 BC Associative Property of Scalar Multiplication (p. 197) TABLES Associative Property of Addition (p. 188) k(AB) 5 (kA)B 5 A(kB) Multiplicative Identity (p. 210) An n 3 n matrix with 1’s on the main diagonal and 0’s elsewhere is an identity matrix, denoted I. For any n 3 n matrix A, AI 5 IA 5 A. Inverse Matrices (p. 210) If the determinant of an n 3 n matrix A is nonzero, then A has an inverse, denoted A21, such that AA21 5 A21 A 5 I. Properties of Exponents Let a and b be real numbers, and let m and n be integers. Product of Powers Property (p. 330) am p an 5 am 1 n Power of a Power Property (p. 330) (am ) n 5 amn Power of a Product Property (p. 330) (ab) m 5 ambm Negative Exponent Property (p. 330) 1 ,aÞ0 a2m 5 } m a 0 Zero Exponent Property (p. 330) a 5 1, a Þ 0 Quotient of Powers Property (p. 330) } n 5a Power of a Quotient Property (p. 330) 1 }b 2 am a a m m2n ,aÞ0 m a 5} m, b Þ 0 b Tables n2pe-9040.indd 1033 1033 10/14/05 11:06:11 AM Properties of Radicals and Rational Exponents Number of Real nth Roots Let n be an integer greater than 1, and let a be a real number. (p. 414) Radicals and Rational Exponents (p. 415) • • • • n} If n is odd, then a has one real nth root: Ï a 5 a1/n n} If n is even and a > 0, then a has two real nth roots: 6 Ï a 5 6a1/n } n If n is even and a 5 0, then a has one nth root: Ï0 5 01/n 5 0 If n is even and a < 0, then a has no real nth roots. Let a1/n be an nth root of a, and let m be a positive integer. n} • am/n 5 (a1/n ) m 5 1 Ï a 2m 1 1 1 • a2m/n 5 } 5} 5} ,aÞ0 n} m m/n 1 a1/n 2m a 1 Ïa 2 Properties of Rational Exponents (p. 420) All of the properties of exponents listed on the previous page apply to rational exponents as well as integer exponents. Product and Quotient Properties of Radicals (p. 421) Let n be an integer greater than 1, and let a and b be positive real n} n} n} numbers. Then Ïa p b 5 Ï a p Ïb and n } n} Ïa a }5} n }. Ïb Ïb Properties of Logarithms TABLES Let a, b, c, m, n, x, and y be positive real numbers such that b Þ 1 and c Þ 1. Logarithms and Exponents (p. 499) log b y 5 x if and only if b x 5 y Special Logarithm Values (p. 499) log b 1 5 0 because b 0 5 1 and log b b 5 1 because b1 5 b Common and Natural Logarithms (p. 500) log10 x 5 log x and loge x 5 ln x Product Property of Logarithms (p. 507) log b mn 5 log b m 1 log b n Quotient Property of Logarithms (p. 507) m log b } n 5 log b m 2 log b n Power Property of Logarithms (p. 507) log b mn 5 n log b m Change of Base (p. 508) logc a 5 } logb a logb c Properties of Functions Operations on Functions (pp. 428, 430) Let f and g be any two functions. A new function h can be defined using any of the following operations. Addition: Subtraction: Multiplication: h(x) 5 f(x) 1 g(x) h(x) 5 f(x) 2 g(x) h(x) 5 f(x) p g(x) Division: h(x) 5 } Composition: h(x) 5 g(f(x)) f(x) g(x) For addition, subtraction, multiplication, and division, the domain of h consists of the x-values that are in the domains of both f and g. Additionally, the domain of the quotient does not include x-values for which g(x) 5 0. For composition, the domain of h is the set of all x-values such that x is in the domain of f and f(x) is in the domain of g. Inverse Functions (p. 438) Functions f and g are inverses of each other provided: f(g(x)) 5 x and g(f(x)) 5 x 1034 Student Resources n2pe-9040.indd 1034 10/14/05 11:06:12 AM English–Spanish Glossary A 2 2 2 2 absolute value (p. 51) The absolute value of a number x, represented by the symbol x, is the distance the number is from 0 on a number line. }3 5 }3, 24.3 5 4.3, and 0 5 0. valor absoluto (pág. 51) El valor absoluto de un número x, representado por el símbolo x, es la distancia a la que está el número de 0 en una recta numérica. }3 5 }3, 24.3 5 4.3 y 0 5 0. absolute value function (p. 123) A function that contains an absolute value expression. y 5 x, y 5 x 2 3, and y 5 4x 1 8 – 9 are absolute value functions. función de valor absoluto (pág. 123) Función que contiene una expresión de valor absoluto. y 5 x, y 5 x 2 3e y 5 4x 1 8 – 9 son funciones de valor absoluto. absolute value of a complex number (p. 279) If z 5 a 1 bi, then the absolute value of z, denoted z, is a nonnegative } real number defi ned as z 5 Ïa2 1 b2 . } } 2 2 24 1 3i 5 Ï (24) 1 3 5 Ï25 5 5 valor absoluto de un número complejo (pág. 279) Si z 5 a 1 bi, entonces el valor absoluto de z, denotado por z, } es un número real no negativo defi nido como z 5 Ï a2 1 b2 . expresión algebraica (pág. 11) Expresión formada por números, variables, operaciones y signos de agrupación. 2 3 8 72r 2 }p, }, k – 5, and n + 2n are algebraic expressions. 2 3 8 72r 2 }p, }, k – 5 y n 1 2n son expresiones algebraicas. amplitude (p. 908) The amplitude of the graph of a sine or 4 1 cosine function is } (M – m), where M is the maximum value 2 y M54 of the function and m is the minimum value of the function. π 2 amplitud (pág. 908) La amplitud de la grafica de una 3π 2 1 función seno o coseno es } (M – m), donde M es el valor 2 x m 5 24 máximo de la función y m es el valor mínimo de la función. The graph of y 5 4 sin x has an amplitude of ENGLISH-SPANISH GLOSSARY algebraic expression (p. 11) An expression that consists of numbers, variables, operations, and grouping symbols. Also called variable expression. 1 2 }(4 – (–4)) 5 4. La gráfica de y 5 4 sen x tiene una amplitud de }1 (4 – (–4)) 5 4. 2 angle of depression (p. 855) The angle by which an observer’s line of sight must be depressed from the horizontal to the point observed. See angle of elevation. ángulo de depresión (pág. 855) El ángulo con el que se debe bajar la línea de visión de un observador desde la horizontal hasta el punto observado. Ver ángulo de elevación. English-Spanish Glossary n2pe-9050.indd 1035 1035 10/14/05 10:05:01 AM angle of elevation (p. 855) The angle by which an observer’s line of sight must be elevated from the horizontal to the point observed. angulo ´ de depresión ángulo de elevación (pág. 855) El ángulo con el que se debe elevar la línea de visión de un observador desde la horizontal hasta el punto observado. ´ angulo de elevación angle of depression angle of elevation arithmetic sequence (p. 802) A sequence in which the difference of consecutive terms is constant. 24, 1, 6, 11, 16, . . . is an arithmetic sequence with common difference 5. progresión aritmética (pág. 802) Progresión en la que la diferencia entre los términos consecutivos es constante. 24, 1, 6, 11, 16, . . . es una progresión aritmética con una diferencia común de 5. arithmetic series (p. 804) The expression formed by adding the terms of an arithmetic sequence. serie aritmética (pág. 804) La expresión formada al sumar los términos de una progresión aritmética. 5 ∑ 2i 5 2 1 4 1 6 1 8 1 10 i=1 asymptote (p. 478) A line that a graph approaches more and more closely. asíntota (pág. 478) Recta a la que se aproxima una gráfica cada vez más. y asymptote asíntota 1 ENGLISH-SPANISH GLOSSARY 1 x The asymptote for the graph shown is the line y 5 3. La asíntota para la gráfica que se muestra es la recta y 5 3. axis of symmetry of a parabola (pp. 236, 620) The line perpendicular to the parabola’s directrix and passing through its focus. See parabola. eje de simetría de una parábola (págs. 236, 620) La recta perpendicular a la directriz de la parábola y que pasa por su foco. Ver parábola. B base of a power (p. 10) The number or expression that is used as a factor in a repeated multiplication. In the power 25, the base is 2. base de una potencia (pág. 10) El número o la expresión que se usa como factor en la multiplicación repetida. En la potencia 25, la base es 2. 1036 Student Resources n2pe-9050.indd 1036 10/14/05 10:05:04 AM best-fitting line (p. 114) The line that lies as close as possible to all the data points in a scatter plot. mejor recta de regresión (pág. 114) La recta que se ajusta lo más posible a todos los puntos de datos de un diagrama de dispersión. best-fitting quadratic model (p. 311) The model given by using quadratic regression on a set of paired data. 600 500 modelo cuadrático con mejor ajuste (pág. 311) El modelo dado al realizar una regresión cuadrática sobre un conjunto de pares de datos. 400 300 200 100 0 0 10 20 30 40 50 60 70 80 “Would you rather see an exciting laser show or a boring movie?” is a biased question. pregunta capciosa (pág. 772) Pregunta que no refleja con exactitud las opiniones o acciones de los encuestados. “¿Preferirías ver un emocionante espectáculo de láser o una película aburrida?” es una pregunta capciosa. biased sample (p. 767) A sample that overrepresents or underrepresents part of a population. The members of a school’s basketball team would form a biased sample for a survey about whether to build a new gym. muestra sesgada (pág. 767) Muestra que representa de forma excesiva o insuficiente a parte de una población. Los miembros del equipo de baloncesto de una escuela formarían una muestra sesgada si participaran en una encuesta sobre si quieren que se construya un nuevo gimnasio. binomial (p. 252) The sum of two monomials. 3x 2 1 and t 3 2 4t are binomials. binomio (pág. 252) La suma de dos monomios. 3x 2 1 y t 3 2 4t son binomios. distribución binomial (pág. 725) La distribución de probabilidades asociada a un experimento binomial. 0.30 Probability Probabilidad binomial distribution (p. 725) The probability distribution associated with a binomial experiment. 0.20 ENGLISH-SPANISH GLOSSARY biased question (p. 772) A question that does not accurately reflect the opinions or actions of the people surveyed. 0.10 0 0 1 2 3 4 5 6 7 8 Number of successes Número de éxitos Binomial distribution for 8 trials with p 5 0.5. Distribución binomial de 8 ensayos con p 5 0.5. English-Spanish Glossary n2pe-9050.indd 1037 1037 10/14/05 10:05:05 AM binomial experiment (p. 725) An experiment that meets the following conditions. (1) There are n independent trials. (2) Each trial has only two possible outcomes: success and failure. (3) The probability of success is the same for each trial. experimento binomial (pág. 725) Experimento que satisface las siguientes condiciones. (1) Hay n pruebas independientes. (2) Cada prueba tiene sólo dos resultados posibles: éxito y fracaso. (3) La probabilidad de éxito es igual para cada prueba. binomial theorem (p. 693) The binomial expansion of (a 1 b) n for any positive integer n: (a 1 b) n 5 nC 0 anb 0 1 nC1an – 1b1 1 nC2an – 2b2 1 . . . 1 nCna 0bn . teorema binomial (pág. 693) La expansión binomial de (a 1 b) n para cualquier número entero positivo n: (a 1 b) n 5 nC 0 anb 0 1 nC1an – 1b1 1 nC2an – 2b2 1 . . . 1 nCna 0bn . A fair coin is tossed 12 times. The probability of getting exactly 4 heads is as follows: Una moneda normal se lanza 12 veces. La probabilidad de sacar exactamente 4 caras es la siguiente: C p k (1 – p) n – k 5 12C4 (0.5)4 (1 – 0.5) 8 5 495(0.5)4 (0.5) 8 ø 0.121 P(k 5 4) 5 n k (x 2 1 y) 3 5 3C 0 (x 2 ) 3y 0 1 3C1(x 2 )2y1 1 3C 2 (x 2 )1y 2 1 C (x 2 ) 0y 3 3 3 5 (1)(x 6 )(1) 1 (3)(x 4)(y) 1 (3)(x 2 )(y 2 ) 1 (1)(1)(y 3) 5 x 6 1 3x 4y 1 3x 2y 2 1 y 3 ENGLISH-SPANISH GLOSSARY C center of a circle (p. 626) See circle. The circle with equation (x – 3)2 1 (y 1 5)2 5 36 has its center at (3, –5). See also circle. centro de un círculo (pág. 626) Ver círculo. El círculo con la ecuación (x – 3)2 1 (y 1 5)2 5 36 tiene el centro en (3, –5). Ver también círculo. center of a hyperbola (p. 642) The midpoint of the transverse axis of a hyperbola. See hyperbola. centro de una hipérbola (pág. 642) El punto medio del eje transverso de una hipérbola. Ver hipérbola. center of an ellipse (p. 634) The midpoint of the major axis of an ellipse. See ellipse. centro de una elipse (pág. 634) El punto medio del eje mayor de una elipse. Ver elipse. central angle (p. 861) An angle formed by two radii of a circle. See sector. ángulo central (pág. 861) Ángulo formado por dos radios de un círculo. Ver sector. 1038 Student Resources n2pe-9050.indd 1038 10/14/05 10:05:06 AM circle (p. 626) The set of all points (x, y) in a plane that are of distance r from a fi xed point, called the center of the circle. círculo (pág. 626) El conjunto de todos los puntos (x, y) de un plano que están a una distancia r de un punto fijo, llamado centro del círculo. y center centro r (x, y) x circle círculo x2 1 y2 5 r 2 coefficient (p. 12) When a term is the product of a number and a power of a variable, the number is the coefficient of the power. In the algebraic expression 2x 2 1 (24x) 1 (21), the coefficient of 2x 2 is 2 and the coefficient of 24x is 24. coeficiente (pág. 12) Cuando un término es el producto de un número y una potencia de una variable, el número es el coeficiente de la potencia. En la expresión algebraica 2x 2 1 (24x) 1 (21), el coeficiente de 2x 2 es 2 y el coeficiente de 24x es 24. coefficient matrix (p. 205) The coefficient matrix of the linear system ax 1 by 5 e, cx 1 dy 5 f is a b . c d F G matriz coeficiente (pág. 205) La matriz coeficiente del sistema lineal ax 1 by 5 e, cx 1 dy 5 f es a b . c d F G n! where nCr 5 } . coefficient matrix: matriz coeficiente: F 93 254 G matrix of constants: matriz de constantes: 26 F 221 G matrix of variables: matriz de variables: Fxy G There are 6 combinations of the n 5 4 letters A, B, C, and D selected r 5 2 at a time: AB, AC, AD, BC, BD, and CD. (n 2 r)! p r! combinación (pág. 690) Selección de r objetos de un grupo de n objetos en el que el orden no importa, denotado nCr, n! donde nCr 5 } . Hay 6 combinaciones de las letras n 5 4 A, B, C y D seleccionadas r 5 2 cada vez: AB, AC, AD, BC, BD y CD. (n 2 r)! p r! common difference (p. 802) The constant difference of consecutive terms of an arithmetic sequence. See arithmetic sequence. diferencia común (pág. 802) La diferencia constante entre los términos consecutivos de una progresión aritmética. Ver progresión aritmética. common logarithm (p. 500) A logarithm with base 10. It is denoted by log10 or simply by log. log10 100 5 log 100 5 2 because 102 5 100. logaritmo común (pág. 500) Logaritmo con base 10. Se denota por log10 ó simplemente por log. log10 100 5 log 100 5 2 ya que 102 5 100. English-Spanish Glossary n2pe-9050.indd 1039 ENGLISH-SPANISH GLOSSARY combination (p. 690) A selection of r objects from a group of n objects where the order is not important, denoted nCr 9x 1 4y 5 26 3x 2 5y 5 221 1039 10/14/05 10:05:07 AM common ratio (p. 810) The constant ratio of consecutive terms of a geometric sequence. See geometric sequence. razón común (pág. 810) La razón constante entre los términos consecutivos de una progresión geométrica. Ver progresión geométrica. complement of a set (p. 715) The complement of a set A, } written A , is the set of all elements in the universal set U that are not in A. Let U be the set of all integers from 1 to 10 } and let A 5 {1, 2, 4, 8}. Then A 5 {3, 5, 6, 7, 9, 10}. complemento de un conjunto (pág. 715) El complemento } de un conjunto A, escrito A, es el conjunto de todos los elementos del conjunto universal U que no están en A. Sea U el conjunto de todos los números enteros entre 1 y 10 y sea A 5 {1, 2, 4, 8}. Por } lo tanto, A 5 {3, 5, 6, 7, 9, 10}. completing the square (p. 284) The process of adding a term to a quadratic expression of the form x2 1 bx to make it a perfect square trinomial. To complete the square for x 2 1 16x, add completar el cuadrado (pág. 284) El proceso de sumar un término a una expresión cuadrática de la forma x2 1 bx, de modo que sea un trinomio cuadrado perfecto. Para completar el cuadrado para x 2 1 16x, complex conjugates (p. 276) Two complex numbers of the form a 1 bi and a 2 bi. ENGLISH-SPANISH GLOSSARY números complejos conjugados (pág. 276) Dos números complejos de la forma a 1 bi y a 2 bi. complex fraction (p. 584) A fraction that contains a fraction in its numerator or denominator. fracción compleja (pág. 584) Fracción que tiene una fracción en su numerador o en su denominador. complex number (p. 276) A number a 1 bi where a and b are real numbers and i is the imaginary unit. número complejo (pág. 276) Un número a 1 bi, donde a y b son números reales e i es la unidad imaginaria. complex plane (p. 278) A coordinate plane in which each point (a, b) represents a complex number a 1 bi. The horizontal axis is the real axis and the vertical axis is the imaginary axis. plano complejo (pág. 278) Plano de coordenadas en el que cada punto (a, b) representa un número complejo a 1 bi. El eje horizontal es el eje real, y el eje vertical es el eje imaginario. composition of functions (p. 430) The composition of a function g with a function f is h(x) 5 g(f(x)). composición de funciones (pág. 430) La composición de una función g con una función f es h(x) 5 g(f(x)). 2 16 1} 2 2 5 64: x 2 1 16x 1 64 5 (x 1 8)2 . 2 16 suma 1 } 2 5 64: x2 1 16x 1 64 5 (x 1 8)2. 2 2 1 4i, 2 2 4i 5 x14 } 1 },} 1 1 6x } } p1} q 2 3x } 0, 2.5, Ï3 , π, 5i, 2 2 i imaginary imaginario 22 1 4i 3i i 1 real real 3 2 2i 24 2 3i f(x) 5 5x 2 2, g(x) 5 4x21 4 2 g(f(x)) 5 g(5x 2 2) 5 4(5x 2 2) 21 5 } ,xÞ} 5x 2 2 5 1040 Student Resources n2pe-9050.indd 1040 10/14/05 10:05:09 AM compound event (p. 707) The union or intersection of two events. When you roll a six-sided die, the event “roll a 2 or an odd number” is a compound event. suceso compuesto (pág. 707) La unión o la intersección de dos sucesos. Cuando lanzas un cubo numerado de seis lados, el suceso “salir el 2 ó un número impar” es un suceso compuesto. compound inequality (p. 41) Two simple inequalities joined by “and” or “or.” 2x > 0 or x 1 4 < 21 is a compound inequality. desigualdad compuesta (pág. 41) Dos desigualdades simples unidas por “y” u “o”. 2x > 0 ó x 1 4 < 21 es una desigualdad compuesta. conditional probability (p. 718) The conditional probability of B given A, written P(B | A), is the probability that event B will occur given that event A has occurred. Two cards are randomly selected from a standard deck of 52 cards. Let event A be “the fi rst card is a club” and let event B be “the second card is a club.” Then 12 4 5} because there are 12 (out of P(B | A) 5 } 51 17 13) clubs left among the remaining 51 cards. probabilidad condicional (pág. 718) La probabilidad condicional de B dado A, escrito P(B | A), es la probabilidad de que ocurra el suceso B dado que ha ocurrido el suceso A. Dos cartas se seleccionan al azar de una baraja normal de 52 cartas. Sea el suceso A “la primera carta es de tréboles” y sea el suceso B “la segunda carta es de tréboles”. 12 4 Entonces P(B | A) 5 } 5} ya que quedan 51 17 12 (del total de 13) cartas de tréboles entre las 51 cartas restantes. See conic section. cónica (pág. 650) Ver sección cónica. Ver sección cónica. conic section (p. 650) A curve formed by the intersection of a plane and a double-napped cone. Conic sections are also called conics. See circle, ellipse, hyperbola, and parabola. sección cónica (pág. 650) Una curva formada por la intersección de un plano y un cono. Las secciones cónicas también se llaman cónicas. Ver círculo, elipse, hipérbola y parábola. } } conjugates (p. 267) The expressions a 1 Ïb and a 2 Ïb where a and b are rational numbers. } } } ENGLISH-SPANISH GLOSSARY conic (p. 650) See conic section. } The conjugate of 7 1 Ï2 is 7 2 Ï2 . } } El conjugado de 7 1 Ï2 es 7 2 Ï2 . conjugados (pág. 267) Las expresiones a 1 Ïb y a 2 Ï b cuando a y b son números racionales. consistent system (p. 154) A system of equations that has at least one solution. sistema compatible (pág. 154) Sistema de ecuaciones que tiene al menos una solución. y 5 2 1 3x 6x 1 2y 5 4 The system above is consistent, with solution (0, 2). El sistema de arriba es compatible, con la solución (0, 2). English-Spanish Glossary n2pe-9050.indd 1041 1041 10/14/05 10:05:10 AM constant of variation (pp. 107, 551, 553) The nonzero constant a in a direct variation equation y 5 ax, an inverse a , or a joint variation equation variation equation y 5 } x z 5 axy. constante de variación (págs. 107, 551, 553) La constante distinta de cero a de una ecuación de variación directa ENGLISH-SPANISH GLOSSARY a o de una y 5 ax, de una ecuación de variación inversa y 5 } x ecuación de variación conjunta z 5 axy. 5 In the direct variation equation y 5 – } x, the 2 5 constant of variation is – } . 2 5 En la ecuación de variación directa y 5 – } x, 2 5 . la constante de variación es – } 2 constant term (pp. 12, 337) A term that has a number part but no variable part. The constant term of the algebraic expression 3x 2 1 5x 1 (27) is 27. término constante (págs. 12, 337) Término que tiene una parte numérica pero sin variable. El término constante de la expresión algebraica 3x 2 1 5x 1 (27) es 27. constraints (p. 174) In linear programming, the linear inequalities that form a system. See linear programming. restricciones (pág. 174) En la programación lineal, las desigualdades lineales que forman un sistema. Ver programación lineal. continuous function (p. 80) A function whose graph is unbroken. Any linear function, such as y 5 2x 1 4, is a continuous function. función continua (pág. 80) Función que tiene una gráfica no interrumpida. Cualquier función lineal, como y 5 2x 1 4, es una función continua. control group (p. 773) A group that does not undergo a procedure or treatment when an experiment is conducted. See also experimental group. See experimental group. grupo de control (pág. 773) Grupo que no se somete a ningún procedimiento o tratamiento durante la realización de un experimento. Ver también grupo experimental. Ver grupo experimental. correlation coefficient (p. 114) A measure, denoted by r where –1 ≤ r ≤ 1, of how well a line fits a set of data pairs (x, y). A data set that shows a strong positive correlation has a correlation coefficient of r ≈ 1. See also positive correlation and negative correlation. coeficiente de correlación (pág. 114) Medida denotada por r, donde –1 ≤ r ≤ 1, y que describe el ajuste de una recta a un conjunto de pares de datos (x, y). Un conjunto de datos que muestra una correlación positiva fuerte tiene un coeficiente de correlación de r ≈ 1. Ver también correlación positiva y correlación negativa. cosecant function (p. 852) If θ is an acute angle of a right triangle, the cosecant of θ is the length of the hypotenuse divided by the length of the side opposite θ. See sine function. función cosecante (pág. 852) Si θ es un ángulo agudo de un triángulo rectángulo, la cosecante de θ es la longitud de la hipotenusa dividida por la longitud del lado opuesto a θ. Ver función seno. 1042 Student Resources n2pe-9050.indd 1042 10/14/05 10:05:12 AM cosine function (p. 852) If θ is an acute angle of a right triangle, the cosine of θ is the length of the side adjacent to θ divided by the length of the hypotenuse. See sine function. función coseno (pág. 852) Si θ es un ángulo agudo de un triángulo rectángulo, el coseno de θ es la longitud del lado adyacente a θ dividida por la longitud de la hipotenusa. Ver función seno. cotangent function (p. 852) If θ is an acute angle of a right triangle, the cotangent of θ is the length of the side adjacent to θ divided by the length of the side opposite θ. See sine function. función cotangente (pág. 852) Si θ es un ángulo agudo de un triángulo rectángulo, la cotangente de θ es la longitud del lado adyacente a θ dividida por la longitud del lado opuesto a θ. Ver función seno. coterminal angles (p. 860) Angles in standard position with terminal sides that coincide. y 1408 ángulos coterminales (pág. 860) Ángulos en posición normal cuyos lados terminales coinciden. x 5008 The angles with measures 500° and 140° are coterminal. Los ángulos que miden 500° y 140° son coterminales. See ellipse. puntos extremos del eje menor de una elipse (pág. 634) Los puntos de intersección de una elipse y la recta perpendicular al eje mayor en el centro. Ver elipse. Cramer’s rule (p. 205) A method for solving a system of linear equations using determinants: For the linear system ax 1 by 5 e, cx 1 dy 5 f, let A be the coefficient matrix. If det A Þ 0, the solution of the system is as follows: e b f d a e c f x 5 }, y 5 } det A det A regla de Cramer (pág. 205) Método para resolver un sistema de ecuaciones lineales usando determinantes: Para el sistema lineal ax 1 by 5 e, cx 1 dy 5 f, sea A la matriz coeficiente. Si det A Þ 0, la solución del sistema es la siguiente: e b a e det A det A 9x 1 4y 5 26 3x 2 5y 5 221; 9 3 4 5 257 25 Applying Cramer’s rule gives the following: ENGLISH-SPANISH GLOSSARY co-vertices of an ellipse (p. 634) The points of intersection of an ellipse and the line perpendicular to the major axis at the center. Al aplicar la regla de Cramer se obtiene lo siguiente: 26 4 221 25 114 5 22 x5}5} 257 257 9 26 3 221 2171 5 3 y5}5} 257 257 f d c f x 5 }, y 5 } English-Spanish Glossary n2pe-9050.indd 1043 1043 10/14/05 10:05:13 AM cross multiplying (p. 589) A method for solving a simple rational equation for which each side of the equation is a single rational expression. 3 9 To solve } 5} , cross multiply. 4x 1 5 3 9 , multiplica en Para resolver } 5 } x 1 1 4x 1 5 x11 cruz. multiplicar en cruz (pág. 589) Método para resolver una ecuación racional simple en la que cada miembro es una sola expresión racional. 3(4x 1 5) 5 9(x 1 1) 12x 1 15 5 9x 1 9 3x 5 26 x 5 22 cycle (p. 908) The shortest repeating portion of the graph of a periodic function. See periodic function. ciclo (pág. 908) En una función periódica, la parte más corta de la gráfica que se repite. Ver función periódica. ENGLISH-SPANISH GLOSSARY D decay factor (p. 486) The quantity b in the exponential decay function y 5 abx with a . 0 and 0 , b , 1. The decay factor for the function y 5 3(0.5) x is 0.5. factor de decrecimiento (pág. 486) La cantidad b de la función de decrecimiento exponencial y 5 abx, con a . 0 y 0 , b , 1. El factor de decrecimiento de la función y 5 3(0.5) x es 0.5. degree of a polynomial function (p. 337) The exponent in the term of a polynomial function where the variable is raised to the greatest power. See polynomial function. grado de una función polinómica (pág. 337) En una función polinómica, el exponente del término donde la variable se eleva a la mayor potencia. Ver función polinómica. dependent events (p. 718) Two events such that the occurrence of one event affects the occurrence of the other event. Two cards are drawn from a deck without replacement. The events “the fi rst is a 3” and “the second is a 3” are dependent. sucesos dependientes (pág. 718) Dos sucesos tales que la ocurrencia de uno de ellos afecta a la ocurrencia del otro. Se sacan dos cartas de una baraja y no se reemplazan. Los sucesos “la primera es un 3” y “la segunda es un 3” son dependientes. dependent system (p. 154) A consistent system of equations that has infi nitely many solutions. sistema dependiente (pág. 154) Sistema compatible de ecuaciones que tiene infi nitas soluciones. 2x 2 y 5 3 4x 2 2y 5 6 Any ordered pair (x, 2x 2 3) is a solution of the system above. Cualquier par ordenado (x, 2x 2 3) es una solución del sistema que figura arriba. 1044 Student Resources n2pe-9050.indd 1044 10/14/05 10:05:14 AM dependent variable (p. 74) The output variable in an equation in two variables. See independent variable. variable dependiente (pág. 74) La variable de salida de una ecuación con dos variables. Ver variable independiente. determinant (p. 203) A real number associated with any square matrix A, denoted by det A or A. determinante (pág. 203) Número real asociado a toda matriz cuadrada A, denotada por det A o A. F 41 G 5 5(1) 2 3(4) 5 27 det 5 3 det a c F db G 5 ad 2 cb dimensions of a matrix (p. 187) The dimensions of a matrix with m rows and n columns are m 3 n. A matrix with 2 rows and 3 columns has the dimensions 2 3 3 (read “2 by 3”). dimensiones de una matriz (pág. 187) Las dimensiones de una matriz con m fi las y n columnas son m 3 n. Una matriz con 2 fi las y 3 columnas tiene por dimensiones 2 3 3 (leído “2 por 3”). direct variation (p. 107) Two variables x and y show direct variation provided that y 5 ax where a is a nonzero constant. The equation 5x 1 2y 5 0 represents direct variation because it is equivalent to the 5 equation y 5 – } x. 2 variación directa (pág. 107) Dos variables x e y indican una variación directa siempre que y 5 ax, donde a es una constante distinta de cero. La ecuación 5x 1 2y 5 0 representa una variación directa ya que es equivalente a la 5 ecuación y 5 – } x. 2 directrix of a parabola (p. 620) See parabola. See parabola. directriz de una parábola (pág. 620) Ver parábola. Ver parábola. función discreta (pág. 80) Función cuya gráfica consiste en puntos aislados. discriminant of a general second-degree equation (p. 653) The expression B2 – 4AC for the equation Ax2 1 Bxy 1 Cy 2 1 Dx 1 Ey 1 F 5 0. Used to identify which type of conic the equation represents. y x For the equation 4x 2 1 y 2 – 8x – 8 5 0, A 5 4, B 5 0, and C 5 1. B2 – 4AC 5 02 – 4(4)(1) 5 –16 ENGLISH-SPANISH GLOSSARY discrete function (p. 80) A function whose graph consists of separate points. Because B2 – 4AC < 0, B 5 0, and A Þ C, the conic is an ellipse. discriminante de una ecuación general de segundo grado (pág. 653) La expresión B2 – 4AC para la ecuación Ax2 1 Bxy 1 Cy 2 1 Dx 1 Ey 1 F 5 0. Se usa para identificar qué tipo de cónica representa la ecuación. Para la ecuación 4x 2 1 y 2 – 8x – 8 5 0, A 5 4, B 5 0 y C 5 1. B2 – 4AC 5 02 – 4(4)(1) 5 –16 Debido a que B2 – 4AC < 0, B 5 0 y A Þ C, la cónica es un elipse. English-Spanish Glossary n2pe-9050.indd 1045 1045 10/14/05 10:05:16 AM discriminant of a quadratic equation (p. 294) The expression b2 – 4ac for the quadratic equation ax2 1 bx 1 c 5 0; also the expression under the radical sign in the quadratic formula. The value of the discriminant of 2x 2 2 3x 2 7 5 0 is b2 2 4ac 5 (23)2 2 4(2)(27) 5 65. discriminante de una ecuación cuadrática (pág. 294) La expresión b2 – 4ac para la ecuación cuadrática ax2 1 bx 1 c 5 0; es también la expresión situada bajo el signo radical de la fórmula cuadrática. El valor del discriminante de 2x 2 2 3x 2 7 5 0 es b2 2 4ac 5 (23)2 2 4(2)(27) 5 65. disjoint events (p. 707) Events A and B are disjoint if they have no outcomes in common; also called mutually exclusive events. When you randomly select a card from a standard deck of 52 cards, selecting a club and selecting a heart are disjoint events. sucesos disjuntos (pág. 707) Los sucesos A y B son disjuntos si no tienen casos en común; también se llaman sucesos mutuamente excluyentes. Al seleccionar al azar una carta de una baraja normal de 52 cartas, sacar una de tréboles y sacar una de corazones son sucesos disjuntos. distance formula (p. 614) The distance d between any two The distance between (–3, 5) and (4, –1) is }} 2 2 } } Ï(4 2 (23))2 1 (21 2 5)2 5 Ï49 1 36 5 Ï85 . fórmula de la distancia (pág. 614) La distancia d entre dos puntos cualesquiera (x1, y1) y (x2, y 2) es La distancia entre (–3, 5) y (4, –1) es }} 2 2 d 5 Ï(x2 2 x1) 1 (y2 2 y1) . ENGLISH-SPANISH GLOSSARY }}} points (x1, y1) and (x2, y 2) is d 5 Ï(x2 2 x1) 1 (y2 2 y1) . }}} } } Ï(4 2 (23))2 1 (21 2 5)2 5 Ï49 1 36 5 Ï85 . domain (p. 72) The set of input values of a relation. See relation. dominio (pág. 72) El conjunto de valores de entrada de una relación. Ver relación. E (x 1 4)2 36 (y 2 2)2 16 eccentricity of a conic section (p. 665) The eccentricity e For the ellipse } 1 } 5 1, of a hyperbola or an ellipse is }c where c is the distance from c 5 Ï36 2 16 5 2Ï 5 , so the eccentricity is a } } } } each focus to the center and a is the distance from each vertex to the center. The eccentricity of a circle is e 5 0. The eccentricity of a parabola is e 5 1. c 5 2Ï5 5 Ï5 ≈ 0.745. e5} } } } excentricidad de una sección cónica (pág. 665) La Para la elipse } 1 } 5 1, excentricidad e de una hipérbola o de una elipse es }c , donde c 5 Ï36 2 16 5 2Ï 5 , por lo tanto la c es la distancia entre cada foco y el centro y a es la distancia entre cada vértice y el centro. La excentricidad de un círculo es e 5 0. La excentricidad de una parábola es e 5 1. Ï5 2Ï 5 c excentricidad es e 5 } } 5 } ≈ 0.745. a 5} element of a matrix (p. 187) Each number in a matrix. See matrix. elemento de una matriz (pág. 187) Cada número de una matriz. Ver matriz. a a Ï36 } 3 (x 1 4)2 36 } (y 2 2)2 16 } Ï36 } 3 1046 Student Resources n2pe-9050.indd 1046 10/14/05 10:05:17 AM element of a set (p. 715) Each object in a set; also called a member of the set. The elements of the set A 5 {1, 2, 3, 4} are 1, 2, 3, and 4. elemento de un conjunto (pág. 715) Cada objeto de un conjunto; también se llama miembro del conjunto. Los elementos del conjunto A 5 {1, 2, 3, 4} son 1, 2, 3 y 4. elimination method (p. 161) A method of solving a system of equations by multiplying equations by constants, then adding the revised equations to eliminate a variable. To use the elimination method to solve the system with equations 3x 2 7y 5 10 and 6x 2 8y 5 8, multiply the fi rst equation by 22 and add the equations to eliminate x. método de eliminación (pág. 161) Método para resolver un sistema de ecuaciones en el que se multiplican ecuaciones por constantes y se agregan luego las ecuaciones revisadas para eliminar una variable. Para usar el método de eliminación a fi n de resolver el sistema con las ecuaciones 3x 2 7y 5 10 y 6x 2 8y 5 8, multiplica la primera ecuación por 22 y suma las ecuaciones para eliminar x. ellipse (p. 634) The set of all points P in a plane such that the sum of the distances between P and two fi xed points, called the foci, is a constant. elipse (pág. 634) El conjunto de todos los puntos P de un plano tales que la suma de las distancias entre P y dos puntos fijos, llamados focos, es una constante. y center centro vertex vértice (2a, 0) co-vertex puntos extremos (0, b) P vertex d1 vértice d2 (a, 0) x (c, 0) focus major axis foco eje mayor minor axis (0, 2b) eje menor co-vertex constant puntos extremos d1 1 d2 5 constante (2c, 0) focus foco The set of positive integers less than 0 is the empty set, Ø. conjunto vacío (pág. 715) El conjunto que no tiene elementos, indicado Ø. El conjunto de los números enteros positivos menores que 0 es el conjunto vacío, Ø. end behavior (p. 339) The behavior of the graph of a function as x approaches positive infi nity (1`) or negative infi nity (2`). comportamiento (pág. 339) El comportamiento de la gráfica de una función al aproximarse x a infi nito positivo (1`) o a infi nito negativo (2`). ENGLISH-SPANISH GLOSSARY empty set (p. 715) The set with no elements, denoted Ø. f (x) → 1` as x → 2` or as x → 1`. f (x) → 1` según x → 2` o según x → 1`. equal matrices (p. 187) Matrices that have the same dimensions and equal elements in corresponding positions. matrices iguales (pág. 187) Matrices que tienen las mismas dimensiones y elementos iguales en posiciones correspondientes. F G 6 0 4 2} 4 3 } 4 5 F 3p2 21 1 1 21 0.75 G English-Spanish Glossary n2pe-9050.indd 1047 1047 10/14/05 10:05:19 AM equation (p. 18) A statement that two expressions are equal. 2x 2 3 5 7, 2x 2 5 4x ecuación (pág. 18) Enunciado que establece la igualdad de dos expresiones. equation in two variables (p. 74) An equation that contains two variables. y 5 3x – 5, d 5 –16t 2 1 64 ENGLISH-SPANISH GLOSSARY ecuación con dos variables (pág. 74) Ecuación que tiene dos variables. equivalent equations (p. 18) Equations that have the same solution(s). x 1 8 5 3 and 4x 5 220 are equivalent because both have the solution 25. ecuaciones equivalentes (pág. 18) Ecuaciones que tienen la misma solución o soluciones. x 1 8 5 3 y 4x 5 220 son equivalentes porque tienen ambas la solución 25. equivalent expressions (p. 12) Two algebraic expressions that have the same value for all values of their variable(s). 8x 1 3x and 11x are equivalent expressions, as are 2(x 2 3) and 2x 2 6. expresiones equivalentes (pág. 12) Dos expresiones algebraicas que tienen el mismo valor para todos los valores de la variable o variables. 8x 1 3x y 11x son expresiones equivalentes, como también lo son 2(x 2 3) y 2x 2 6. equivalent inequalities (p. 42) Inequalities that have the same solution. 3n 2 1 ≤ 8 and n 1 1.5 ≤ 4.5 are equivalent inequalities because the solution of both inequalities is all numbers less than or equal to 3. desigualdades equivalentes (pág. 42) Desigualdades que tienen la misma solución. 3n 2 1 ≤ 8 y n 1 1.5 ≤ 4.5 son desigualdades equivalentes ya que la solución de ambas son todos los números menores o iguales a 3. experimental group (p. 773) A group that undergoes some procedure or treatment when an experiment is conducted. See also control group. One group of headache sufferers, the experimental group, is given pills containing medication. Another group, the control group, is given pills containing no medication. grupo experimental (pág. 773) Grupo que se somete a algún procedimiento o tratamiento durante la realización de un experimento. Ver también grupo de control. Un grupo de personas que sufren de dolores de cabeza, el grupo experimental, recibe píldoras que contienen el medicamento. Otro grupo, el grupo de control, recibe píldoras sin el medicamento. experimental probability (p. 700) A probability based on performing an experiment, conducting a survey, or looking at the history of an event. You roll a six-sided die 100 times and get a 4 nineteen times. The experimental probability of rolling a 4 with the die 19 is } 5 0.19. 100 probabilidad experimental (pág. 700) Probabilidad basada en la realización de un experimento o una encuesta o en el estudio de la historia de un suceso. Lanzas 100 veces un dado de seis caras y sale diecinueve veces el 4. La probabilidad experimental de que salga el 4 al lanzar el 19 dado es } 5 0.19. 100 1048 Student Resources n2pe-9050.indd 1048 10/14/05 10:05:20 AM explicit rule (p. 827) A rule for a sequence that gives the nth term an as a function of the term’s position number n in the sequence. The rules an 5 211 1 4n and an 5 3(2) n 2 1 are explicit rules for sequences. regla explícita (pág. 827) Regla de una progresión que expresa el término enésimo an en función del número de posición n del término en la progresión. Las reglas an 5 211 1 4n y an 5 3(2) n 2 1 son reglas explícitas de progresiones. exponent (p. 10) The number or variable that represents the number of times the base of a power is used as a factor. In the power 25, the exponent is 5. exponent (pág. 10) El número o la variable que representa la cantidad de veces que la base de una potencia se usa como factor. En la potencia 25, el exponente es 5. exponential decay function (p. 486) If a > 0 and 0 < b < 1, then the function y 5 ab x is an exponential decay function with decay factor b. y y52 función de decrecimiento exponencial (pág. 486) Si a > 0 y 0 < b < 1, entonces la función y 5 ab x es una función de decrecimiento exponencial con factor de decrecimiento b. x 14 cx 1 x 1 exponential equation (p. 515) An equation in which a variable expression occurs as an exponent. 1 4x 5 1 } 2 x23 ecuación exponencial (pág. 515) Ecuación que tiene como exponente una expresión algebraica. 1 4x 5 1 } 2 x23 exponential function (p. 478) A function of the form y 5 abx, where a Þ 0, b > 0, and b Þ 1. See exponential growth function and exponential decay function. función exponencial (pág. 478) Función de la forma y 5 abx, donde a Þ 0, b > 0 y b Þ 1. Ver función de crecimiento exponencial y función de decrecimiento exponencial. función de crecimiento exponencial (pág. 478) Si a > 0 y b > 1, entonces la función y 5 abx es una función de crecimiento exponencial con factor de crecimiento b. 2 is an exponential equation. es una ecuación exponencial. y y5 3 1 1 2 ? 4x x extraneous solution (p. 51) An apparent solution that must be rejected because it does not satisfy the original equation. Solving 2x 1 12 5 4x gives the apparent solutions x 5 6 and x 5 22. The apparent solution 22 is extraneous because it does not satisfy the original equation. solución extraña (pág. 51) Solución aparente que debe rechazarse ya que no satisface la ecuación original. Al resolver 2x 1 12 5 4x se obtienen las soluciones aparentes x 5 6 y x 5 22. La solución aparente 22 es extraña ya no satisface la ecuación original. English-Spanish Glossary n2pe-9050.indd 1049 ENGLISH-SPANISH GLOSSARY exponential growth function (p. 478) If a > 0 and b > 1, then the function y 5 abx is an exponential growth function with growth factor b. 2 1049 10/14/05 10:05:22 AM F factor by grouping (p. 354) To factor a polynomial with four terms by grouping, factor common monomials from pairs of terms, and then look for a common binomial factor. x 3 2 3x 2 2 16x 1 48 5 x 2 (x 2 3) 2 16(x 2 3) 5 (x 2 2 16)(x 2 3) 5 (x 1 4)(x 2 4)(x 2 3) factorizar por grupos (pág. 354) Para factorizar por grupos un polinomio con cuatro términos, factoriza unos monomios comunes a partir de los pares de términos y luego busca un factor binómico común. factored completely (p. 353) A factorable polynomial with integer coefficients is factored completely if it is written as a product of unfactorable polynomials with integer coefficients. 3x(x 2 5) is factored completely. completamente factorizado (pág. 353) Un polinomio que puede factorizarse y que tiene coeficientes enteros está completamente factorizado si está escrito como producto de polinomios que no pueden factorizarse y que tienen coeficientes enteros. 3x(x 2 5) está completamente factorizado. (x 1 2)(x 2 2 6x 1 8) is not factored completely because x 2 2 6x 1 8 can be factored as (x 2 2)(x 2 4). (x 1 2)(x 2 2 6x 1 8) no está completamente factorizado ya que x 2 2 6x 1 8 puede factorizarse como (x 2 2)(x 2 4). factorial (p. 684) For any positive integer n, the expression n!, read “n factorial,” is the product of all the integers from 1 to n. Also, 0! is defi ned to be 1. 6! 5 6 · 5 · 4 · 3 · 2 · 1 5 720 ENGLISH-SPANISH GLOSSARY factorial (pág. 684) Para cualquier número entero positivo n, la expresión n!, leída “factorial de n”, es el producto de todos los números enteros entre 1 y n. También, 0! se defi ne como 1. feasible region (p. 174) In linear programming, the graph of the system of constraints. See linear programming. región factible (pág. 174) En la programación lineal, la gráfica del sistema de restricciones. Ver programación lineal. finite differences (p. 393) When the x-values in a data set are equally spaced, the differences of consecutive y-values are called fi nite differences. diferencias finitas (pág. 393) Cuando los valores de x de un conjunto de datos están a igual distancia entre sí, las diferencias entre los valores de y consecutivos se llaman diferencias fi nitas. f (x) 5 x 2 f(1) 1 f(2) 4 42153 f(3) 9 92455 f (4) 16 16 2 9 5 7 The first-order finite differences are 3, 5, and 7. Las diferencias fi nitas de primer orden son 3, 5 y 7. foci of a hyperbola (p. 642) See hyperbola. See hyperbola. focos de una hipérbola (pág. 642) Ver hipérbola. Ver hipérbola. foci of an ellipse (p. 634) See ellipse. See ellipse. focos de una elipse (pág. 634) Ver elipse. Ver elipse. 1050 Student Resources n2pe-9050.indd 1050 10/14/05 10:05:23 AM focus of a parabola (p. 620) See parabola. See parabola. foco de una parábola (pág. 620) Ver parábola. Ver parábola. formula (p. 26) An equation that relates two or more quantities, usually represented by variables. The formula P 5 2l 1 2w relates the length and width of a rectangle to its perimeter. fórmula (pág. 26) Ecuación que relaciona dos o más cantidades que generalmente se representan por variables. La fórmula P 5 2l 1 2w relaciona el largo y el ancho de un rectángulo con su perímetro. frequency of a periodic function (p. 910) The reciprocal of the period. Frequency is the number of cycles per unit of time. 2p 1 P 5 2 sin 4000pt has period } 5} , frecuencia de una función periódica (pág. 910) El recíproco del período. La frecuencia es el número de ciclos por unidad de tiempo. 4000p 2000 so its frequency is 2000 cycles per second (hertz) when t represents time in seconds. 2p 1 P 5 2 sen 4000pt tiene período } 5} , 4000p 2000 por lo que su frecuencia es de 2000 ciclos por segundo (hertzios) cuando t representa el tiempo en segundos. The relation (–4, 6), (3, –9), and (7, –9) is a function. The relation (0, 3), (0, 6), and (10, 8) is not a function because the input 0 is mapped onto both 3 and 6. función (pág. 73) Relación para la que cada entrada tiene exactamente una salida. La relación (–4, 6), (3, –9) y (7, –9) es una función. La relación (0, 3), (0, 6) y (10, 8) no es una función ya que la entrada 0 se hace corresponder tanto con 3 como con 6. function notation (p. 75) Using f(x) (or a similar symbol such as g(x) or h(x)) to represent the dependent variable of a function. The linear function y 5 mx 1 b can be written using function notation as f(x) 5 mx 1 b. notación de función (pág. 75) Usar f(x) (o un símbolo semejante como g(x) o h(x)) para representar la variable dependiente de una función. La función lineal y 5 mx 1 b escrita en notación de función es f(x) 5 mx 1 b. G general second-degree equation in x and y (p. 653) The form Ax2 1 Bxy 1 Cy 2 1 Dx 1 Ey 1 F 5 0. 16x 2 – 9y 2 – 96x 1 36y – 36 5 0 and 4x 2 1 y 2 – 8x – 8 5 0 are second-degree equations in x and y. ecuación general de segundo grado en x e y (pág. 653) La forma Ax2 1 Bxy 1 Cy 2 1 Dx 1 Ey 1 F 5 0. 16x 2 – 9y 2 – 96x 1 36y – 36 5 0 y 4x 2 1 y 2 – 8x – 8 5 0 son ecuaciones de segundo grado en x e y. English-Spanish Glossary n2pe-9050.indd 1051 ENGLISH-SPANISH GLOSSARY function (p. 73) A relation for which each input has exactly one output. 1051 10/14/05 10:05:25 AM geometric probability (p. 701) A probability found by calculating a ratio of two lengths, areas, or volumes. 14 probabilidad geométrica (pág. 701) Probabilidad hallada al calcular una razón entre dos longitudes, áreas o volúmenes. 14 The probability that a dart that hits the square p7 at random lands inside the circle is p < 0.785. } 2 2 14 La probabilidad de que un dardo que da con el blanco cuadrado, dé al azar en el interior del p7 círculo es p < 0.785. } 2 2 14 geometric sequence (p. 810) A sequence in which the ratio of any term to the previous term is constant. 219, 38, 276, 152 is a geometric sequence with common ratio 22. progresión geométrica (pág. 810) Progresión en la que la razón entre cualquier término y el término precedente es constante. 219, 38, 276, 152 es una progresión geométrica con una razón común de 22. geometric series (p. 812) The expression formed by adding the terms of a geometric sequence. serie geométrica (pág. 812) La expresión formada al sumar los términos de una progresión geométrica. 5 ∑ 4(3)i 2 1 5 4 1 12 1 36 1 108 1 324 i51 ENGLISH-SPANISH GLOSSARY graph of a linear inequality in two variables (p. 132) The set of all points in a coordinate plane that represent solutions of the inequality. y y > 4x 2 3 1 2x gráfica de una desigualdad lineal con dos variables (pág. 132) El conjunto de todos los puntos de un plano de coordenadas que representan las soluciones de la desigualdad. graph of a system of linear inequalities (p. 168) The graph of all solutions of the system. y graph of system gráfica del sistema y≥x23 1 gráfica de un sistema de desigualdades lineales (pág. 168) La gráfica de todas las soluciones del sistema. x 1 y < 22x 1 3 graph of an equation in two variables (p. 74) The set of all points (x, y) that represent solutions of the equation. gráfica de una ecuación con dos variables (pág. 74) El conjunto de todos los puntos (x, y) que representan soluciones de la ecuación. y y 5 2 12 x 1 4 1 x 1 1052 Student Resources n2pe-9050.indd 1052 10/14/05 10:05:26 AM graph of an inequality in one variable (p. 41) All points on a number line that represent solutions of the inequality. gráfica de una desigualdad con una variable (pág. 41) Todos los puntos de una recta numérica que representan soluciones de la desigualdad. 0 21 1 2 3 4 x<3 growth factor (p. 478) The quantity b in the exponential growth function y 5 abx with a > 0 and b > 1. The growth factor for the function y 5 8(3.4) x is 3.4. factor de crecimiento (pág. 478) La cantidad b de la función de crecimiento exponencial y 5 abx, con a > 0 y b > 1. El factor de crecimiento de la función y 5 8(3.4) x es 3.4. H half-planes (p. 132) The two regions into which the boundary line of a linear inequality divides the coordinate plane. The solution of y , 3 is the half-plane consisting of all the points below the line y 5 3. semiplanos (pág. 132) Las dos regiones en que la recta límite de una desigualdad lineal divide al plano de coordenadas. La solución de y , 3 es el semi-plano que consta de todos los puntos que se encuentran debajo de la recta y 5 3. hyperbola (pp. 558, 642) The set of all points P in a plane such that the difference of the distances from P to two fi xed points, called the foci, is constant. center centro vertex vértice (2a, 0) y (0, b) d2 P d1 vertex vértice (a, 0) (c, 0) focus foco (2c, 0) focus foco x (0, 2b) transverse axis eje transverso constant d2 2 d1 5 constante I identity (p. 12) A statement that equates two equivalent expressions. 8x 1 3x 5 11x and 2(x 2 3) 5 2x 2 6 are identities. identidad (pág. 12) Enunciado que hace iguales a dos expresiones equivalentes. 8x 1 3x 5 11x y 2(x 2 3) 5 2x 2 6 son identidades. identity matrix (p. 210) The n 3 n matrix that has 1’s on the main diagonal and 0’s elsewhere. The 2 3 2 identity matrix is matriz identidad (pág. 210) La matriz n 3 n que tiene los 1 en la diagonal principal y los 0 en las otras posiciones. La matriz identidad 2 3 2 es F G F G 1 0 0 1 1 0 0 1 . . English-Spanish Glossary n2pe-9050.indd 1053 ENGLISH-SPANISH GLOSSARY hipérbola (págs. 558, 642) El conjunto de todos los puntos P de un plano tales que la diferencia de distancias entre P y dos puntos fijos, llamados focos, es constante. branches of hyperbola ramas de una hypérbola 1053 10/14/05 10:05:27 AM imaginary number (p. 276) A complex number a 1 bi where b Þ 0. 5i and 2 2 i are imaginary numbers. número imaginario (pág. 276) Un número complejo a 1 bi, donde b Þ 0. 5i y 2 2 i son números imaginarios. } imaginary unit i (p. 275) i 5 Ï21 , so i 2 5 21. } } 2 unidad imaginaria i (pág. 275) i 5 Ï21 , por lo que i 5 21. inconsistent system (p. 154) A system of equations that has no solution. sistema incompatible (pág. 154) Sistema de ecuaciones que no tiene solución. } Ï 23 5 i Ï 3 x1y54 x1y51 The system above has no solution because the sum of two numbers cannot be both 4 and 1. ENGLISH-SPANISH GLOSSARY El sistema de arriba no tiene ninguna solución porque la suma de dos números no puede ser 4 y 1. independent events (p. 717) Two events such that the occurrence of one event has no effect on the occurrence of the other event. If a coin is tossed twice, the outcome of the fi rst toss (heads or tails) and the outcome of the second toss are independent events. sucesos independientes (pág. 717) Dos sucesos tales que la ocurrencia de uno de ellos no afecta a la ocurrencia del otro. Al lanzar una moneda dos veces, el resultado del primer lanzamiento (cara o cruz) y el resultado del segundo lanzamiento son sucesos independientes. independent system (p. 154) A consistent system that has exactly one solution. The system consisting of 4x 1 y 5 8 and 2x 2 3y 5 18 has exactly one solution, (3, 24). sistema independiente (pág. 154) Sistema compatible que tiene exactamente una solución. El sistema que consiste de 4x 1 y 5 8 y 2x 2 3y 5 18 tiene exactamente una solución, (3, 24). independent variable (p. 74) The input variable in an equation in two variables. In y 5 3x – 5, the independent variable is x. The dependent variable is y because the value of y depends on the value of x. variable independiente (pág. 74) La variable de entrada de una ecuación con dos variables. En y 5 3x – 5, la variable independiente es x. La variable dependiente es y ya que el valor de y depende del valor de x. index of a radical (p. 414) The integer n, greater than 1, in n} the expression Ïa . The index of Ï2216 is 3. índice de un radical (pág. 414) El número entero n, que es n} mayor que 1 y aparece en la expresión Ïa . El índice de Ï2216 es 3. initial side of an angle (p. 859) See terminal side of an angle. See standard position of an angle. lado inicial de un ángulo (pág. 859) Ver lado terminal de un ángulo. Ver posición normal de un ángulo. 3} 3} 1054 Student Resources n2pe-9050.indd 1054 10/14/05 10:05:28 AM intercept form of a quadratic function (p. 246) The form y 5 a(x 2 p)(x 2 q), where the x-intercepts of the graph are p and q. The function y 5 2(x 1 3)(x 2 1) is in intercept form. forma de intercepto de una función cuadrática (pág. 246) La forma y 5 a(x 2 p)(x 2 q), donde los interceptos en x de la gráfica son p y q. La función y 5 2(x 1 3)(x 2 1) está en la forma de intercepto. intersection of sets (p. 715) The intersection of two sets A and B, written A ∩ B, is the set of all elements in both A and B. If A 5 {1, 2, 4, 8} and B 5 {2, 4, 6, 8, 10}, then A ∩ B 5 {2, 4, 8}. intersección de conjuntos (pág. 715) La intersección de dos conjuntos A y B, escrita A ∩ B, es el conjunto de todos los elementos que están tanto en A como en B. Si A 5 {1, 2, 4, 8} y B 5 {2, 4, 6, 8, 10}, entonces A ∩ B 5 {2, 4, 8}. inverse cosine function (p. 875) If 21 ≤ a ≤ 1, then the inverse cosine of a is an angle θ, written θ 5 cos21 a, where cos θ 5 a and 0 ≤ θ ≤ π (or 08 ≤ θ ≤ 1808). When 08 ≤ θ ≤ 1808, the angle θ whose cosine 1 1 is } is 608, so θ 5 cos21 } 5 608 2 2 p 1 } (or θ 5 cos21 } 5 3 ). 2 función inversa del coseno (pág. 875) Si 21 ≤ a ≤ 1, entonces el coseno inverso de a es un ángulo θ, escrito θ 5 cos21 a, donde cos θ 5 a y 0 ≤ θ ≤ π (ó 08 ≤ θ ≤ 1808 ). Cuando 08 ≤ θ ≤ 1808, el ángulo θ cuyo coseno 1 1 es } es de 608, por lo que θ 5 cos21 } 5 608 2 2 p 1 } (ó θ 5 cos21 } 5 3 ). 2 inverse function (p. 438) An inverse relation that is a function. Functions f and g are inverses provided that f(g(x)) 5 x and g(f(x)) 5 x. inverse matrices (p. 210) Two n 3 n matrices are inverses of each other if their product (in both orders) is the n 3 n identity matrix. See also identity matrix. matrices inversas (pág. 210) Dos matrices n 3 n son inversas entre sí si su producto (de ambos órdenes) es la matriz identidad n 3 n. Ver también matriz identidad. f(g(x)) 5 (x 2 5) 1 5 5 x g(f(x)) 5 (x 1 5) 2 5 5 x So, f and g are inverse functions. Entonces, f y g son funciones inversas. F G F G F GF G F G F GF G F G 21 25 8 2 23 5 3 8 25 8 2 5 2 23 25 8 3 8 2 23 2 5 3 8 because 2 5 ya que 5 5 1 0 and 0 1 y 1 0 0 1 . inverse relation (p. 438) A relation that interchanges the input and output values of the original relation. The graph of an inverse relation is a reflection of the graph of the original relation, with y 5 x as the line of reflection. To fi nd the inverse of y 5 3x 2 5, switch x and y to obtain x 5 3y 2 5. Then solve for y relación inversa (pág. 438) Relación en la que se intercambian los valores de entrada y de salida de la relación original. La gráfica de una relación inversa es una reflexión de la gráfica de la relación original, con y 5 x como eje de reflexión. Para hallar la inversa de y 5 3x 2 5, intercambia x e y para obtener x 5 3y 2 5. Luego resuelve para y para obtener la 5 1 to obtain the inverse relation y 5 } x1} . 3 3 5 1 relación inversa y 5 } x1} . 3 3 English-Spanish Glossary n2pe-9050.indd 1055 ENGLISH-SPANISH GLOSSARY función inversa (pág. 438) Relación inversa que es una función. Las funciones f y g son inversas siempre que f(g(x)) 5 x y g(f(x)) 5 x. f(x) 5 x 1 5; g(x) 5 x 2 5 1055 10/14/05 10:05:30 AM inverse sine function (p. 875) If 21 ≤ a ≤ 1, then the inverse sine of a is an angle θ, written θ 5 sin21 a, where sin θ 5 a π π ≤θ≤} (or 2908 ≤ θ ≤ 908). and 2} 2 2 función inversa del seno (pág. 875) Si 21 ≤ a ≤ 1, entonces el seno inverso de a es un ángulo θ, escrito θ 5 sen21 a, inverse tangent function (p. 875) If a is any real number, then the inverse tangent of a is an angle θ, written π θ 5 tan21 a, where tan θ 5 a and – } <θ<} (or 2908 < θ < 908). 2 2 función inversa de la tangente (pág. 875) Si a es un número real cualquiera, entonces la tangente inversa de a π π es un ángulo θ, escrito θ 5 tan21 a, donde tan θ 5 a y – } <θ<} 2 2 (ó 2908 < θ < 908 ). inverse variation (p. 551) The relationship of two variables a x and y if there is a nonzero number a such that y 5 } x. variación inversa (pág. 551) La relación entre dos variables a x e y si hay un número a distinto de cero tal que y 5 } x. ENGLISH-SPANISH GLOSSARY 1 1 is } is 308, so θ 5 sin21 } 5 308 2 2 p 21 1 (or θ 5 sin } 5 }). 6 2 Cuando –908 ≤ θ ≤ 908, el ángulo θ cuyo seno 1 1 es } es de 308, por lo que θ 5 sen21 } 5 308 2 2 p 1 } (ó θ 5 sen21 } 5 ). 6 2 π π donde sen θ 5 a y 2} ≤θ≤} (ó 2908 ≤ θ ≤ 908 ). 2 2 π When –908 ≤ θ ≤ 908, the angle θ whose sine iteration (p. 830) The repeated composition of a function with itself. The result of one iteration is f(f(x)), and of two iterations is f(f(f(x))). iteración (pág. 830) La composición repetida de una función usando la función misma. El resultado de una iteración es f(f(x)), y el de dos iteraciones es f(f(f(x))). When –908 < θ < 908, the angle θ whose } tangent is 2Ï3 is 260°, so } 21 θ 5 tan (2Ï3 ) 5 2608 } p 21 (or θ 5 tan (2Ï3 ) 5 2}). 3 Cuando –908 < θ < 908, el ángulo θ cuya } tangente es 2Ï3 es de 260°, por lo que } 21 θ 5 tan (2Ï3 ) 5 2608 } p (ó θ 5 tan21 (2Ï 3 ) 5 2}). 3 3 represent The equations xy 5 7 and y 5 – } x inverse variation. 3 representan la Las ecuaciones xy 5 7 e y 5 – } x variación inversa. f(x) 5 23x 1 1; x 0 5 2 x1 5 f(x 0 ) 5 f(2) 5 23(2) 1 1 5 25 x 2 5 f(x1) 5 f(25) 5 23(25) 1 1 5 16 x 3 5 f(x 2 ) 5 f(16) 5 23(16) 1 1 5 247 J joint variation (p. 553) A relationship that occurs when a quantity varies directly with the product of two or more other quantities. The equation z 5 5xy represents joint variation. variación conjunta (pág. 553) Relación producida cuando una cantidad varía directamente con el producto de dos o más otras cantidades. La ecuación z 5 5xy representa la variación conjunta. 1056 Student Resources n2pe-9050.indd 1056 10/14/05 10:05:32 AM L law of cosines (p. 889) If TABC has sides of length a, b, and c as shown, then a2 5 b2 1 c 2 2 2bc cos A, b2 5 a2 1 c 2 2 2ac cos B, and c 2 5 a2 1 b2 2 2ab cos C. ley de los cosenos (pág. 889) Si TABC tiene lados de longitud a, b y c como se indica, entonces a2 5 b2 1 c 2 2 2bc cos A, b2 5 a2 1 c 2 2 2ac cos B y c 2 5 a2 1 b2 2 2ab cos C. c 5 14 A 348 a c b a 5 11 A C b C b2 5 a2 1 c 2 2 2ac cos B b2 5 112 1 142 2 2(11)(14) cos 348 b2 ≈ 61.7 b ≈ 7.85 law of sines (p. 882) If TABC has sides of length a, b, and c as shown, then sin A sin B sin C 5} } a 5} c . b B B B b 5 15 a c C 1078 a 258 A ley de los senos (pág. 882) Si TABC tiene lados de longitud a, b y c como se indica, entonces b A C c sin 258 }5 15 sen 258 }5 15 sen A sen B sen C 5} } a 5} c . b B sin 1078 c sen 1078 → c < 33.9 } c } → c < 33.9 leading coefficient (p. 337) The coefficient in the term of a polynomial function that has the greatest exponent. See polynomial function. coeficiente inicial (pág. 337) En una función polinómica, el coeficiente del término con el mayor exponente. Ver función polinómica. like radicals (p. 422) Radical expressions with the same index and radicand. Ï10 and 7Ï10 are like radicals. radicales semejantes (pág. 422) Expresiones radicales con el mismo índice y el mismo radicando. Ï10 y 7Ï10 son radicales semejantes. like terms (p. 12) Terms that have the same variable parts. Constant terms are also like terms. In the algebraic expression 5x 2 1 (23x) 1 7 1 4x 1 (22), 23x and 4x are like terms, and 7 and 22 are like terms. términos semejantes (pág. 12) Términos que tienen las mismas variables. Los términos constantes también son términos semejantes. En la expresión algebraica 5x 2 1 (23x) 1 7 1 4x 1 (22), 23x y 4x son términos semejantes, y 7 y 22 también lo son. linear equation in one variable (p. 18) An equation that can be written in the form ax 1 b 5 0 where a and b are constants and a Þ 0. 4 The equation } x 1 8 5 0 is a linear equation ecuación lineal con una variable (pág. 18) Ecuación que puede escribirse en la forma ax 1 b 5 0, donde a y b son constantes y a Þ 0. 4 La ecuación } x 1 8 5 0 es una ecuación 4} 4} 5 in one variable. 5 lineal con una variable. English-Spanish Glossary n2pe-9050.indd 1057 ENGLISH-SPANISH GLOSSARY 4} 4} 1057 10/14/05 10:05:34 AM linear equation in three variables (p. 178) An equation of the form ax 1 by 1 cz 5 d where a, b, and c are not all zero. 2x 1 y 2 z 5 5 is a linear equation in three variables. ecuación lineal con tres variables (pág. 178) Ecuación de la forma ax 1 by 1 cz 5 d, donde a, b y c no son todos cero. 2x 1 y 2 z 5 5 es una ecuación lineal con tres variables. linear function (p. 75) A function that can be written in the form y 5 mx 1 b where m and b are constants. The function y 5 –2x – 1 is a linear function with m 5 –2 and b 5 –1. función lineal (pág. 75) Función que puede escribirse en la forma y 5 mx 1 b, donde m y b son constantes. La función y 5 –2x – 1 es una función lineal con m 5 –2 y b 5 –1. linear inequality in one variable (p. 41) An inequality that can be written in one of the following forms: ax 1 b < 0, ax 1 b ≤ 0, ax 1 b > 0, or ax 1 b ≥ 0. 5x 1 2 > 0 is a linear inequality in one variable. desigualdad lineal con una variable (pág. 41) Desigualdad que puede escribirse de una de las siguientes formas: ax 1 b < 0, ax 1 b ≤ 0, ax 1 b > 0 ó ax 1 b ≥ 0. 5x 1 2 > 0 es una desigualdad lineal con una variable. linear inequality in two variables (p. 132) An inequality that can be written in one of the following forms: Ax 1 By < C, Ax 1 By ≤ C, Ax 1 By > C, or Ax 1 By ≥ C. 5x – 2y ≥ –4 is a linear inequality in two variables. ENGLISH-SPANISH GLOSSARY desigualdad lineal con dos variables (pág. 132) Desigualdad que puede escribirse de una de las siguientes formas: Ax 1 By < C, Ax 1 By ≤ C, Ax 1 By > C o Ax 1 By ≥ C. 5x – 2y ≥ –4 es una desigualdad lineal con dos variables. linear programming (p. 174) The process of maximizing or minimizing a linear objective function subject to a system of linear inequalities called constraints. The graph of the system of constraints is called the feasible region. programación lineal (pág. 174) El proceso de maximizar o minimizar una función objetivo lineal sujeta a un sistema de desigualdades lineales llamadas restricciones. La gráfica del sistema de restricciones se llama región factible. y (4, 5) feasible region región factible 1 1 (4, 0) (8, 0) x To maximize the objective function P 5 35x 1 30y subject to the constraints x ≥ 4, y ≥ 0, and 5x 1 4y ≤ 40, evaluate P at each vertex. The maximum value of 290 occurs at (4, 5). Para maximizar la función objetivo P 5 35x 1 30y sujeta a las restricciones x ≥ 4, y ≥ 0 y 5x 1 4y ≤ 40, evalúa P en cada vértice. El valor máximo de 290 ocurre en (4, 5). 1058 Student Resources n2pe-9050.indd 1058 10/14/05 10:05:35 AM local maximum (p. 388) The y-coordinate of a turning point of a function if the point is higher than all nearby points. máximo local (pág. 388) La coordenada y de un punto crítico de una función si el punto está situado más alto que todos los puntos cercanos. Maximum Maximo X=0 Y=6 The function f(x) 5 x 3 2 3x 2 1 6 has a local maximum of y 5 6 when x 5 0. La función f(x) 5 x 3 2 3x 2 1 6 tiene un máximo local de y 5 6 cuando x 5 0. local minimum (p. 388) The y-coordinate of a turning point of a function if the point is lower than all nearby points. mínimo local (pág. 388) La coordenada y de un punto crítico de una función si el punto está situado más bajo que todos los puntos cercanos. Minimum Minimo X=-.56971 Y=-6.50858 The function f(x ) 5 x 4 2 6x 3 1 3x 2 1 10x 2 3 has a local minimum of y < 26.51 when x < 20.57. La función f(x ) 5 x 4 2 6x 3 1 3x 2 1 10x 2 3 tiene un mínimo local de y < 26.51 cuando x < 20.57. log 2 8 5 3 because 23 5 8. logaritmo de y con base b (pág. 499) Sean b e y números positivos, con b Þ 1. El logaritmo de y con base b, denotado por log b y y leído “log base b de y”, se defi ne de esta manera: log b y 5 x si y sólo si bx 5 y. log 2 8 5 3 ya que 2 3 5 8. logarithmic equation (p. 517) An equation that involves a logarithm of a variable expression. log5 (4x 2 7) 5 log5 (x 1 5) is a logarithmic equation. ecuación logarítmica (pág. 517) Ecuación en la que aparece el logaritmo de una expresión algebraica. log5 (4x 2 7) 5 log5 (x 1 5) es una ecuación logarítmica. 21 1 log1/4 4 5 21 because 1 } 2 4 1 log1/4 4 5 21 ya que 1 } 2 4 21 5 4. 5 4. ENGLISH-SPANISH GLOSSARY logarithm of y with base b (p. 499) Let b and y be positive numbers with b Þ 1. The logarithm of y with base b, denoted log b y and read “log base b of y,” is defi ned as follows: log b y 5 x if and only if bx 5 y. M major axis of an ellipse (p. 634) The line segment joining the vertices of an ellipse. See ellipse. eje mayor de una elipse (pág. 634) El segmento de recta que une los vértices de una elipse. Ver elipse. English-Spanish Glossary n2pe-9050.indd 1059 1059 10/14/05 10:05:37 AM margin of error (p. 768) The margin of error gives a limit on how much the response of a sample would be expected to differ from the response of the population. If 40% of the people in a poll prefer candidate A, and the margin of error is ±4%, then it is expected that between 36% and 44% of the entire population prefer candidate A. margen de error (pág. 768) El margen de error indica un límite acerca de cuánto se prevé que diferirían las respuestas obtenidas en una muestra de las obtenidas en la población. Si el 40% de los encuestados prefiere al candidato A y el margen de error es ±4%, entonces se prevé que entre el 36% y el 44% de la población total prefiere al candidato A. matrix, matrices (p. 187) A rectangular arrangement of numbers in rows and columns. Each number in a matrix is an element. matriz, matrices (pág. 187) Disposición rectangular de números colocados en fi las y columnas. Cada numero de la matriz es un elemento. A5 F 4 21 5 0 3 6 G Matrix A has 2 rows and 3 columns. The element in the second row and fi rst column is 0. La matriz A tiene 2 fi las y 3 columnas. El elemento en la segunda fi la y en la primera columna es 0. matrix of constants (p. 212) The matrix of constants of the linear system ax 1 by 5 e, cx 1 dy 5 f is e . f See coefficient matrix. matriz de constantes (pág. 212) La matriz de constantes del sistema lineal ax 1 by 5 e, cx 1 dy 5 f es e . f Ver matriz coeficiente. matrix of variables (p. 212) The matrix of variables of the linear system ax 1 by 5 e, cx 1 dy 5 f is xy . See coefficient matrix. matriz de variables (pág. 212) La matriz de variables del sistema lineal ax 1 by 5 e, cx 1 dy 5 f es xy . Ver matriz coeficiente. FG ENGLISH-SPANISH GLOSSARY FG FG FG maximum value of a quadratic function (p. 238) The y-coordinate of the vertex for y 5 ax2 1 bx 1 c when a < 0. valor máximo de una función cuadrática (pág. 238) La coordenada y del vértice para y 5 ax2 1 bx 1 c cuando a < 0. 2 y y 5 2x 2 1 2x 2 1 (1, 0) 3 x The maximum value of y 5 2x 2 1 2x 2 1 is 0. El valor máximo de y 5 2x 2 1 2x 2 1 es 0. mean (p. 744) For the data set x1, x2, . . . , xn, the mean is _ x1 1 x2 1 . . . 1 xn x 5 }} . Also called average. n See measure of central tendency. media (pág. 744) Para el conjunto de datos x1, x2, . . . , xn, la _ x1 1 x2 1 . . . 1 xn media es x 5 }} . También se llama promedio. n Ver medida de tendencia central. 1060 Student Resources n2pe-9050.indd 1060 10/14/05 10:05:38 AM measure of central tendency (p. 744) A number used to represent the center or middle of a set of data values. Mean, median, and mode are three measures of central tendency. 14, 17, 18, 19, 20, 24, 24, 30, 32 1 17 1 18 1 . . . 1 32 198 The mean is 14 }} 5 } 5 22. 9 9 The median is the middle number, 20. The mode is 24 because 24 occurs the most frequently. medida de tendencia central (pág. 744) Número usado para representar el centro o la posición central de un conjunto de valores de datos. La media, la mediana y la moda son tres medidas de tendencia central. 1 17 1 18 1 . . . 1 32 198 La media es 14 }} 5 } 5 22. measure of dispersion (p. 745) A statistic that tells you how dispersed, or spread out, data values are. Range and standard deviation are measures of dispersion. See range and standard deviation. medida de dispersión (pág. 745) Estadística que te indica cómo se dispersan, o distribuyen, los valores de datos. El rango y la desviación típica son medidas de dispersión. Ver rango y desviación típica. median (p. 744) The median of n numbers is the middle number when the numbers are written in numerical order. If n is even, the median is the mean of the two middle numbers. See measure of central tendency. mediana (pág. 744) La mediana de n números es el número central cuando los números se escriben en orden numérico. Si n es par, la mediana es la media de los dos números centrales. Ver medida de tendencia central. midpoint formula (p. 615) The midpoint M of the line The midpoint of the line segment joining 2 segment joining A(x1, y1) and B(x2, y 2) is M } , } . 2 2 fórmula del punto medio (pág. 615) El punto medio M del segmento de recta que une A(x1, y1) y B(x2, y 2) es 1 x1 1 x2 y1 1 y2 2 22 1 8 3 1 6 9 (–2, 3) and (8, 6) is 1 } , } 2 5 1 3, } 2. 2 2 minimum value of a quadratic function (p. 238) The y-coordinate of the vertex for y 5 ax2 1 bx 1 c when a > 0. 2 2 El punto medio del segmento de recta que 22 1 8 3 1 6 9 une (–2, 3) y (8, 6) es 1 } , } 2 5 1 3, } 2. 2 M },} . 2 9 2 2 y x 2 valor mínimo de una función cuadrática (pág. 238) La coordenada y del vértice para y 5 ax2 1 bx 1 c cuando a > 0. y 5 x 2 2 6x 1 5 ENGLISH-SPANISH GLOSSARY 1 x1 1 x2 y1 1 y2 9 La mediana es el número central, 20. La moda es 24 ya que 24 ocurre más veces. (3, 24) The minimum value of y 5 x 2 2 6x 1 5 is 24. El valor mínimo de y 5 x 2 2 6x 1 5 es 24. minor axis of an ellipse (p. 634) The line segment joining the co-vertices of an ellipse. See ellipse. eje menor de una elipse (pág. 634) El segmento de recta que une los puntos extremos de una elipse. Ver elipse. English-Spanish Glossary n2pe-9050.indd 1061 1061 10/14/05 10:05:39 AM mode (p. 744) The mode of n numbers is the number or numbers that occur most frequently. See measure of central tendency. moda (pág. 744) La moda de n números es el número o números que ocurren más veces. Ver medida de tendencia central. monomial (p. 252) An expression that is either a number, a variable, or the product of a number and one or more variables with whole number exponents. 1 6, 0.2x, } ab, and –5.7n 4 are monomials. monomio (pág. 252) Expresión que es un número, una variable o el producto de un número y una o más variables con exponentes enteros. 1 6, 0.2x, } ab y –5.7n 4 son monomios. mutually exclusive events (p. 707) See disjoint events. See disjoint events. sucesos mutuamente excluyentes (pág. 707) Ver sucesos disjuntos. Ver sucesos disjuntos. 2 2 N natural base e (p. 492) An irrational number defi ned as See natural logarithm. 1 n follows: As n approaches 1`, 1 1 1 } n 2 approaches e ≈ 2.718281828. base natural e (pág. 492) Número irracional defi nido de Ver logaritmo natural. 1 n esta manera: Al aproximarse n a 1`, 1 1 1 } n 2 se aproxima a ENGLISH-SPANISH GLOSSARY e ≈ 2.718281828. natural logarithm (p. 500) A logarithm with base e. It can be denoted loge, but is more often denoted by ln. ln 0.3 ≈ 21.204 because e21.204 ≈ (2.7183) 21.204 ≈ 0.3. logaritmo natural (pág. 500) Logaritmo con base e. Puede denotarse loge, pero es más frecuente que se denote ln. ln 0.3 ≈ 21.204 ya que e21.204 ≈ (2.7183) 21.204 ≈ 0.3. negative correlation (p. 113) The paired data (x, y) have a negative correlation if y tends to decrease as x increases. y correlación negativa (pág. 113) Los pares de datos (x, y) presentan una correlación negativa si y tiende a disminuir al aumentar x. x normal curve (p. 757) A smooth, symmetrical, bell-shaped curve that can model normal distributions and approximate some binomial distributions. See normal distribution. curva normal (pág. 757) Curva lisa, simétrica y con forma de campana que puede representar distribuciones normales y aproximar a algunas distribuciones binomiales. Ver distribución normal. 1062 Student Resources n2pe-9050.indd 1062 10/14/05 10:05:40 AM normal distribution (p. 757) A probability distribution with mean } x and standard deviation σ modeled by a bell-shaped curve with the area properties shown at the right. 95% x1 σ 2 x1 σ 3σ x x1 3 x2 σ 2σ x2 σ 99.7% x2 distribución normal (pág. 757) Una distribución de probabilidad con media } x y desviación normal σ representada por una curva en forma de campana y que tiene las propiedades vistas a la derecha. 68% 3} nth root of a (p. 414) For an integer n greater } than 1, if n bn 5 a, then b is an nth root of a. Written as Ïa. Ï2216 5 26 because (26) 3 5 2216. raíz enésima de a (pág. 414) Para un número entero n mayor que 1,}si bn 5 a, entonces b es una raíz enésima de a. n Se escribe Ïa . Ï2216 5 26 ya que (26) 3 5 2216. numerical expression (p. 10) An expression that consists of numbers, operations, and grouping symbols. 24(23)2 2 6(23) 1 11 is a numerical expression. expresión numérica (pág. 10) Expresión formada por números, operaciones y signos de agrupación. 24(23)2 2 6(23) 1 11 es una expresión numérica. 3} O See linear programming. función objetivo (pág. 174) En la programación lineal, la función lineal que se maximiza o minimiza. Ver programación lineal. odds against (p. 699) When all outcomes are equally likely, The odds against rolling a 4 using a Odds against Number of outcomes not in A 5 }}} . event A Number of outcomes in A 5 standard six-sided die are } , or 5 : 1, because probabilidad en contra (pág. 699) Cuando todos los casos son igualmente posibles, La probabilidad en contra de sacar el 4 al Número de casos no del A Probabilidad en contra 5 }}} . del suceso A Número de casos del A 1 5 outcomes correspond to not rolling a 4 and only 1 outcome corresponds to rolling a 4. 5 lanzar un dado normal de seis caras es } ,ó 1 5 : 1, ya que 5 casos corresponden a un número que no sea el 4 y sólo 1 caso corresponde al 4. odds in favor (p. 699) When all outcomes are equally likely, The odds in favor of rolling a 4 using a Number of outcomes in A Odds in favor 5 }}} . of event A Number of outcomes not in A 1 standard six-sided die are } , or 1 : 5, because probabilidad a favor (pág. 699) Cuando todos los casos son igualmente posibles, La probabilidad a favor de sacar el 4 al lanzar Número de casos del A Probabilidad a favor 5 }}} . del suceso A Número de casos no del A 5 only 1 outcome corresponds to rolling a 4 and 5 outcomes correspond to not rolling a 4. 1 un dado normal de seis caras es } , ó 1 : 5, ya 5 que sólo 1 caso corresponde al 4 y 5 casos corresponden a un número que no sea el 4. English-Spanish Glossary n2pe-9050.indd 1063 ENGLISH-SPANISH GLOSSARY objective function (p. 174) In linear programming, the linear function that is maximized or minimized. 1063 10/14/05 10:05:42 AM opposite (p. 4) The opposite, or additive inverse, of any number b is 2b. 6.2 and 26.2 are opposites. opuesto (pág. 4) El opuesto, o inverso aditivo, de cualquier número b es 2b. 6.2 y 26.2 son opuestos. ordered triple (p. 178) A set of three numbers of the form (x, y, z) that represents a point in space. The ordered triple (2, 1, 23) is a solution of the equation 4x 1 2y 1 3z 5 1. terna ordenada (pág. 178) Un conjunto de tres números de la forma (x, y, z) que representa un punto en el espacio. La terna ordenada (2, 1, 23) es una solución de la ecuación 4x 1 2y 1 3z 5 1. outlier (p. 746) A value that is much greater than or much less than most of the other values in a data set. 3 is an outlier in the data set 3, 11, 12, 13, 13, 14, 15, 15, 15, 15, 17. valor extremo (pág. 746) Valor que es mucho mayor o mucho menor que la mayoría de los otros valores de un conjunto de datos. 3 es un valor extremo del conjunto de datos 3, 11, 12, 13, 13, 14, 15, 15, 15, 15, 17. P parabola (pp. 236, 620) The set of all points equidistant from a point called the focus and a line called the directrix. The graph of a quadratic function y 5 ax2 1 bx 1 c is a parabola. ENGLISH-SPANISH GLOSSARY parábola (págs. 236, 620) El conjunto de todos los puntos equidistantes de un punto, llamado foco, y de una recta, llamada directriz. La gráfica de una función cuadrática y 5 ax2 1 bx 1 c es una parábola. axis of symmetry eje de simetría focus foco vertex vértice directrix directriz parallel lines (p. 84) Two lines in the same plane that do not intersect. y rectas paralelas (pág. 84) Dos rectas del mismo plano que no se cortan. x parent function (p. 89) The most basic function in a family of functions. The parent function for the family of all linear functions is y 5 x. función básica (pág. 89) La función más fundamental de una familia de funciones. La función básica de la familia de todas las funciones lineales es y 5 x. partial sum (p. 820) The sum Sn of the fi rst n terms of an infi nite series. suma parcial (pág. 820) La suma Sn de los n primeros términos de una serie infi nita. 1 2 1 4 1 8 1 16 1 32 }1}1}1}1}1... The series above has the partial sums S1 5 0.5, S2 5 0.75, S3 ≈ 0.88, S4 ≈ 0.94, . . . . La serie de arriba tiene las sumas parciales S1 5 0.5, S2 5 0.75, S3 ≈ 0.88, S4 ≈ 0.94, . . . . 1064 Student Resources n2pe-9050.indd 1064 10/14/05 10:05:43 AM Pascal’s triangle (p. 692) An arrangement of the values of C in a triangular pattern in which each row corresponds to n r a value of n. C 0 0 C C 1 0 C triángulo de Pascal (pág. 692) Disposición de los valores de C en un patrón triangular en el que cada fi la corresponde a n r un valor de n. C 2 0 C C C C C 5 2 C período (pág. 908) La longitud horizontal de cada ciclo de una función periódica. Ver función periódica. C 4 3 5 3 See periodic function. función periódica (pág. 908) Función cuya gráfica tiene un patrón que se repite. 3 3 C 4 2 period (p. 908) The horizontal length of each cycle of a periodic function. periodic function (p. 908) A function whose graph has a repeating pattern. C 3 2 C 4 1 5 1 2 2 C 3 1 C 5 0 C 2 1 C 3 0 4 0 1 1 4 4 C 5 4 C 5 5 y 2 1 π 4 π π 2 3π 2 x 2π period: π período: π The graph shows 3 cycles of y 5 tan x, a periodic function with a period of p. permutation (p. 684) An ordering of objects. The number of permutations of r objects taken from a group of n distinct n! objects is denoted nPr where nPr 5 } . There are 6 permutations of the n 5 3 letters A, B, and C taken r 5 3 at a time: ABC, ACB, BAC, BCA, CAB, and CBA. (n 2 r)! permutación (pág. 684) Ordenación de objetos. El número de permutaciones de r objetos tomados de un grupo de n! n objetos diferenciados se indica nPr , donde nPr 5 } . Hay 6 permutaciones de las letras n 5 3 A, B y C tomadas r 5 3 cada vez: ABC, ACB, BAC, BCA, CAB y CBA. (n 2 r)! perpendicular lines (p. 84) Two lines in the same plane that intersect to form a right angle. y rectas perpendiculares (pág. 84) Dos rectas del mismo plano que al cortarse forman un ángulo recto. x piecewise function (p. 130) A function defi ned by at least two equations, each of which applies to a different part of the function’s domain. función definida a trozos (pág. 130) Función defi nida por al menos dos ecuaciones, cada una de las cuales se aplica a una parte diferente del dominio de la función. g(x) 5 5 3x 2 1, if x , 1 0, if x 5 1 2x 1 4, if x . 1 g(x) 5 5 3x 2 1, si x , 1 0, si x 5 1 2x 1 4, si x . 1 English-Spanish Glossary n2pe-9050.indd 1065 ENGLISH-SPANISH GLOSSARY La gráfica muestra 3 ciclos de y 5 tan x, función periódica con período p. 1065 10/14/05 10:05:44 AM point-slope form (p. 98) An equation of a line written in the form y – y1 5 m(x – x1) where the line passes through the point (x1, y1) and has a slope of m. The equation y 1 2 5 –4(x – 5) is in pointslope form. forma punto-pendiente (pág. 98) Ecuación de una recta escrita en la forma y – y1 5 m(x – x1), donde la recta pasa por el punto (x1, y1) y tiene pendiente m. La ecuación y 1 2 5 –4(x – 5) está en la forma punto-pendiente. polynomial (p. 337) A monomial or a sum of monomials, each of which is called a term of the polynomial. See also monomial. 1 2 214, x 4 2 } x 1 3, and 7b 2 Ï3 1 pb2 polinomio (pág. 337) Monomio o suma de monomios, cada uno de los cuales se llama término del polinomio. Ver también monomio. 1 2 x 1 3 y 7b 2 Ï3 1 pb2 son 214, x 4 2 } are polynomials. } 4 polinomios. polynomial function (p. 337) A function of the form f(x) 5 anxn 1 a n 2 1xn 2 1 1 . . . 1 a1x 1 a 0 where an Þ 0, the exponents are all whole numbers, and the coefficients are all real numbers. f(x) 5 11x 5 2 0.4x 2 1 16x 2 7 is a polynomial function. The degree of f(x) is 5, the leading coefficient is 11, and the constant term is 27. función polinómica (pág. 337) Función de la forma f(x) 5 anxn 1 a n 2 1xn 2 1 1 . . . 1 a1x 1 a 0 donde an Þ 0, los exponentes son todos números enteros y los coeficientes son todos números reales. f(x) 5 11x 5 2 0.4x 2 1 16x 2 7 es una función polinómica. El grado de f(x) es 5, el coeficiente inicial es 11 y el término constante es 27. polynomial long division (p. 362) A method used to divide polynomials similar to the way you divide numbers. ENGLISH-SPANISH GLOSSARY } 4 división desarrollada polinómica (pág. 362) Método utilizado para dividir polinomios semejante a la manera en que divides números. x 2 1 7x 1 7 x 2 2 q x3 1 5x2 2 7x 1 2 x 3 2 2x 2 7x2 2 7x 7x 2 2 14x 7x 1 2 7x 2 14 16 }} x3 1 5x2 2 7x 1 2 x22 16 x22 2 }} 5 x 1 7x 1 7 1 } population (p. 766) A group of people or objects that you want information about. A sportswriter randomly selects 5% of college baseball coaches for a survey. The population is all college baseball coaches. The 5% of coaches selected is the sample. población (pág. 766) Grupo de personas u objetos acerca del cual deseas informarte. Un periodista deportiva selecciona al azar al 5% de los entrenadores universitarios de béisbol para que participe en una encuesta. La población son todos los entrenadores universitarios de béisbol. El 5% de los entrenadores que resultó seleccionado es la muestra. 1066 Student Resources n2pe-9050.indd 1066 10/14/05 10:05:45 AM positive correlation (p. 113) The paired data (x, y) have a positive correlation if y tends to increase as x increases. y correlacion positiva (pág. 113) Los pares de datos (x, y) presentan una correlación positiva si y tiende a aumentar al aumentar x. x power (p. 10) An expression that represents repeated multiplication of the same factor. 32 is the fi fth power of 2 because 32 5 2 p 2 p 2 p 2 p 2 5 25. potencia (pág. 10) Expresión que representa la multiplicación repetida del mismo factor. 32 es la quinta potencia de 2 ya que 32 5 2 p 2 p 2 p 2 p 2 5 25. power function (p. 428) A function of the form y 5 axb, where a is a real number and b is a rational number. f(x) 5 4x 3/2 is a power function. función potencial (pág. 428) Función de la forma y 5 axb, donde a es un número real y b es un número racional. f(x) 5 4x 3/2 es una función potencial. probability distribution (p. 724) A function that gives the probability of each possible value of a random variable. The sum of all the probabilities in a probability distribution must equal 1. Let the random variable X represent the number showing after rolling a standard sixsided die. distribución de probabilidades (pág. 724) Función que indica la probabilidad de cada valor posible de una variable aleatoria. La suma de todas las probabilidades de una distribución de probabilidades debe ser igual a 1. Sea la variable aleatoria X el número que salga al lanzar un dado normal de seis caras. Probability Distribution for Rolling a Die Distribución de probabilidad al lanzar un dado 1 2 3 4 5 6 P(X) 1 } 6 1 } 6 1 } 6 1 } 6 1 } 6 } 1 6 probability of an event (p. 698) A number from 0 to 1 that indicates the likelihood that the event will occur. See experimental probability, geometric probability, and theoretical probability. probabilidad de un suceso (pág. 698) Número entre 0 y 1 que indica la probabilidad de que ocurra el suceso. Ver probabilidad experimental, probabilidad geométrica y probabilidad teórica. pure imaginary number (p. 276) A complex number a 1 bi where a 5 0 and b Þ 0. 24i and 1.2i are pure imaginary numbers. número imaginario puro (pág. 276) Número complejo a 1 bi, donde a 5 0 y b Þ 0. 24i y 1.2i son números imaginarios puros. ENGLISH-SPANISH GLOSSARY X Q quadrantal angle (p. 867) An angle in standard position whose terminal side lies on an axis. ángulo cuadrantal (pág. 867) Ángulo en posición normal cuyo lado terminal se encuentra en un eje. y u x English-Spanish Glossary n2pe-9050.indd 1067 1067 10/14/05 10:05:47 AM quadratic equation in one variable (p. 253) An equation that can be written in the form ax2 1 bx 1 c 5 0 where a Þ 0. The equation x 2 2 5x 5 36 is a quadratic equation in one variable because it can be written in the form x 2 – 5x – 36 5 0. ecuación cuadrática con una variable (pág. 253) Ecuación que puede escribirse en la forma ax2 1 bx 1 c 5 0, donde a Þ 0. La ecuación x 2 2 5x 5 36 es una ecuación cuadrática con una variable ya que puede escribirse en la forma x 2 – 5x – 36 5 0. quadratic form (p. 355) The form au2 1 bu 1 c, where u is any expression in x. The expression 16x 4 2 8x 2 2 8 is in quadratic form because it can be written as u22 2u 2 8 where u 5 4x 2 . forma cuadrática (pág. 355) La forma au2 1 bu 1 c, donde u es cualquier expresión en x. La expresión 16x 4 2 8x 2 2 8 está en la forma cuadrática ya que puede escribirse u22 2u 2 8, donde u 5 4x 2 . } 2b 6 Ï b2 2 4ac quadratic formula (p. 292) The formula x 5 }} 2a used to fi nd the solutions of the quadratic equation ax2 1 bx 1 c 5 0 when a, b, and c are real numbers and a Þ 0. fórmula cuadrática (pág. 292) La fórmula } 2 x 5 2b 6 Ïb 2 4ac que se usa para hallar las soluciones de }} 2a 2 ENGLISH-SPANISH GLOSSARY la ecuación cuadrática ax 1 bx 1 c 5 0 cuando a, b y c son números reales y a Þ 0. To solve 3x 2 1 6x 1 2 5 0, substitute 3 for a, 6 for b, and 2 for c in the quadratic formula. Para resolver 3x 2 1 6x 1 2 5 0, sustituye a por 3, b por 6 y c por 2 en la fórmula cuadrática. }} 26 6 62 2 4(3)(2) 2(3) } Ï 23 6 Ï 3 5} x 5 }} 3 quadratic function (p. 236) A function that can be written in the form y 5 ax2 1 bx 1 c where a Þ 0. The functions y 5 3x 2 2 5 and y 5 x 2 2 4x 1 6 are quadratic functions. función cuadrática (pág. 236) Función que puede escribirse en la forma y 5 ax2 1 bx 1 c, donde a Þ 0. Las funciones y 5 3x 2 2 5 e y 5 x 2 2 4x 1 6 son funciones cuadráticas. quadratic inequality in one variable (p. 302) An inequality that can be written in the form ax2 1 bx 1 c < 0, ax2 1 bx 1 c ≤ 0, ax2 1 bx 1 c > 0, or ax2 1 bx 1 c ≥ 0. x 2 1 x ≤ 0 and 2x 2 1 x 2 4 > 0 are quadratic inequalities in one variable. desigualdad cuadrática con una variable (pág. 302) Desigualdad que se puede escribir en la forma ax2 1 bx 1 c < 0, ax2 1 bx 1 c ≤ 0, ax2 1 bx 1 c > 0 ó ax2 1 bx 1 c ≥ 0. x 2 1 x ≤ 0 y 2x 2 1 x 2 4 > 0 son desigualdades cuadráticas con una variable. quadratic inequality in two variables (p. 300) An inequality that can be written in the form y < ax2 1 bx 1 c, y ≤ ax2 1 bx 1 c, y > ax2 1 bx 1 c, or y ≥ ax2 1 bx 1 c. y > x 2 1 3x 2 4 is a quadratic inequality in two variables. desigualdad cuadrática con dos variables (pág. 300) Desigualdad que se puede escribir en la forma y < ax2 1 bx 1 c, y ≤ ax2 1 bx 1 c, y > ax2 1 bx 1 c ó y ≥ ax2 1 bx 1 c. y > x 2 1 3x 2 4 es una desigualdad cuadrática con dos variables. 1068 Student Resources n2pe-9050.indd 1068 10/14/05 10:05:48 AM quadratic system (p. 658) A system of equations that includes one or more equations of conics. sistema cuadrático (pág. 658) Sistema de ecuaciones que incluye una o más ecuaciones de cónicas. y 2 – 7x 1 3 5 0 2x – y 5 3 x 2 1 4y 2 1 8y 5 16 2x 2 – y 2 – 6x – 4 5 0 The systems above are quadratic systems. Los sistemas de arriba son sistemas cuadráticos. R radian (p. 860) In a circle with radius r and center at the origin, one radian is the measure of an angle in standard position whose terminal side intercepts an arc of length r. y r radián (pág. 860) En un círculo con radio r y cuyo centro está en el origen, un radián es la medida de un ángulo en posición normal cuyo lado terminal intercepta un arco de longitud r. } 1 radian 1 radián x n} radical (p. 266) An expression of the form Ïs or Ïs where s is a number or an expression. } r } 3} Ï 5 , Ï 2x 1 1 n} radical (pág. 266) Expresión de la forma Ï s o Ï s, donde s es un número o una expresión. radical equation (p. 452) An equation with one or more radicals that have variables in their radicands. 3} ecuación radical (pág. 452) Ecuación con uno o más radicales en cuyo radicando aparecen variables. radical function (p. 446) A function that contains a radical with a variable in its radicand. } 3} 1 Ïx, g(x) 5 23Ïx 1 5 f(x) 5 } 2 función radical (pág. 446) Función que tiene un radical con una variable en su radicando. } radicand (p. 266) The number or expression beneath a radical sign. The radicand of Ï5 is 5, and the radicand of radicando (pág. 266) El número o la expresión que aparece bajo el signo radical. El radicando de Ï5 es 5, y el radicando de } Ï8y2 is 8y2. } } Ï8y2 es 8y2. radius of a circle (p. 626) The distance from the center of a circle to a point on the circle. Also, a line segment that connects the center of a circle to a point on the circle. See also circle. The circle with equation (x – 3)2 1 } (y 1 5)2 5 36 has radius Ï36 5 6. See also circle. radio de un círculo (pág. 626) La distancia desde el centro de un círculo hasta un punto del círculo. También, es un segmento de recta que une el centro de un círculo con un punto del círculo. Ver también círculo. El círculo con la ecuación (x – 3)2 1 } (y 1 5)2 5 36 tiene el radio Ï 36 5 6. Ver también círculo. English-Spanish Glossary n2pe-9050.indd 1069 ENGLISH-SPANISH GLOSSARY Ï 2x 1 7 5 3 1069 10/14/05 10:05:50 AM random variable (p. 724) A variable whose value is determined by the outcomes of a random event. The random variable X representing the number showing after rolling a six-sided die has possible values of 1, 2, 3, 4, 5, and 6. variable aleatoria (pág. 724) Variable cuyo valor viene determinado por los resultados de un suceso aleatorio. La variable aleatoria X que representa el número que sale al lanzar un dado de seis caras tiene como valores posibles 1, 2, 3, 4, 5 y 6. range of a relation (p. 72) The set of output values of a relation. See relation. rango de una relación (pág. 72) El conjunto de los valores de salida de una relación. Ver relación. range of data values (p. 745) A measure of dispersion equal to the difference between the greatest and least data values. rango de valores de datos (pág. 745) Medida de dispersión igual a la diferencia entre el valor máximo y el valor mínimo de los datos. rate of change (p. 85) A comparison of how much one quantity changes, on average, relative to the change in another quantity. 14, 17, 18, 19, 20, 24, 24, 30, 32 The range of the data set above is 32 – 14 5 18. El rango del conjunto de datos de arriba es 32 – 14 5 18. The temperature rises from 75°F at 8 A .M. to 91°F at 12 P.M. The average rate of change in 918F 2 758F 5 16°F 5 4°/h. temperature is } } 12 P.M. 2 8 A.M. ENGLISH-SPANISH GLOSSARY relación de cambio (pág. 85) Comparación entre el cambio producido, por término medio, en una cantidad y el cambio producido en otra cantidad. 4h La temperatura sube de 75°F a las 8 de la mañana a 91°F a las 12 del mediodía. La relación de cambio media en la temperatura 918F 2 758F 5 16°F 5 4°/h. es } } 12 P.M. 2 8 A.M. 4h rational function (p. 558) A function of the form 6 and y 5 2x 1 1 are The functions y 5 } } p(x) f(x) 5 }, where p(x) and q(x) are polynomials and q(x) Þ 0. q(x) rational functions. función racional (pág. 558) Función de la forma 6 e y 5 2x 1 1 son Las funciones y 5 } } p(x) q(x) x x x23 x23 f(x) 5 }, donde p(x) y q(x) son polinomios y q(x) Þ 0. funciones racionales. rationalizing the denominator (p. 267) The process of eliminating a radical expression in the denominator of a fraction by multiplying both the numerator and denominator by an appropriate radical expression. Ï5 To rationalize the denominator of } }, racionalizar el denominador (pág. 267) El proceso de eliminar una expresión radical del denominador de una fracción al multiplicar tanto el numerador como el denominador por una expresión radical adecuada. Ï5 Para racionalizar el denominador de } }, reciprocal (p. 4) The reciprocal, or multiplicative inverse, of 1 5 2} are reciprocals. 22 and } 2 } Ï2 multiply the numerator and denominator } by Ï2 . } 1 any nonzero number b is } . b recíproco (pág. 4) El recíproco, o inverso multiplicativo, de 1 cualquier número b distinto de cero es } . Ï2 multiplica el numerador y el denominador } por Ï2 . 1 22 1 1 5 2} son recíprocos. 22 y } 2 22 b 1070 Student Resources n2pe-9050.indd 1070 10/14/05 10:05:51 AM recursive rule (p. 827) A rule for a sequence that gives the beginning term or terms of the sequence and then a recursive equation that tells how the nth term an is related to one or more preceding terms. The recursive rule a 0 5 1, an 5 an 2 1 1 4 gives the arithmetic sequence 1, 5, 9, 13, … . regla recursiva (pág. 827) Regla de una progresión que da el primer término o términos de la progresión y luego una ecuación recursiva que indica qué relación hay entre el término enésimo an y uno o más de los términos precedentes. La regla recursiva a 0 5 1, an 5 an 2 1 1 4 da la progresión aritmética 1, 5, 9, 13, … . reference angle (p. 868) If θ is an angle in standard position, its reference angle is the acute angle θ´ formed by the terminal side of θ and the x-axis. y u u' ángulo de referencia (pág. 868) Si θ es un ángulo en posición normal, su ángulo de referencia es el ángulo agudo θ´ formado por el lado terminal de θ y el eje de x. x The acute angle θ´ is the reference angle for angle θ. El ángulo agudo θ´ es el ángulo de referencia para el ángulo θ. reflection (p. 124) A transformation that fl ips a graph or figure in a line. (2, 3) (5, 3) f (x) 1 x 1 g(x) (2, 23) (5, 23) The graph of g(x) is the reflection of the graph of f(x) in the x-axis. La gráfica de g(x) es la reflexión de la gráfica de f(x) en el eje de x. relation (p. 72) A mapping, or pairing, of input values with output values. The ordered pairs (22, 22), (22, 2), (0, 1), and (3, 1) represent the relation with inputs (domain) of –2, 0, and 3 and outputs (range) of –2, 1, and 2. relación (pág. 72) Correspondencia entre los valores de entrada y los valores de salida. Los pares ordenados (22, 22), (22, 2), (0, 1) y (3, 1) representan la relación con entradas (dominio) de –2, 0 y 3 y salidas (rango) de –2, 1 y 2. English-Spanish Glossary n2pe-9050.indd 1071 ENGLISH-SPANISH GLOSSARY reflexión (pág. 124) Transformación que vuelca una gráfica o una figura en una recta. y 1071 10/14/05 10:05:53 AM repeated solution (p. 379) For the polynomial equation f(x) 5 0, k is a repeated solution if and only if the factor x – k has an exponent greater than 1 when f(x) is factored completely. 21 is a repeated solution of the equation (x 1 1)2 (x 2 2) 5 0. solución repetida (pág. 379) Para la ecuación polinómica f(x) 5 0, k es una solución repetida si y sólo si el factor x – k tiene un exponente mayor que 1 cuando f(x) está completamente factorizado. 21 es una solución repetida de la ecuación (x 1 1)2 (x 2 2) 5 0. root of an equation (p. 253) The solutions of a quadratic equation are its roots. The roots of the quadratic equation x 2 2 5x 2 36 5 0 are 9 and 24. raíz de una ecuación (pág. 253) Las soluciones de una ecuación cuadrática son sus raíces. Las raíces de la ecuación cuadrática x 2 2 5x 2 36 5 0 son 9 y 24. sample (p. 766) A subset of a population. See population. muestra (pág. 766) Subconjunto de una población. Ver población. scalar (p. 188) A real number by which you multiply a matrix. See scalar multiplication. escalar (pág. 188) Número real por el que se multiplica una matriz. Ver multiplicación escalar. scalar multiplication (p. 188) Multiplication of each element of a matrix by a real number, called a scalar. F GF 28 2 4 21 0 22 1 0 5 22 24 214 2 7 multiplicación escalar (pág. 188) Multiplicación de cada elemento de una matriz por un número real llamado escalar. scatter plot (p. 113) A graph of a set of data pairs (x, y) used to determine whether there is a relationship between the variables x and y. diagrama de dispersión (pág. 113) Gráfica de un conjunto de pares de datos (x, y) que sirve para determinar si hay una relación entre las variables x e y. Test scores Resultados de las pruebas ENGLISH-SPANISH GLOSSARY S G y 90 70 50 0 0 2 4 6 8 Hours of studying Horas de estudio scientific notation (p. 331) The representation of a number in the form c 3 10n where 1 ≤ c < 10 and n is an integer. 0.693 is written in scientific notation as 6.93 3 1021. notación científica (pág. 331) La representación de un número de la forma c 3 10n, donde 1 ≤ c < 10 y n es un número entero. 0.693 escrito en notación científica es 6.93 3 1021. x 1072 Student Resources n2pe-9050.indd 1072 10/14/05 10:05:54 AM secant function (p. 852) If θ is an acute angle of a right triangle, the secant of θ is the length of the hypotenuse divided by the length of the side adjacent to θ. See sine function. función secante (pág. 852) Si θ es un ángulo agudo de un triángulo rectángulo, la secante de θ es la longitud de la hipotenusa dividida por la longitud del lado adyacente a θ. Ver función seno. sector (p. 861) A region of a circle that is bounded by two radii and an arc of the circle. The central angle θ of a sector is the angle formed by the two radii. sector (pág. 861) Región de un círculo delimitada por dos radios y un arco del círculo. El ángulo central θ de un sector es el ángulo formado por dos radios. sector sector r arc length s longitud de un arco s central angle u ángulo central u For the domain n 5 1, 2, 3, and 4, the sequence defi ned by an 5 2n has the terms 2, 4, 6, and 8. progresión (pág. 794) Función cuyo dominio es un conjunto de números enteros consecutivos. El dominio da la posición relativa de cada término de la secuencia. El rango da los términos de la secuencia. Para el dominio n 5 1, 2, 3 y 4, la secuencia defi nida por an 5 2n tiene los términos 2, 4, 6 y 8. series (p. 796) The expression formed by adding the terms of a sequence. A series can be fi nite or infi nite. Finite series: 2 1 4 1 6 1 8 Infinite series: 2 1 4 1 6 1 8 1 . . . serie (pág. 796) La expresión formada al sumar los términos de una progresión. La serie puede ser fi nita o infi nita. Serie fi nita: 2 1 4 1 6 1 8 Serie infinita: 2 1 4 1 6 1 8 1 . . . set (p. 715) A collection of distinct objects. If A is the set of positive integers less than 5, then A 5 {1, 2, 3, 4}. conjunto (pág. 715) Colección de objetos diferenciados. Si A es el conjunto de números enteros positivos menores que 5, entonces A 5 {1, 2, 3, 4}. sigma notation (p. 796) See summation notation. See summation notation. notación sigma (pág. 796) Ver notación de sumatoria. Ver notación de sumatoria. simplest form of a radical (p. 422) A radical with index n is in simplest form if the radicand has no perfect nth powers as factors and any denominator has been rationalized. Ï 135 in simplest form is 3Ï5 . forma más simple de un radical (pág. 422) Un radical con índice n está escrito en la forma más simple si el radicando no tiene como factor ninguna potencia enésima perfecta y el denominador ha sido racionalizado. 3} 5} Ï7 Ï8 3} 5} Ï28 2 in simplest form is }. } 5} 3} 3} Ï 135 en la forma más simple es 3Ï5 . 5} Ï7 Ï8 5} Ï28 2 en la forma más simple es }. } 5} English-Spanish Glossary n2pe-9050.indd 1073 ENGLISH-SPANISH GLOSSARY sequence (p. 794) A function whose domain is a set of consecutive integers. The domain gives the relative position of each term of the sequence. The range gives the terms of the sequence. 1073 10/14/05 10:05:55 AM simplified form of a rational expression (p. 573) A rational expression in which the numerator and denominator have no common factors other than 61. x2 2 2x 2 15 x 29 (x 1 3)(x 2 5) (x 1 3)(x 2 3) ↑ Simplified form Forma simplificada forma simplificada de una expresión racional (pág. 573) Expresión racional en la que el numerador y el denominador no tienen factores comunes además de 61. sine function (p. 852) If θ is an acute angle of a right triangle, the sine of θ is the length of the side opposite θ divided by the length of the hypotenuse. función seno (pág. 852) Si θ es un ángulo agudo de un triángulo rectángulo, el seno de θ es la longitud del lado opuesto a θ dividida por la longitud de la hipotenusa. 13 5 u 12 hyp opp 13 hyp adj 12 cos θ 5 } 5 } 13 hyp opp 5 tan θ 5 } 5 } 12 adj 5 sin θ 5 } 5 } 13 csc θ 5 } opp 5 } 5 hyp 13 sec θ 5 } 5 } 12 adj adj 12 cot θ 5 } opp 5 } 5 op 13 hip ady 12 cos θ 5 } 5 } 13 hip op 5 tan θ 5 } 5 } 12 ady hip 5 sen θ 5 } 5 } sinusoids (p. 941) Graphs of sine and cosine functions. y sinusoides (pág. 941) Gráficas de funciones seno y coseno. 13 cosec θ 5 } op 5 } 5 hip 13 sec θ 5 } 5 } 12 ady ady 12 cot θ 5 } op 5 } 5 y 5 2 sin 4x 1 3 y 5 2 sen 4x 1 3 1 π 4 skewed distribution (p. 727) A probability distribution that is not symmetric. See also symmetric distribution. distribución asimétrica (pág. 727) Distribución de probabilidades que no es simétrica. Ver también distribución simétrica. Probability Probabilidad ENGLISH-SPANISH GLOSSARY x25 x23 5}5} } 2 π x 2 0.40 0.20 0 0 1 2 3 4 5 6 7 8 Number of successes Número de éxitos 1074 Student Resources n2pe-9050.indd 1074 10/14/05 10:05:56 AM slope (p. 82) The ratio of vertical change (the rise) to horizontal change (the run) for a nonvertical line. For a nonvertical line passing through the points (x1, y1) and y 2y 2 1 (x2, y 2), the slope is m 5 } x 2x . 2 1 pendiente (pág. 82) Para una recta no vertical, la razón entre el cambio vertical (distancia vertical) y el cambio horizontal (distancia horizontal). Para una recta no vertical que pasa por los puntos (x1, y1) y (x2, y 2), la pendiente es The slope of the line that passes through the points (23, 0) and (3, 4) is: La pendiente de la recta que pasa por los puntos (23, 0) y (3, 4) es: y 2y 420 2 1 4 2 m5} x 2 x 5 }5 } 5 } 2 3 2 (23) 1 6 3 y 2y 2 1 m5} x 2x . 2 1 2 slope-intercept form (p. 90) A linear equation written in the form y 5 mx 1 b where m is the slope and b is the y-intercept of the equation’s graph. The equation y 5 2}x 2 1 is in slope3 intercept form. forma pendiente-intercepto (pág. 90) Ecuación lineal escrita en la forma y 5 mx 1 b, donde m es la pendiente y b es el intercepto en y de la gráfica de la ecuación. La ecuación y 5 2}x 2 1 está en la forma 3 pendiente-intercepto. solution of a linear inequality in two variables (p. 132) An ordered pair (x, y) that produces a true statement when the values of x and y are substituted into the inequality. The ordered pair (1, 2) is a solution of 3x 1 4y > 8 because 3(1) 1 4(2) 5 11, and 11 > 8. solución de una desigualdad lineal con dos variables El par ordenado (1, 2) es una solución de 3x 1 4y > 8 ya que 3(1) 1 4(2) 5 11, y 11 > 8. (pág. 132) Par ordenado (x, y) que produce una expresión 2 verdadera cuando x e y se sustituyen por sus valores en la desigualdad. solución de un sistema de ecuaciones lineales en tres variables (pág. 178) Terna ordenada (x, y, z) cuyas coordenadas hacen que cada ecuación del sistema sea verdadera. solution of a system of linear equations in two variables (p. 153) An ordered pair (x, y) that satisfies each equation of the system. solución de un sistema de ecuaciones lineales en dos variables (pág. 153) Par ordenado (x, y) que satisface cada ecuación del sistema. solution of a system of linear inequalities in two variables (p. 168) An ordered pair (x, y) that is a solution of each inequality in the system. solución de un sistema de desigualdades lineales en dos variables (pág. 168) Par ordenado (x, y) que es una solución de cada desigualdad del sistema. 4x 1 2y 1 3z 5 1 2x 2 3y 1 5z 5 214 6x 2 y 1 4z 5 21 (2, 1, 23) is the solution of the system above. (2, 1, 23) es la solución del sistema de arriba. 4x 1 y 5 8 2x 2 3y 5 18 (3, 24) is the solution of the system above. (3, 24) es la solución del sistema de arriba. y > 22x 2 5 y≤x13 (–1, 1) is a solution of the system above. (–1, 1) es una solución del sistema de arriba. English-Spanish Glossary n2pe-9050.indd 1075 ENGLISH-SPANISH GLOSSARY solution of a system of linear equations in three variables (p. 178) An ordered triple (x, y, z) whose coordinates make each equation in the system true. 1075 10/14/05 10:05:57 AM ENGLISH-SPANISH GLOSSARY solution of an equation in one variable (p. 18) A number that produces a true statement when substituted for the variable in the equation. 4 x 1 8 5 20 The solution of the equation } 5 is 15. solución de una ecuación con una variable (pág. 18) Número que produce una expresión verdadera al sustituir la variable por él en la ecuación. 4 x 1 8 5 20 es 15. La solución de la ecuación } solution of an equation in two variables (p. 74) An ordered pair (x, y) that produces a true statement when the values of x and y are substituted in the equation. (–2, 3) is a solution of y 5 –2x – 1. solución de una ecuación con dos variables (pág. 74) Par ordenado (x, y) que produce una expresión verdadera al sustituir x e y por sus valores en la ecuación. (–2, 3) es una solución de y 5 –2x – 1. solution of an inequality in one variable (p. 41) A number that produces a true statement when substituted for the variable in the inequality. 21 is a solution of the inequality 5x 1 2 > 7x 2 4. solución de una desigualdad con una variable (pág. 41) Número que produce una expresión verdadera al sustituir la variable por él en la desigualdad. 21 es una solución de la desigualdad 5x 1 2 > 7x 2 4. solve for a variable (p. 26) Rewrite an equation as an equivalent equation in which the variable is on one side and does not appear on the other side. When you solve the circumference formula resolver para una variable (pág. 26) Escribir una ecuación como ecuación equivalente que tenga la variable en uno de sus miembros pero no en el otro. Al resolver para r la fórmula de circunferencia C 5 2p r, el resultado es 5 C C 5 2p r for r, the result is r 5 } . 2p C r5} . 2p square root (p. 266) If b2 5 a, then b is a square root of a. The } radical symbol Ï represents a nonnegative square root. The square roots of 9 are 3 and 23 because } 32 5 9 and (23)2 5 9. So, Ï9 5 3 and } 2Ï9 5 23. raíz cuadrada (pág. 266) Si b2 5 a, entonces b es una } raíz cuadrada de a. El signo radical Ï representa una raíz cuadrada no negativa. Las raíces cuadradas de 9 son 3 y 23 ya que } 32 5 9 y (23)2 5 9. Así pues, Ï9 5 3 y } 2Ï9 5 23. standard deviation (p. 745) The typical difference (or deviation) between a data value and the mean. The standard deviation s of a numerical data set x1, x2, . . . , xn is given by the following formula: }}}} s5 Ï (x1 2 } x)2 1 (x2 2 } x)2 1 . . . 1 (xn 2 } x)2 }} n desviación típica (pág. 745) La diferencia (o desviación) más común entre un valor de los datos y la media. La desviación típica s de un conjunto de datos numéricos x1, x2, . . . , xn viene dada por la siguiente fórmula: 14, 17, 18, 19, 20, 24, 24, 30, 32 Because the mean of the data set is 22, the standard deviation is: Como la media del conjunto de datos es 22, la desviación típica es: Î }}}} s5 Î (14 2 22)2 1 (17 2 22)2 1 . . . 1 (32 2 22)2 9 }} } 290 < 5.7 5 } 9 }}} s5 Ï (x1 2 } x)2 1 (x2 2 } x)2 1 . . . 1 (xn 2 } x)2 }}} n 1076 Student Resources n2pe-9050.indd 1076 10/14/05 10:05:58 AM standard form of a complex number (p. 276) The form a 1 bi where a and b are real numbers and i is the imaginary unit. The standard form of the complex number i(1 1 i) is 21 1 i. forma general de un número complejo (pág. 276) La forma a 1 bi, donde a y b son números reales e i es la unidad imaginaria. La forma general del número complejo i(1 1 i) es 21 1 i. standard form of a linear equation (p. 91) A linear equation written in the form Ax 1 By 5 C where A and B are not both zero. The linear equation y 5 –3x 1 4 can be written in standard form as 3x 1 y 5 4. forma general de una ecuación lineal (pág. 91) Ecuación lineal escrita en la forma Ax 1 By 5 C, donde A y B no son ambos cero. La ecuación lineal y 5 –3x 1 4 escrita en la forma general es 3x 1 y 5 4. standard form of a polynomial function (p. 337) The form of a polynomial function that has terms written in descending order of exponents from left to right. The function g(x) 5 7x 2 Ï 3 1 p x 2 can be written in standard form as } g(x) 5 p x 2 1 7x 2 Ï3 . forma general de una función polinómica (pág. 337) La forma de una función polinómica en la que los términos se ordenan de tal modo que los exponentes disminuyen de izquierda a derecha. La función g(x) 5 7x 2 Ï3 1 p x 2 escrita en } la forma general es g(x) 5 p x 2 1 7x 2 Ï3 . standard form of a quadratic equation in one variable (p. 253) The form ax 2 1 bx 1 c 5 0 where a Þ 0. The quadratic equation x 2 2 5x 5 36 can be written in standard form as x 2 2 5x 2 36 5 0. forma general de una ecuación cuadrática con una variable (pág. 253) La forma ax2 1 bx 1 c 5 0, donde a Þ 0. La ecuación cuadrática x 2 2 5x 5 36 escrita en la forma general es x 2 2 5x 2 36 5 0. standard form of a quadratic function (p. 236) The form y 5 ax2 1 bx 1 c where a Þ 0. The quadratic function y 5 2(x 1 3)(x 2 1) can be written in standard form as y 5 2x 2 1 4x 2 6. forma general de una función cuadrática (pág. 236) La forma y 5 ax2 1 bx 1 c, donde a Þ 0. La función cuadrática y 5 2(x 1 3)(x 2 1) escrita en la forma general es y 5 2x 2 1 4x 2 6. } } 2 z5 3 2 z5 2 2 1 z5 0 z5 1 z5 2 z5 3 z5 distribución normal típica (pág. 758) La distribución normal con media 0 y desviación típica 1. Ver también puntuación z. ENGLISH-SPANISH GLOSSARY standard normal distribution (p. 758) The normal distribution with mean 0 and standard deviation 1. See also z-score. English-Spanish Glossary n2pe-9050.indd 1077 1077 10/14/05 10:06:00 AM standard position of an angle (p. 859) In a coordinate plane, the position of an angle whose vertex is at the origin and whose initial side lies on the positive x-axis. 908 y terminal side lado terminal 08 posición normal de un ángulo (pág. 859) En un plano de coordenadas, la posición de un ángulo cuyo vértice está en el origen y cuyo lado inicial se sitúa en el eje de x positivo. 1808 vertex vértice x initial side lado inicial 3608 2708 statistics (p. 744) Numerical values used to summarize and compare sets of data. See mean, median, mode, range, and standard deviation. estadística (pág. 744) Valores numéricos utilizados para resumir y comparar conjuntos de datos. Ver media, mediana, moda, rango y desviación típica. step function (p. 131) A piecewise function defi ned by a constant value over each part of its domain. Its graph resembles a series of stair steps. 5 función escalonada (pág. 131) Función defi nida a trozos y por un valor constante en cada parte de su dominio. Su gráfica parece un grupo de escalones. ENGLISH-SPANISH GLOSSARY 1, if 0 ≤ x , 1 f(x) 5 2, if 1 ≤ x , 2 3, if 2 ≤ x , 3 5 1, si 0 ≤ x , 1 f(x) 5 2, si 1 ≤ x , 2 3, si 2 ≤ x , 3 subset (p. 716) If every element of a set A is also an element of a set B, then A is a subset of B. This is written as A ⊆ B. For any set A, ∅ ⊆ A and A ⊆ A. If A 5 {1, 2, 4, 8} and B is the set of all positive integers, then A is a subset of B, or A ⊆ B. subconjunto (pág. 716) Si cada elemento de un conjunto A es también un elemento de un conjunto B, entonces A es un subconjunto de B. Esto se escribe A ⊆ B. Para cualquier conjunto A, ∅ ⊆ A y A ⊆ A. Si A 5 {1, 2, 4, 8} y B es el conjunto de todos los números enteros positivos, entonces A es un subconjunto de B, o A ⊆ B. substitution method (p. 160) A method of solving a system of equations by solving one of the equations for one of the variables and then substituting the resulting expression in the other equation(s). método de sustitución (pág. 160) Método para resolver un sistema de ecuaciones mediante la resolución de una de las ecuaciones para una de las variables seguida de la sustitución de la expresión resultante en la(s) otra(s) ecuación (ecuaciones). summation notation (p. 796) Notation for a series that uses the uppercase Greek letter sigma, o. Also called sigma notation. notación de sumatoria (pág. 796) Notación de una serie que usa la letra griega mayúscula sigma, o. También se llama notación sigma. 2x 1 5y 5 25 x 1 3y 5 3 Solve equation 2 for x: x 5 23y 1 3. Substitute the expression for x in equation 1 and solve for y: y 5 11. Use the value of y to fi nd the value of x: x 5 230. Resuelve la ecuación 2 para x: x 5 23y 1 3. Sustituye la expresión para x en la ecuación 1 y resuelve para y: y 5 11. Usa el valor de y para hallar el valor de x: x 5 230. 5 ∑ 7i 5 7(1) 1 7(2) 1 7(3) 1 7(4) 1 7(5) i51 5 7 1 14 1 21 1 28 1 35 1078 Student Resources n2pe-9050.indd 1078 10/14/05 10:06:01 AM distribución simétrica (pág. 727) Distribución de probabilidad representada por un histograma en la que se puede trazar una recta vertical que divida al histograma en dos partes; éstas son imágenes especulares entre sí. 0.30 Probability Probabilidad symmetric distribution (p. 727) A probability distribution, represented by a histogram, in which you can draw a vertical line that divides the histogram into two parts that are mirror images. 0.20 0.10 0 synthetic division (p. 363) A method used to divide a polynomial by a divisor of the form x – k. 23 0 1 2 3 4 5 6 7 8 Number of successes Número de éxitos 2 división sintética (pág. 363) Método utilizado para dividir un polinomio por un divisor en la forma x – k. 2 1 28 5 26 15 221 25 7 216 2x 3 1 x 2 2 8x 1 5 5 2x 2 2 5x 1 7 2 16 } } x13 x13 synthetic substitution (p. 338) A method used to evaluate a polynomial function. sustitución sintética (pág. 338) Método utilizado para evaluar una función polinómica. 3 2 2 25 0 24 8 6 3 9 15 1 3 5 23 The synthetic substitution above indicates that for f(x) 5 2x 4 2 5x 3 2 4x 1 8, f(3) 5 23. La sustitución sintética de arriba indica que para f(x) 5 2x 4 2 5x 3 2 4x 1 8, f(3) 5 23. sistema de desigualdades lineales con dos variables (pág. 168) Sistema que consiste de dos o más desigualdades lineales con dos variables. Ver también desigualdad lineal con dos variables. system of three linear equations in three variables (p. 178) A system consisting of three linear equations in three variables. See also linear equation in three variables. sistema de tres ecuaciones lineales en tres variables (pág. 178) Sistema formado por tres ecuaciones lineales con tres variables. Ver también ecuación lineal con tres variables. x1y≤8 4x 2 y > 6 2x 1 y 2 z 5 5 3x 2 2y 1 z 5 16 4x 1 3y 2 5z 5 3 English-Spanish Glossary ENGLISH-SPANISH GLOSSARY system of linear inequalities in two variables (p. 168) A system consisting of two or more linear inequalities in two variables. See also linear inequality in two variables. 1079 system of two linear equations in two variables (p. 153) A system consisting of two equations that can be written in the form Ax 1 By 5 C and Dx 1 Ey 5 F where x and y are variables, A and B are not both zero, and D and E are not both zero. sistema de dos ecuaciones lineales con dos variables (pág. 153) Un sistema que consiste en dos ecuaciones que se pueden escribir de la forma Ax 1 By 5 C y Dx 1 Ey 5 F, donde x e y son variables, A y B no son ambos cero, y D y E tampoco son ambos cero. 4x 1 y 5 8 2x 2 3y 5 18 ENGLISH-SPANISH GLOSSARY T tangent function (p. 852) If θ is an acute angle of a right triangle, the tangent of θ is the length of the side opposite θ divided by the length of the side adjacent to θ. See sine function. función tangente (pág. 852) Si θ es un ángulo agudo de un triángulo rectángulo, la tangente de θ es la longitud del lado opuesto a θ dividida por la longitud del lado adyacente a θ. Ver función seno. terminal side of an angle (p. 859) In a coordinate plane, an angle can be formed by fi xing one ray, called the initial side, and rotating the other ray, called the terminal side, about the vertex. See standard position of an angle. lado terminal de un ángulo (pág. 859) En un plano de coordenadas, un ángulo puede formarse al fijar un rayo, llamado lado inicial, y al girar el otro rayo, llamado lado terminal, en torno al vértice. Ver posición normal de un ángulo. terms of a sequence (p. 794) The values in the range of a sequence. The fi rst 4 terms of the sequence 1, 23, 9, 227, 81, 2243, . . . are 1, 23, 9, and 227. términos de una progresión (pág. 794) Los valores del rango de una progresión. Los 4 primeros términos de la progresión 1, 23, 9, 227, 81, 2243, . . . son 1, 23, 9 y 227. terms of an expression (p. 12) The parts of an expression that are added together. The terms of the algebraic expression 3x 2 1 5x 1 (27) are 3x 2, 5x, and 27. términos de una expresión (pág. 12) Las partes de una expresión que se suman. Los términos de la expresión algebraica 3x 2 1 5x 1 (27) son 3x 2, 5x y 27. 1080 Student Resources n2pe-9050.indd 1080 10/14/05 10:06:04 AM theoretical probability (p. 698) When all outcomes are equally likely, the theoretical probability that an event A will The theoretical probability of rolling an even number using a standard six-sided die Number of outcomes in event A occur is P(A) 5 }}} . 3 5 1 because 3 outcomes correspond is } } Total number of outcomes 6 2 to rolling an even number out of 6 total outcomes . probabilidad teórica (pág. 698) Cuando todos los casos son igualmente posibles, la probabilidad teórica de que ocurra La probabilidad teórica de sacar un número par al lanzar un dado normal de seis caras Número de casos del suceso A . un suceso A es P(A) 5 }}} 3 5 1 ya que 3 casos corresponden a un es } } Número total de casos 6 2 número par del total de 6 casos. transformation (p. 123) A transformation changes a graph’s size, shape, position, or orientation. Translations, vertical stretches and shrinks, reflections, and rotations are transformations. transformación (pág. 123) Una transformación cambia el tamaño, la forma, la posición o la orientación de una gráfica. Las traslaciones, las expansiones y contracciones verticales, las reflexiones y las rotaciones son transformaciones. translation (p. 123) A transformation that shifts a graph horizontally and/or vertically, but does not change its size, shape, or orientation. traslación (pág. 123) Transformación que desplaza una gráfica horizontal o verticalmente, o de ambas maneras, pero que no cambia su tamaño, forma u orientación. y y5zx14z22 y 5 zx z 24 21 22 x 24 La gráfica de y 5x 1 4 2 2 es la gráfica de y 5x al trasladar ésta 2 unidades hacia abajo y 4 unidades hacia la izquierda. transverse axis of a hyperbola (p. 642) The line segment joining the vertices of a hyperbola. See hyperbola. eje transverso de una hipérbola (pág. 642) El segmento de recta que une los vértices de una hipérbola. Ver hipérbola. trigonometric identity (p. 924) A trigonometric equation that is true for all domain values. sin (– θ) 5 –sin θ sin2 θ 1 cos2 θ 5 1 identidad trigonométrica (pág. 924) Ecuación trigonométrica que es verdadera para todos los valores del dominio. sen (– θ) 5 –sen θ sen2 θ 1 cos2 θ 5 1 trinomial (p. 252) The sum of three monomials. 4x 2 1 3x 2 1 is a trinomial. trinomio (pág. 252) La suma de tres monomios. 4x 2 1 3x 2 1 es un trinomio. English-Spanish Glossary n2pe-9050.indd 1081 ENGLISH-SPANISH GLOSSARY The graph of y 5x 1 4 2 2 is the graph of y 5x translated down 2 units and left 4 units. 1081 10/14/05 10:06:05 AM U unbiased sample (p. 767) A sample that is representative of the population you want information about. You want to poll members of the senior class about where to hold the prom. If every senior has an equal chance of being polled, then the sample is unbiased. muestra no sesgada (pág. 767) Muestra que es representativa de la población acerca de la cual deseas informarte. Quieres encuestar a algunos estudiantes de último curso sobre el lugar donde organizar el baile de fi n de año. Si cada estudiante de último curso tiene iguales posibilidades de ser encuestado, entonces es una muestra no sesgada. union of sets (p. 715) The union of two sets A and B, written A ∪ B, is the set of all elements in either A or B. If A 5 {1, 2, 4, 8} and B 5 {2, 4, 6, 8, 10}, then A ∪ B 5 {1, 2, 4, 6, 8, 10}. unión de conjuntos (pág. 715) La unión de dos conjuntos A y B, escrita A ∪ B, es el conjunto de todos los elementos que están en A o B. Si A 5 {1, 2, 4, 8} y B 5 {2, 4, 6, 8, 10}, entonces A ∪ B 5 {1, 2, 4, 6, 8, 10}. unit circle (p. 867) The circle x2 1 y 2 5 1, which has center (0, 0) and radius 1. For an angle θ in standard position, the terminal side of θ intersects the unit circle at the point (cos θ, sin θ ). ENGLISH-SPANISH GLOSSARY círculo unidad (pág. 867) El círculo x2 1 y 2 5 1, que tiene centro (0, 0) y radio 1. Para un ángulo θ en posición normal, el lado terminal de θ corta al círculo unidad en el punto (cos θ, sen θ ). y u x (cos u, sin u ) (cos u, sen u) r51 universal set (p. 715) The set of all elements under consideration; denoted U. See complement of a set. conjunto universal (pág. 715) El conjunto de todos los elementos tenidos en cuenta; se indica U. Ver complemento de un conjunto. V variable (p. 11) A letter that is used to represent one or more numbers. In the expressions 6x, 3x 2 1 1, and 12 2 5x, the letter x is the variable. variable (pág. 11) Letra utilizada para representar uno o más números. En las expresiones 6x, 3x 2 1 1 y 12 2 5x, la letra x es la variable. variable term (p. 12) A term that has a variable part. The variable terms of the algebraic expression 3x 2 1 5x 1 (27) are 3x 2 and 5x. término algebraico (pág. 12) Término que tiene variable. Los términos algebraicos de la expresión algebraica 3x 2 1 5x 1 (27) son 3x 2 y 5x. verbal model (p. 34) A word equation that represents a reallife problem. Distance (miles) 5 Rate p Time (miles/hour) (hours) modelo verbal (pág. 34) Ecuación expresada mediante palabras que representa un problema de la vida real. Distancia (millas) 5 Velocidad p Tiempo (millas/hora) (horas) 1082 Student Resources n2pe-9050.indd 1082 10/14/05 10:06:06 AM 1 vertex form of a quadratic function (p. 245) The form y 5 a(x 2 h)2 1 k, where the vertex of the graph is (h, k) and the axis of symmetry is x 5 h. The quadratic function y 5 2} (x 1 2)2 1 5 4 is in vertex form. forma de vértice de una función cuadrática (pág. 245) La forma y 5 a(x 2 h)2 1 k, donde el vértice de la gráfica es (h, k) y el eje de simetría es x 5 h. (x 1 2)2 1 5 La función cuadrática y 5 2} 4 está en la forma de vértice. vertex of a parabola (pp. 236, 620) The point on a parabola that lies on the axis of symmetry. See parabola. vértice de una parábola (págs. 236, 620) El punto de una parábola que se encuentra en el eje de simetría. Ver parábola. 1 vertex of an absolute value graph (p. 123) The highest or lowest point on the graph of an absolute value function. y vértice de una gráfica de valor absoluto (pág. 123) El punto más alto o más bajo de la gráfica de una función de valor absoluto. (4, 3) 1 x 1 The vertex of the graph of y 5 x 2 4 1 3 is the point (4, 3). El vértice de la gráfica de y 5 x 2 4 1 3 es el punto (4, 3). See hyperbola. vértices de una hipérbola (pág. 642) Los puntos de intersección de una hipérbola y la recta que pasa por los focos de la hipérbola. Ver hipérbola. vertices of an ellipse (p. 634) The points of intersection of an ellipse and the line through the foci of the ellipse. See ellipse. vértices de una elipse (pág. 634) Los puntos de intersección de una elipse y la recta que pasa por los focos de la elipse. Ver elipse. ENGLISH-SPANISH GLOSSARY vertices of a hyperbola (p. 642) The points of intersection of a hyperbola and the line through the foci of the hyperbola. X x-intercept (p. 91) The x-coordinate of a point where a graph intersects the x-axis. intercepto en x (pág. 91) La coordenada x de un punto donde una gráfica corta al eje de x. y (0, 3) x 1 2y 5 6 1 (6, 0) x 1 The x-intercept is 6. El intercepto en x es 6. English-Spanish Glossary n2pe-9050.indd 1083 1083 10/14/05 10:06:08 AM Y y-intercept (p. 89) The y-coordinate of a point where a graph intersects the y-axis. y (0, 3) x 1 2y 5 6 intercepto en y (pág. 89) La coordenada y de un punto donde una gráfica corta al eje de y. 1 (6, 0) x 1 The y-intercept is 3. El intercepto en y es 3. Z zero of a function (p. 254) A number k is a zero of a function f if f(k) 5 0. The zeros of the function f(x) 5 2(x 1 3)(x 2 1) are 23 and 1. cero de una función (pág. 254) Un número k es un cero de una función f si f(k) 5 0. Los ceros de la función f(x) 5 2(x 1 3)(x 2 1) son 23 y 1. z-score (p. 758) The number z of standard deviations that a data value lies above or below the mean of the data set: A normal distribution has a mean of 76 and a standard deviation of 9. The z-score for x2x z5} . 64 2 76 x 5 64 is z 5 } 5 } ≈ 21.3. puntuación z (pág. 758) El número z de desviaciones típicas que un valor se encuentra por encima o por debajo Una distribución normal tiene una media de 76 y una desviación típica de 9. La puntuación z para x 5 64 es _ σ _ x2x de la media del conjunto de datos: z 5 } . σ ENGLISH-SPANISH GLOSSARY _ x 2x σ 9 _ x 2 x 5 64 2 76 ≈ 21.3. z5} } σ 9 1084 Student Resources n2pe-9050.indd 1084 10/14/05 10:06:09 AM Index A Alternative method, See Another Way; Problem Solving Workshop Ambiguous case, 883–884 Amplitude, 908, 909 Analyze, exercises, 8, 185, 696, 712, 748, 749 And rule, 1000–1001 Angle(s) Brewster’s, 930 central, 861 complementary, 994 coterminal, 860, 863 degree measure of, 859–864 of depression, 855 of elevation, 855 initial side of, 859 quadrantal, 867 radian measure of, 860–864 reference, 868, 871 of repose, 879 special, 963 in standard position, 859 supplementary, 994 terminal side of, 859 Angle bisector, 994 Animated Algebra, Throughout. See for example 1, 71, 151, 235, 329, 413, 477, 549, 613, 681, 743, 793, 851, 907 Another Way, 18, 48–49, 91, 92, 105, 179, 189, 218–219, 272–273, 284, 293, 360–361, 396, 460–461, 493, 523–525, 575, 596–597, 640, 660, 714, 720, 781, 834–835, 867, 889, 895, 938–939, 950 Applications advertising, 170, 181, 220 apparel, 410, 461, 681, 720 archaeology, 105, 358, 470, 619, 890 archery, 743, 748 art, 200, 264, 358, 373, 400, 404, 676, 729, 808 astronomy, 8, 270, 332, 369, 426, 459, 477, 519, 524, 525, 535, 632, 638, 639, 641, 648, 654, 666, 673, 677, 800, 864, 888, 904 aviation, 34, 215, 398, 426, 570, 587, 631, 639, 667, 790, 879, 894, 961 baseball, 54, 57, 69, 78, 173, 233, 287, 315, 327, 345, 442, 475, 571, 595, 663, 766, 790, 846, 847, 862, 896 Index n2pe-9060.indd 1085 INDEX Absolute value, 50, 51 of a complex number, 279, 280 equations, 50–58, 60, 64 functions, 121–129, 144 graphing, 50, 51 inequalities, 50–58, 60, 64, 135, 136 in a system, 169–173 ACT, See Standardized Test Preparation Activities, See also Geometry software; Graphing calculator absolute value equations and inequalities, 50 collect and model trigonometric data, 948 completing the square, 283 end behavior of polynomial functions, 336 exploring inverse functions, 437 exploring recursive rules, 826 exploring transformations, 121–122 fitting a line to data, 112 fitting a model to data, 774 graphing linear equations in three variables, 177 infinite geometric series, 819 intersections of planes and cones, 649 inverse trigonometric functions, 874 inverse variation, 550 using the Location Principle, 378 modeling data with an exponential function, 528 probability using Venn diagrams, 706 solve linear systems using tables, 152 trigonometric identities, 923 Addition of complex numbers, 276–277, 279 with fractions, 979 of functions, 428–435 integer, 975 matrix, 187, 189–192, 194 as opposite of subtraction, 4 of polynomials, 346–352, 403 properties, 3, 18 for matrices, 188 of rational expressions, 582–588, 602, 605 Addition property of equality, 18 Additive inverse, 4 of a complex number, 280 Algebra, formulas and theorems from, 1027–1028 Algebra tiles to model binomial products, 985 to model completing the square, 283 Algorithm for adding or subtracting a rational expression, 582, 583 for evaluating a trigonometric function, 868 for graphing an absolute value function, 124, 125 for graphing an equation of a circle, 626 translated, 650 for graphing an equation of an ellipse, 635 translated, 652 for graphing an equation of a hyperbola, 643 translated, 651 for graphing an equation of a parabola, 621 translated, 651 for graphing an equation in slopeintercept form, 90 for graphing an equation in standard form, 91 for graphing an equation in two variables, 74 for graphing a horizontal translation, 916 for graphing a linear inequality, 133 for graphing a quadratic function in intercept form, 247 for graphing a quadratic inequality in two variables, 300 for graphing a rational function, 558 for graphing a system of linear inequalities, 168 for graphing a vertical translation, 915 for order of operations, 10 for solving an absolute value equation, 52 for solving a radical equation, 452 for translating a trigonometric graph, 915 for writing an exponential function, 529 for writing a power function, 531 1085 9/28/05 9:28:03 AM INDEX basketball, 57, 74, 215, 241, 368, 458, 606, 679, 691, 721, 729, 740, 770, 846, 962, 973 bicycling, 78, 351, 472, 601, 683, 756, 879, 940, 946 biology, 44, 88, 91, 108, 186, 250, 298, 319, 335, 344, 369, 400, 416, 421, 426, 429, 433, 458, 485, 491, 494, 497, 505, 512, 532, 533, 534, 536, 541, 624, 711, 722, 759, 761, 782, 832 botany, 38, 46, 85, 762, 779, 782 bowling, 94, 418, 436, 469 business, 32, 103, 117, 128, 137, 165, 166, 174, 175, 176, 185, 186, 192, 194, 201, 202, 209, 213, 218, 219, 220, 262, 264, 274, 290, 348, 365, 367, 385, 398, 400, 405, 464, 475, 498, 563, 631, 679, 689, 732, 769, 779, 824, 849 camping, 94, 368 chemistry, 59, 69, 206, 209, 491, 497, 504, 506, 521, 588, 589, 601, 818, 832, 872, 904 computers and Internet, 15, 47, 65, 106, 139, 145, 274, 480, 484, 535, 547, 553, 562, 607, 657, 713, 741, 767, 769, 782, 787, 791, 841, 847 construction, 39, 145, 306, 323, 377, 392, 458, 557, 594, 754, 818, 896, 901, 902, 904 consumer economics, 16, 23, 33, 40, 49, 68, 103, 111, 155, 157, 185, 223, 230, 316, 431, 433, 473, 611, 679, 784 contests, 46, 269, 272, 273, 303, 696, 699, 942 crafts, 137, 148, 174, 176, 186, 216, 226, 261, 264, 274, 291, 326, 843, 905 design, 24, 31, 39, 258, 316, 326, 358, 360, 361, 369, 376, 377, 560, 570, 618, 667, 688, 849 diving, 38, 76, 110, 270, 597 earth science, 3, 6, 57, 945 education, 119, 201, 230, 351, 392, 485, 537, 546, 611, 688, 692, 695, 705, 708, 737, 754, 773, 775, 817 electricity, 144, 277, 281, 282, 556, 587, 749, 849, 945 employment, 7, 19, 23, 33, 49, 157, 172, 436, 735, 753, 755, 764, 770, 818, 864, 922 engineering, 129, 246, 250 entertainment, 59, 103, 120, 200, 208, 227, 351, 475, 579, 699, 808, 843 1086 Student Resources equipment, 58, 198, 484, 504, 552, 556, 572 exercising, 15, 439 fairs, 11, 42, 165, 176, 258, 809 finance, 15, 33, 62, 66, 144, 219, 227, 352, 386, 407, 418, 475, 481, 483, 484, 485, 488, 489, 490, 491, 495, 497, 498, 506, 521, 525, 526, 537, 543, 547, 570, 588, 679, 817, 832, 838, 847, 849 fitness, 94, 105, 157, 184, 705 food, 7, 33, 62, 137, 149, 176, 184, 193, 216, 251, 351, 361, 403, 521, 547, 572, 697, 705, 709, 712, 730, 735, 765, 873, 940 football, 59, 106, 247, 250, 298, 314, 441, 485, 638, 723, 741, 748, 787, 791, 871, 962, 970 fundraising, 81, 104, 129, 139, 162, 166, 475, 600, 717, 849 games, 128, 172, 231, 290, 316, 326, 592, 596, 687, 696, 701, 702, 703, 704, 732, 738, 746, 800, 865 gardening, 67, 103, 139, 208, 230, 258, 264, 274, 522, 543, 631, 676, 678, 872, 887, 895 geography, 204, 208, 209, 320, 704, 858, 872, 951, 961 geology, 8, 334, 335, 419, 505, 521, 543, 616, 619, 631, 663 golf, 149, 756, 893, 961 government, 79, 106, 146, 147, 149, 352, 407, 410, 703, 723, 737, 740, 769, 771, 772 gymnastics, 54, 55, 57, 233 history, 139, 148, 158, 233, 359, 385, 392, 400, 407, 410, 598, 600, 611, 679, 779, 846 hockey, 33, 198, 231 law enforcement, 157 manufacturing, 57, 176, 189, 225, 358, 367, 368, 369, 376, 433, 475, 567, 572, 574, 601, 732, 761, 780 measurement, 31, 33, 58, 59, 68, 81, 104, 257, 265, 343, 383, 389, 391, 400, 408, 411, 436, 444, 450, 451, 569, 601, 611, 647, 664, 679, 855, 857, 901, 927 medicine, 490, 498, 639, 723, 729 movies, 15, 29, 59, 134, 148, 159, 271, 343, 398, 399, 433, 547, 696, 705, 790, 813 music, 101, 224, 242, 316, 367, 537, 552, 556, 594, 688, 718, 767, 808, 829, 844, 864, 913, 948, 963, 972 nutrition, 216 oceanography, 932, 933, 938, 939 Olympics, 119, 201, 684, 705, 748, 749, 752, 780, 864 parabolic reflectors, 622–625, 641 pets, 219, 713, 749 photography, 13, 106, 242, 426, 512, 585, 644, 683, 688, 735, 755, 880, 896, 954 physics, 119, 125, 143, 272, 273, 306, 311, 340, 418, 444, 447, 450, 451, 453, 457, 458, 464, 469, 475, 497, 509, 511, 512, 513, 519, 522, 524, 535, 547, 585, 654, 673, 864, 913, 929, 953, 957, 972 physiology, 57, 78, 173, 306, 385, 444, 536, 540, 572, 765, 782, 790, 832, 910, 921 population, 79, 106, 119, 385, 410, 484, 546, 842 racing, 38, 451, 463, 718 recreation, 8, 35, 36, 46, 68, 79, 95, 137, 138, 166, 186, 239, 299, 556, 562, 656, 669, 679, 737, 815, 855, 870, 871, 873, 894, 916, 921, 922, 940 repairs, 23, 65 running, 39, 314, 579 safety, 105, 662, 720 skateboarding, 82, 290, 343, 564 soccer, 68, 307, 322, 676, 726, 815, 904, 957, 961 softball, 200, 290 space travel, 243, 396, 570, 656, 751, 752 structures, 87, 88, 104, 242, 246, 250, 257, 258, 289, 299, 311, 314, 315, 356, 358, 359, 386, 419, 498, 631, 716, 808, 847, 851, 864, 873, 887, 888, 918, 921, 953 swimming, 46, 158, 243, 392 telephone, 94, 113, 137, 298, 369, 497, 595, 600, 628, 630, 954 television, 138, 184, 323, 407, 600, 607, 731, 768, 770, 788 temperature, 32, 44, 57, 59, 69, 444, 521, 535, 562, 570, 754, 756, 781, 943, 946, 947, 963, 972, 1005 tennis, 325, 335, 679, 723, 904 track and field, 32, 184, 418, 689, 695, 756, 864 travel, 5, 7, 15, 33, 62, 63, 81, 88, 158, 159, 243, 601, 641, 703, 857 vehicles, 7, 36, 38, 76, 95, 115, 128, 166, 192, 227, 233, 298, 307, 410, 444, 445, 469, 472, 530, 537, 563, 572, 618, 619, 641, Associative property, 3 for matrix operations, 188, 197 Asymptote for exponential decay functions, 486, 487 for exponential growth functions, 478, 479 horizontal, 558 of a hyperbola, 642 for logarithmic functions, 502 for rational functions, 558, 565 vertical, 558 @Home Tutor, Throughout. See for example xxiv, 8, 15, 17, 23, 25, 31, 38, 46, 57, 61, 63, 70, 78, 94, 97, 103, 110, 122, 141, 143 Augmented matrices, 218–219 Average, See Mean Avoid Errors, 12, 20, 28, 43, 52, 73, 82, 108, 115, 126, 133, 153, 163, 187, 205, 238, 247, 252, 260, 267, 277, 287, 292, 330, 338, 347, 354, 355, 362, 370, 371, 415, 416, 423, 430, 441, 480, 488, 507, 553, 573, 584, 599, 644, 654, 658, 683, 691, 693, 700, 708, 719, 726, 744, 795, 797, 803, 811, 821, 828, 862, 877, 890 Axis (Axes) coordinate, 987 of an ellipse, 634 of symmetry for a conic section, 652, 655 of symmetry for a parabola, 236, 620 B Bar graph, 1006–1007 Base of a logarithm, 499 of a power, 10 Best-fitting line, 112–120, 143 correlation coefficient, 114 linear regression and, 116 Best-fitting quadratic model, 311 Bias in sampling, 767, 769, 770–773 Biased question, 772–773 Biased sample, 767 Biconditional statement, 1002–1003 Big Ideas, 1, 60, 71, 140, 151, 221, 235, 317, 329, 401, 413, 465, 477, 538, 549, 602, 613, 668, 681, 733, 743, 783, 793, 839, 851, 964 Binomial(s), See also Polynomial(s), 252 cube of, 347 multiplying, 347–351, 985 square of, 347 Binomial distribution, 725–731, 733, 736 approximating, 763–765 calculating, 731 skewed, 727, 728 symmetric, 727, 728 Binomial expansion, 693–696 Pascal’s triangle and, 693, 695 Binomial experiment, 725 Binomial theorem, 693 using, 693–696, 735 Bisector angle, 994 perpendicular, 615–617 Bounded region, 174 Boundary line, for an inequality, 132 Box-and-whisker plot, 1008–1009 Branches, of a hyperbola, 558 Brewster’s angle, 930 C Calculator, See also Graphing calculator approximating roots, 415, 417 calculating compound interest, 481 entering negative numbers, 17 evaluating expressions, 17 evaluating inverse trigonometric functions, 876 evaluating logarithms, 500–501 evaluating permutations, 685 evaluating trigonometric functions, 854 simplifying natural base expressions, 492 Calculator Based Laboratory (CBL), 948 Center of a circle, 626, 992 of a conic section, 650–652 of a hyperbola, 642 of rotation, 988 Central angle, of a sector, 861 Central tendency, measures of, 744–750, 783, 784, 1005 Chain rule, 1000–1001 Challenge, exercises, Throughout. See for example 7, 9, 15, 16, 23, 24, 31, 32, 38, 39, 45, 47, 56, 58, 78, 79, 87, 88, 94, 96 Change-of-base formula, 508–509, 511 Chapter Review, 61–64, 141–144, 222–226, 318–322, 402–406, 466–468, 539–542, 603–606, 669–672, 734–736, 784–786, 840–842, 898–900, 965–968 Index INDEX 646, 660, 662, 667, 722, 777, 780, 857, 869, 877, 879, 898, 901, 904, 905, 921, 969 volleyball, 64, 290, 321, 327, 594 volunteering, 95, 696 weather, 105, 108, 110, 458, 460, 500, 513, 562, 636, 712, 719, 755, 785, 791, 936, 943, 953, 969 wildlife, 8, 9, 16, 44, 108, 109, 111, 172, 204, 242, 251, 273, 306, 331, 344, 385, 400, 429, 485, 630, 648, 664, 729, 922 winter sports, 343, 556, 682, 734 Approximation, See also Estimation; Prediction of the area of an ellipse, 640 of best-fitting line, 115–120, 146 of binomial distribution, 763–765 of correlation, 113, 114, 117 exercises, 335, 557, 904 of real zeros of a function, 382–383, 384 of roots, 415, 417 Arc length, of a sector, 861–865 Archimedes, 857 Area, See also Formulas, 991 using determinants to find, 204, 208, 209, 217 of an ellipse, 636, 638, 640 Heron’s area formula, 891 of a parallelogram, 991 of a rectangle, 991 of a sector, 861–865 of a trapezoid, 991 of a triangle, 885, 887, 888, 891, 991 Area model for completing the square, 283 for a quadratic equation, 254, 257, 258, 261 Arithmetic sequence, 802–809, 839, 841 recursive rules and, 827–833, 839, 842 Arithmetic series, 804–809, 839, 841 Assessment, See also Online Quiz; State Test Practice Chapter Test, 65, 145, 227, 323, 407, 469, 543, 607, 673, 737, 787, 843, 901, 969 Quiz, Throughout. See for example 40, 58, 96, 120, 138, 167, 193, 217, 265, 291, 315, 352, 377, 399 Standardized Test Practice, 68–69, 148–149, 230–231, 326–327, 410–411, 472–473, 546–547, 610–611, 676–677, 740–741, 790–791, 846–847, 904–905, 972–973 1087 INDEX Chapter Summary, 60, 140, 221, 317, 401, 465, 538, 602, 668, 733, 783, 839, 897, 964 Chapter Test, See Assessment Checking solutions using a calculator, 876 using end behavior, 393 by graphing, 255, 311, 440, 518, 931 using a graphing calculator, 161, 285, 292, 293, 462, 518, 591, 659, 958 using inverse operations, 362 using logical reasoning, 373, 763 using slope-intercept form, 154 using substitution, 18, 19, 20, 36, 52, 91, 133, 153, 160, 179, 205, 267, 285, 381, 452, 454, 455, 468, 517, 518, 591, 934 using unit analysis, 5, 7, 34 Choosing a method exercises, 94, 164, 183, 305, 357, 892, 893 for solving linear systems, 163 Choosing a model, for data, 774–781, 786 Circle, 626, 992 area of, 992 center of, 626, 992 central angle of, 861 circumference of, 992 degree measure of, 860 diameter of, 992 eccentricity of, 665–666 equation of, 626 graphing, 626–633, 668, 670 translated, 650, 652–657 writing, 627–632, 668, 670 finding the center given three points, 616, 619 inequalities and, 628, 630–632 radian measure of, 860 radius of, 626, 992 sector of, 861 translated, 650 unit, 867 Circle graph, 1006–1007 Circular function, 866 Circular model, 628, 630–632 Circular motion, modeling, 916, 921, 922 Circular permutation, 689 Circumference, of a circle, 992 Classifying conics, 653, 654, 656 functions, 75, 80–81, 337, 479, 487, 489 inverse and direct variation, 551 linear systems, 154–157 lines by slope, 83 numbers, 2 1088 Student Resources parallel and perpendicular lines, 84, 86 probability distributions, 727, 728 samples, 766, 769 series, 805 triangles using the distance formula, 614–615, 617 zeros of a polynomial function, 381–382, 384, 385 Closure property, 3 Coefficient leading, 337 of a power, 12 Coefficient matrix, 205–206 Cofunction identities, 924 Combination(s), 690–697, 733, 735 formula, 690 Pascal’s triangle and, 692, 695 probability and, 699, 702 Combinatorics, formulas from, 1028 Common difference, 802 Common factors, 978 Common logarithm, 500 change-of-base formula and, 508–509, 511 Common misconceptions, See Error Analysis Common multiple, 978 Common ratio, 810 Communication describing in words, 7, 13, 17, 22, 30, 38, 45, 50, 56, 68, 77, 86, 93, 94, 102, 117, 118, 119, 122, 127, 128, 136, 149, 156, 164, 171, 180, 186, 190, 199, 207, 209, 214, 216, 240, 256, 263, 270, 280, 281, 289, 296, 297, 304, 308, 313, 316, 334, 344, 349, 357, 366, 375, 390, 397, 418, 424, 436, 442, 450, 456, 483, 489, 496, 503, 505, 510, 511, 520, 534, 555, 557, 562, 568, 570, 578, 586, 623, 624, 629, 637, 645, 646, 655, 694, 702, 710, 722, 747, 748, 753, 754, 769, 770, 778, 806, 814, 823, 829, 831, 835, 856, 863, 878, 886, 893, 898, 905, 913, 919, 920, 928, 935, 942, 944, 952, 959, 960 reading math, 54, 83, 174, 277, 339, 830, 854, 861, 868 writing in math, 6, 13, 21, 25, 37, 44, 49, 55, 61, 76, 86, 93, 101, 109, 117, 127, 128, 135, 136, 156, 164, 171, 182, 190, 199, 207, 214, 240, 249, 255, 263, 269, 279, 288, 296, 304, 312, 318, 333, 341, 349, 356, 374, 383, 390, 392, 397, 402, 417, 424, 432, 442, 449, 456, 466, 482, 489, 510, 519, 525, 533, 539, 555, 561, 568, 577, 586, 592, 617, 624, 629, 654, 661, 686, 694, 710, 714, 721, 734, 747, 753, 760, 769, 778, 798, 806, 807, 814, 823, 840, 856, 862, 870, 878, 886, 892, 898, 912, 919, 927, 935, 939, 944, 945, 952, 959 Commutative property, 3 for matrix operations, 188, 196 Compare, exercises, 8, 39, 61, 135, 137, 208, 243, 251, 271, 289, 327, 458, 472, 528, 572, 580, 619, 623, 645, 649, 702, 714, 740, 749, 756, 787, 791, 800, 807, 818, 872, 896, 912, 948 Comparing graphs of absolute value functions, 121–125 graphs of quadratic functions, 236–237, 240 independent and dependent events, 719 standard and translated equations, 650, 651 types of variation, 554 Complement of an event, 709–713, 718 of a set, 715–716 Complementary angles, 994 Completing the square, 283–291, 317, 321, 653 using algebra tiles, 283 Complex conjugates, 278, 380 Complex conjugates theorem, 380 Complex fraction, 584 simplifying, 584–588 Complex number(s), 276 absolute value of, 279, 280 additive inverse of, 280 Julia set and, 282 Mandelbrot set and, 281 multiplicative inverse of, 280 operations with, 276–282, 317, 320–321 plotting, 278 standard form of, 276 Complex plane, 278 Julia set on, 282 Mandelbrot set on, 281 Composite number, 978 Composition, of a function, 430–435, 465, 467 Compound event, 707 probability of, 707–713 Compound inequality, 41–47 absolute value form of, 53 Compound interest, 481, 483–485 graphing, 874, 908–914, 964, 965 reflections, 917, 920 translations, 915–917, 919–922, 966 half-angle formula for, 955 using, 955–962 inverse of, 874–879, 897 sinusoids, and, 941–948 sum formula for, 949 using, 949–954 Cosines, law of, 889–895, 897, 900 Cotangent function, See also Trigonometric function(s) evaluating for any angle, 866–872 evaluating for right triangles, 852–858 Cotangent identities, 924, 966 Coterminal angle, 860, 863 Counterexample, 1003 Co-vertices, of an ellipse, 634 Cramer’s rule, 205–209, 221, 226 Critical x-values, 303, 599 Cross multiplication, to solve rational equations, 589–590 Cube root function, 447–451, 465, 468 parent, 446, 465 Cubes difference of, 354 sum of, 354 Cubic function, See also Polynomial function(s), 337 inverse of, 441 writing, 393–399 Cubic regression, 396 Cumulative Review, 232–233, 474–475, 678–679, 848–849 Cycle, of a function, 908 Cylinder surface area of, 63, 567, 572, 580, 993 volume of, 334, 350, 567, 572, 580, 993 D Data, See also Graphs; Modeling; Statistics analyzing using best-fitting line, 112–120 choosing a model for, 774–781, 786 finite differences, 393–399 fitting a model to, 774–781 geometric mean, 749 hypothesis testing, 764–765 margin of error, 768–771 measures of central tendency, 744–750, 783, 784, 1005 measures of dispersion, 744–750, 783, 784, 1005 negative correlation, 113, 114, 117 normal distribution, 757–762, 783, 785 outlier, 746, 747 positive correlation, 113, 114, 117 quartiles, 1008–1009 range, 745–750 standard deviation, 745–750 applying transformations to, 751–755 collecting, 112, 550 biased question, 772–773 biased sample, 767 control group, 773 convenience sample, 766 from an experiment, 308, 528, 772–773, 774 population, 766 random sample, 766 sampling, 766–771, 783, 786 self-selected sample, 766 using simulation, 714 from a survey, 763, 764, 766–771, 772–773, 786 systematic sample, 766 unbiased sample, 767 displaying in a bar graph, 1006–1007 in a circle graph, 1006–1007 in a line graph, 1006–1007 in a scatter plot, 113–120 organizing in a box-and-whisker plot, 1008–1009 in a histogram, 724, 726–731, 1008–1009 in a line plot, 1008–1009 using matrices, 189, 192 in a stem-and-leaf plot, 1008–1009 in a table, 112, 528 in a Venn diagram, 706 Decay factor, 486 Decay function exponential, 486–491, 538, 540 involving e, 493–498 Decimal exponents, 425 Decimals, fractions, percents, and, 976 Degree converting between radians and, 860–864, 899 measure of a circle, 860 of a polynomial function, 337, 339 Dependent events, 718–723 Dependent linear system, 154–157 Dependent variable, 74 Depression, angle of, 855 Index INDEX Compound statement, 1001 Concept Summary, 188, 197, 387, 861 Concepts, See Big Ideas; Concept Summary; Key Concept Condensing a logarithmic expression, 508, 510, 541 Conditional probability, 718–723 Cone, intersected by a plane, See Conics Congruent figures, 996–997 Conics, See also Circle; Ellipse; Hyperbola; Parabola, 649–657 classifying, 653, 656 degenerate, 657 discriminant of, 653, 656 eccentricity of, 665–666 lines of symmetry of, 652, 655 translated, equations of, 650–657, 672 Conjugates, 267 complex, 278 Connections, See Applications Consistent linear system, 154–157 Constant adding to data values, 751–755 common difference, 802 Constant ratio, 810 Constant term, 12 Constant of variation, 107, 551 Constraints, 174 Continuous function, 80–81 Continuously compounded interest, 494–495, 497 Control group, 773 Convenience sample, 766 Converse, of a conditional statement, 1002–1003 Coordinate geometry, formulas from, 1026 Coordinate plane, 987 Corollary to the fundamental theorem of algebra, 379 Correlation, describing, 113–114 Correlation coefficient, 114 Cosecant function, See also Trigonometric function(s) evaluating for any angle, 866–872 evaluating for right triangles, 852–858 Cosine function, See also Trigonometric equation(s); Trigonometric function(s) difference formula for, 949 using, 949–954 double-angle formula for, 955 using, 955–962 evaluating for any angle, 866–872 evaluating for right triangles, 852–858 1089 INDEX Derivation of Snell’s law, 930 of a trigonometric model, 957 Descartes, René, 381 Descartes’ Rule of Signs, 381 using, 381–382, 384, 385 Determinant, of a matrix, 203–209, 226 Diagram drawing, problem solving strategy, 35, 37, 39 interpreting, 324, 326, 608, 609, 610, 844, 846, 847, 937 mapping, 72, 73, 77 Pascal’s triangle, 692, 695 tree, 682, 686, 720, 978 Venn, 2, 430, 706–708, 715–716, 1004 Diameter, of a circle, 992 Difference of two cubes, 354 of two squares, 253 Difference formulas, 949, 964 using, 949–954, 968 Dilation, on the coordinate plane, 989 Dimensions, of a matrix, 187, 195 Direct argument, 1000–1001 Direct substitution, for evaluating polynomial functions, 338 Direct variation, 107–111, 140, 143 Directrix, of a parabola, 620 Discrete function, 80–81 Discrete mathematics counting methods, 682–689 discrete functions, 80–81 finite differences, 393–399 greatest common factor (GCF), 978–979 least common denominator (LCD), 979 least common multiple (LCM), 978–979 matrices, 187–219 mutually exclusive events, 707 Pascal’s triangle, 692, 695 scatter plots, 113–120, 143 sequences, 794–816 set theory, 715–716 tree diagram, 682, 686, 720, 978 triangular numbers, 394 triangular pyramidal numbers, 395 Discriminant, 294, 296 of a conic equation, 653, 656 Disjoint event, 707, 736 Dispersion, measures of, 744–750, 783, 784, 1005 Distance formula, 614, 619, 669 Distribution binomial, 763–765 1090 Student Resources normal, 757–762, 783, 785 standard normal, 758–762 Distributive property, 3 to add and subtract like radicals, 422 for matrix operations, 188, 197 for solving linear equations, 20, 22–24 Division of complex numbers, 278, 280 of functions, 429–435 inequalities and, 42–47 integer, 975 as opposite of multiplication, 4 polynomial, 362–368 properties, 18 with rational expressions, 576–580, 602, 605 synthetic, 363–368 Division property of equality, 18 Domain of a function, 73, 76, 391, 428–430, 446–447, 463, 479, 482, 485, 487, 489, 491 of a relation, 72 of a sequence, 794 Doppler effect, 563 Double-angle formulas, 955, 968 using, 955–962, 968 Draw angles in standard position, 859, 860, 863 Draw conclusions examples, 132 exercises, 50, 112, 122, 152, 177, 283, 308, 336, 437, 528, 550, 649, 688, 706, 774, 819, 826, 874, 881, 923, 948 from samples, 766–771 Draw a diagram exercises, 887, 895 problem solving strategy, 35, 37, 39 Draw a graph exercises, 57, 95, 104, 119, 129, 242, 290, 306, 314, 343, 344, 451, 631, 729, 816, 914, 929, 937 problem solving strategy, 49, 273 E Eccentricity of conic sections, 665–666 Efficiency, 574, 580 Element of a matrix, 187 of a set, 715–716 Elevation, angle of, 855 Eliminate choices, test-taking strategy, 3, 228, 229, 286, 544, 545, 590, 627, 788, 789, 933 Elimination method for solving linear systems, 161–167, 179–185, 221, 223 for solving quadratic systems, 660–664, 668 Ellipse, 634 area of, 636, 638, 640 co-vertices of, 634 eccentricity of, 665–666 equation of, 634 graphing, 634–639, 668, 671 translated, 650, 652–657 writing, 635–639, 668, 671 foci of, 634 major axis of, 634 minor axis of, 634 vertices of, 634 Empty set, 715 End behavior, for a polynomial function, 336, 339–344 Equation(s), See also Formulas; Function(s); Inequalities; Linear equation(s); Polynomial(s); Quadratic equation(s), 18 absolute value, 50–58, 60, 64 of circles, 626–633 of conic sections, 650–657, 668, 670–672 direct variation, 107–111 of ellipses, 634–639 equivalent, 18 exponential, 515–516, 519–525, 542 general second-degree, 653 of hyperbolas, 642–648, 650–657, 668, 671, 672 inverse variation, 551–557 joint variation, 553–557 logarithmic, 499–501, 503–505 matrix, 190–192 of parabolas, 620–625 radical, 452–461, 465, 468 rational, 589–597, 602, 606 with rational exponents, 453, 456, 458–459 recursive, 826–833 rewriting, 26–32, 63 for sequences, 794–795, 798–801 for translated conics, 650–657 trigonometric, 876–880, 931–939, 964, 967 in two variables, 74–79 Equivalent equations, 18 Equivalent expressions, 12 Equivalent inequalities, 42 Error analysis Avoid Errors, 12, 20, 28, 43, 52, 73, 82, 108, 115, 126, 133, 153, 163, 187, 238, 247, 252, 260, 267, 277, 287, 292, 330, 338, Exponential equation(s), 515 modeling with, 516, 521–522 property of equality for, 515 solving, 515–516, 519–525, 538, 542 Exponential function(s), 478 decay, 486–491, 538, 540, 776 graphing, 478–485, 486–491, 538, 539, 776 growth, 478–485, 538, 539 as inverse of logarithmic functions, 501 involving e, 493–498, 540 modeling with, 480–481, 483–485, 488–491, 528, 530, 534–536 natural base, 493–498 writing, 529–531, 533–536, 542 Exponential inequalities, 526, 527 Exponential regression, 528, 530 Exponentiating an equation, 517 Expression(s) combining like terms in, 12–17 equivalent, 12 evaluating, 10–17, 60, 62, 330–335 exponential, 330–335 factorial, 684 logarithmic, 499–501, 503–505, 507–512, 541 natural base, 492–493, 495–496 numerical, 10–17, 330–331, 333 in quadratic form, 355 rational, 573–588, 602, 605 with rational exponents, 415–419, 420–427, 467 simplifying, 10–17, 60, 62, 330–335, 420–427 terms of, 12 trigonometric, 925–926, 928, 955–956, 959, 960 writing, 984 Extended response questions, 146–148, 470–472, 738–740, 970–972 practice, Throughout. See for example 8, 32, 33, 47, 59, 69, 88, 95, 106, 119, 139, 158, 166, 173, 185, 186 Extensions approximate binomial distributions, 763–765 design surveys and experiments, 772–773 determine eccentricity of conic sections, 665–666 discrete and continuous functions, 80–81 linear programming, 174–176 piecewise functions, 130–131 prove statements using mathematical induction, 836–837 set theory, 715–716 solve exponential inequalities, 526, 527 solve logarithmic inequalities, 527 solve radical inequalities, 462–463 solve rational inequalities, 598–600 Extra Practice, 1010–1023 Extraneous solutions for absolute value equations, 52 for logarithmic equations, 518 for radical equations, 454 for rational equations, 591 for trigonometric equations, 934 F Factor(s), 978 common, 978 conversion, 981 decay, 486 growth, 478 scale, 989 Factor theorem, 364, 404 Factor tree, 978 Factorial, See also Combination(s); Permutation(s), 684 Factoring completely, 353 difference of two squares, 253 patterns, 354 perfect square trinomials, 253 polynomials, 353–359, 364–368, 404 by grouping, 354, 357 quadratic equations, 252–265, 317, 319–320 quadratic expressions, 252–253, 255–256, 259–260, 263 with special patterns, 253, 256, 260, 263 the sum or difference of cubes, 354 trinomials, 252–265 zeros and, 262–265 Factorization, prime, 978–979 Feasible region, 174 Fibonacci sequence, 828, 832 recursive rule for, 828 Find the error, See Error analysis Finite differences, 393–399 first-order differences, 393 properties of, 394 second-order differences, 394 third-order differences, 395 Finite sequence, 794 First-order differences, 393 Focal length, 585, 624 Focus (Foci) of an ellipse, 634 of a hyperbola, 642 of a parabola, 620 Index INDEX 347, 354, 355, 362, 370, 371, 415, 416, 423, 430, 441, 480, 488, 507, 553, 573, 584, 599, 644, 654, 658, 683, 691, 693, 700, 708, 719, 726, 744, 795, 797, 803, 811, 821, 828, 862, 877, 890 exercises, Throughout. See for example 7, 13, 17, 22, 30, 38, 45, 56, 77, 86, 93, 102, 110, 118, 128, 136 Estimation, See also Approximation; Prediction of best-fitting line, 113–120, 143 of coordinates of turning points, 390 of correlation coefficients, 114–120 using exponential decay models, 488–491 using exponential growth models, 480, 483–485 using linear graphs, 91, 153 using natural base functions, 494, 497–498 using nth roots, 416 of solutions of linear systems, 153–158 using transformed data, 781 Euler, Leonhard, 492 Euler number e, 492 Even function, 928 Event(s) complement of, 709 compound, 707, 733, 736 dependent, 718–723, 733, 736 disjoint, 707, 736 independent, 717–719, 721–723, 733, 736 mutually exclusive, 707 overlapping, 707, 733, 736 probability, 698 Expanding a logarithmic expression, 508, 510, 541 Experiment, 308, 528 binomial, 725, 728, 729 control groups and, 773 designing, 772–773 Experimental group, 773 Experimental probability, 700, 702 Explicit rule, 827, 839 Exponent(s) decimal, 425 evaluating, 10–17 irrational, 425 properties of, 330, 402, 420, 465, 1034 using, 330–335, 402, 420–427, 467 rational, 415–419, 420–427, 465, 466 1091 INDEX FOIL method, 248, 985 Formulas, 26 area of a circle, 26, 992 of a parallelogram, 991 of a rectangle, 26, 991 of a trapezoid, 26, 991 of a triangle, 26, 885, 891, 991 Beaufort number, 458 change-of-base, 508 circumference, 26, 992 combinations, 690 degrees/radians, 860 distance, 26, 34 to the horizon, 450 between points, 614, 669 double angle, 955 Fahrenheit/Celsius, 26, 44, 69 half angle, 955 interest compound, 481 continuously compounded, 494 interior angle of a regular polygon, 799 Kelvin/Celsius, 450 margin of error, 768 midpoint, 615 Newton’s law of cooling, 516 nth pentagonal number, 394 nth triangular number, 394 perimeter, of a rectangle, 26, 27, 991 permutation, 685 probability, 698, 700 of the complement of an event, 709 of compound events, 707 of dependent events, 718 of disjoint events, 707 of independent events, 717 rewriting, 26–32, 63 slant height, of a truncated pyramid, 459 slope, 82 standard deviation, 748 standard normal distribution, 758 sum of first n positive integers, 797 sum of squares of first n positive integers, 797 surface area of a cone, 451 of a cylinder, 63, 567, 572, 580, 993 of a hemisphere, 472 of a rectangular prism, 993 of a sphere, 427 table of, 1026–1032 trigonometric difference, 949 trigonometric sum, 949 1092 Student Resources volume of a cone, 65 of a cube, 350, 601 of a cylinder, 334, 350, 567, 572, 580, 993 of a dodecahedron, 419 of an icosahedron, 419 of an octahedron, 419 of a pyramid, 350, 373 of a rectangular prism, 68, 334, 350, 993 of a sphere, 332, 409, 427, 436, 475, 601 of a tetrahedron, 419 Forty-five degree angle, trigonometric values for, 853 Fractal geometry fractal tree, 838 Julia set, 282 Mandelbrot set, 281 Sierpinski carpet, 816 Sierpinski triangle, 825 Fraction(s) adding, 979 complex, 584 decimals, percents, and, 976 subtracting, 979 writing repeating decimals as, 822 Fraction bars, as grouping symbols, 14 Frequency, of a periodic function, 910 Function(s), See also Graphs; Linear function(s); Parent function; Quadratic function(s), 73, 140, 141 absolute value, 121–129 classifying, 75, 80–81, 479, 487, 489 composition of, 430–435, 465, 467 continuous, 80–81 cosine, 852–858, 866–872, 949–962 cube root, 446–451, 465, 468 discrete, 80–81 domain of, 73, 76, 428–430 even, 928 exponential growth and decay, 478–491, 528–531, 533–536 family, 89 greatest integer, 131 inverse, 438–445, 465, 467, 501 horizontal line test for, 440 iterating, 830, 831, 833 linear, 75–79, 89–97, 438–439, 442–444 logarithmic, 502–505 logistic, 522 natural base, 493–498 objective, 174 odd, 928 operations on, 428–435, 465, 467 piecewise, 130–131 power, 428–435, 531–535 properties of, 1034 quadratic, 236–243, 245–251, 310–315, 322 radical, 446–451, 465, 468 range of, 73 rational, 548–607, 602, 604 recursive rules and, 827–835 representing, 73–79 rounding, 131 sine, 852–858, 866–872, 949–962 square root, 446–451, 465, 468 step, 131 tangent, 852–858, 866–872, 911–914, 949–962 vertical line test for, 73–74, 77 Function notation, 75 Fundamental counting principle, 682 and permutations, 684–689, 734–735 with repetition, 683, 687–689 using, 682–683, 686–689, 720 Fundamental theorem of algebra, 379 applying, 379–386, 405 corollary to, 379 G Gauss, Karl Friedrich, 379 General rational functions, 565–571, 602, 604 General second-degree equation, 653 Generalize, exercises, 703, 705, 712 Geometric mean, 749 Geometric probability, 701, 703, 704, 738–739 Geometric sequence, 810–816, 839, 841 recursive rules and, 827–835, 839, 842 Geometric series, 812–816, 839, 841 infinite, 819–825, 839, 842 Geometry, See also Angle(s); Circle; Formulas; Triangle(s); Trigonometric function(s) congruent figures, 996–997 conics, 649–657, 665–666 formulas from, 1032 golden rectangle, 594 line symmetry, 990 parallel lines, 84–86, 99, 102 perpendicular bisector, 615–619 Pythagorean theorem, 995 reflection, 988–989 similar figures, 996–997 transformation, 988–989 triangle relationships, 995 modeling natural base functions, 494 modeling the period of a pendulum, 447 sinusoidal regression, 943 solving an equation with two radicals, 455 solving an exponential inequality, 526 solving a linear quadratic system, 658 solving a logarithmic inequality, 527 solving radical inequalities, 462–463 solving rational equations, 596–597 solving rational inequalities, 598 solving trigonometric equations, 938–939 exercises, 118, 243, 251, 305, 434, 444, 483, 489, 491, 496, 497, 521, 534–536, 569–571, 638, 662, 712, 761, 816, 888, 914, 920, 937 exponential regression feature, 528, 530, 776, 781 graphing feature, 49, 97, 121, 122, 336, 361, 455, 461, 514, 523, 525, 526, 527, 567, 597, 598, 633, 658, 775, 776, 777, 801, 834, 923, 939, 943 intersect feature, 159, 305, 361, 455, 461, 463, 523, 525, 526, 527, 597, 598, 658, 939, 950 linear regression feature, 116, 775 list feature, 116, 308, 311, 731, 781, 943 LN feature, 514 LOG feature, 514 matrix feature, 194, 211, 213 maximum feature, 244 minimum feature, 244, 567 power regression feature, 533, 535 quadratic regression feature, 308, 311, 777, 786 random number feature, 714 root feature, 382 sequence mode, 801, 834 sinusoidal regression feature, 943 sort feature, 714 statistics feature, 750 STATPLOT feature, 116, 308, 311, 396, 528, 530, 774, 775, 781, 786, 943 summation feature, 801 table feature, 25, 48, 49, 152, 360, 460, 462, 496, 523, 524, 526, 527, 530, 581, 596, 598, 801, 938 table setup feature, 25, 460, 462, 596 test feature, 49 trace feature, 49, 244, 292, 293, 447, 494 window settings, 97, 159, 345, 361, 463, 923 zero feature, 243, 382 zoom feature, 121, 122, 633 Graphs of absolute value, 50, 51 of absolute value functions, 121–129 of absolute value inequalities, 50, 53, 54, 56, 135, 136, 169, 172 bar, 1006–1007 of best-fitting lines, 115–120 circle, 1006–1007 of continuous functions, 80–81 of cosine functions, 874, 908–914, 964, 965 translations, 915–917, 919–922, 966 of cube root functions, 446–451, 465, 468 of discrete functions, 80–81 of equations of circles, 626–633, 668, 670 translated, 650 of equations of ellipses, 634–639, 668, 671 translated, 652 of equations of hyperbolas, 642–648, 668, 671 translated, 651 of equations of parabolas, 620–625, 668, 670 translated, 651 of equations in two variables, 74–79 of exponential decay functions, 486–491, 538, 539 of exponential growth functions, 478–485, 538, 539 histogram, 724, 726–731 of horizontal and vertical lines, 92, 94 of horizontal and vertical translation, 916 of infinite geometric series, 820 interpreting, 325–326 of inverse functions, 437, 438, 440, 443, 445 line, 1006–1007 of linear equations, 89–97, 142 in three variables, 177 of linear inequalities in one variable, 41–49, 64, 133 in two variables, 132–138, 144 Index INDEX Geometry software activity, explore the law of sines, 881 Golden ratio, 594 Golden rectangle, 594 Graphing calculator activities calculate a binomial distribution, 731 calculate one-variable statistics, 750 end behavior of functions, 336 evaluate expressions, 17 find maximum and minimum values, 244 function operations, 435 graph equations of circles, 633 graph linear equations, 97 graph logarithmic functions, 514 graph rational functions, 564 graph systems of equations, 159 use matrix operations, 194 modeling data with a quadratic function, 308 operations with functions, 435 operations with sequences, 801 set a good viewing window, 345 solving linear systems using tables, 152 use tables to solve equations, 25 transformations, 121–122 trigonometric identities, 923 verify operations with rational expressions, 581 binomial probability feature, 731 checking solutions with, 161, 285, 292, 293, 462, 518, 591, 659, 958 connected mode, 564, 581 cubic regression feature, 396 dot mode, 564 entering equations, 25, 48, 49, 97, 121, 122, 152, 159, 345, 396, 455, 460, 461, 462, 463, 494, 514, 523, 524, 525, 564, 581, 596, 597, 633, 834, 923, 939 examples approximating real zeros of a polynomial function, 382–383 computing inverse matrices, 211–213 drawing a histogram, 731 finding a best-fitting line, 116 finding an exponential model, 530 finding a polynomial model, 396 finding a power model, 532 finding turning points of a polynomial function, 388 maximizing a polynomial model, 589 1093 INDEX of linear-quadratic systems, 658, 661, 662–664 of linear systems, 153–159, 221, 222 of logarithmic functions, 502–505, 514, 541 of natural base functions, 493–494, 496–498 of parallel and perpendicular lines, 84–86 of piecewise functions, 130–131 of polynomial functions, 336, 339–344, 387–392, 401, 403, 406 of quadratic functions in intercept form, 246–251, 317, 319 in standard form, 236–243, 317, 318 in vertex form, 245–246, 249–251, 317, 319 of quadratic inequalities, 300–307 of quadratic systems, 658–664, 668 of radical functions, 446–451, 465, 468 of rational functions, 558–571, 602, 604 of real numbers, 2 of relations, 72, 76 scatter plots, 112–120, 143 of sequences, 795, 798, 800, 801 of sine functions, 874, 908–914, 964, 965 translations, 915–917, 919–922, 966 of square root functions, 446–451, 465, 468 of systems of constraints, 174–176 of systems of linear inequalities, 168–173, 221, 223 of systems of quadratic inequalities, 301, 304, 305 of tangent functions, 911–914, 964, 965 translations, 918, 920, 921 of trigonometric functions, 874, 908–922, 964, 965 vertical shrinking of, 479 vertical stretching of, 479 Greatest common factor (GCF), 978–979 Greatest integer function, 131 Gridded-answer questions, Throughout. See for example 33, 59, 69, 106, 139, 149, 186, 220, 231, 274, 316, 327, 369, 400, 411 Grouping symbols fraction bars, 14 parentheses, 15 Growth factor, for an exponential growth function, 478 Growth function exponential, 478–485, 538, 539 involving e, 493–498, 540 Guess, check, and revise, problem solving strategy, 998–999 H Half-angle formulas, 955, 964 using, 955–962, 968 Half plane, 132 Heron’s area formula, 891 Hexagonal number, 837 Histogram, 1008–1009 on a graphing calculator, 731 probability distribution, 724, 726–731, 733 Hooke’s law, 444 Horizontal asymptote, 558 Horizontal line, graph of, 92 Horizontal line test, 440, 443 Horizontal translation, graphing, 916 Hyperbola, 558, 642 asymptotes of, 642 branches of, 642 center of, 642 eccentricity of, 665–666 equation of graphing, 642–648, 668, 671 translated, 650, 651, 652–657, 672 writing, 643–648, 668, 671 foci of, 642 transverse axis of, 642 vertices of, 642 Hypotenuse, 995 Hypothesis, 1002–1003 Hypothesis testing, 764–765 I Identity, 12 Identity matrix, 210 Identity property, 3 If-then form, of a conditional statement, 1002–1003 Imaginary number, 276 Imaginary unit i, 275 Inconsistent linear system, 154–157 Independent events, 717–719, 721–723 Independent linear system, 154–157 Independent variable, 74 Index of a radical, 414 Index of refraction, 879, 930, 963 Index of summation, 796, 797 Indirect argument, 1000–1001 Indirect measurement, 855, 857–858 Induction, mathematical, 836–837 Inequalities, See also Linear inequalities absolute value, 50–58, 135, 136, 169, 172 compound, 41–47 equivalent, 42 exponential, 526, 527 linear, 41–49, 132–138 systems of, 168–176, 221, 223 logarithmic, 527 quadratic, 300–307, 322 systems of, 301, 304, 305 radical, 462–463 rational, 598–600 Infinite geometric series, 819–825, 839, 842 Infinite sequence, 794 Infinite series, 796 Infinity, positive and negative, 336, 339 Initial side, of an angle, 859 Integers, 2 operations with, 975 Intercept form, of a quadratic function, 246–251, 317, 319 Interest compound, 481, 483, 484, 485 continuously compounded, 494–495, 497 Interpret examples, 91, 124, 132, 429, 494, 560, 869 exercises, 88, 306, 307, 335, 458, 535, 555, 762, 809, 872, 888, 930 probability distributions, 725, 726 Intersection of graphs of linear-quadratic systems, 658 of graphs of quadratic systems, 659 of sets, 707–713, 715–716 Inverse cosine, 874–879 Inverse function(s), 437–445, 465, 467 logarithmic and exponential, 501 trigonometric, 874–880, 897, 899 Inverse matrices, 210–217, 221, 226 Inverse property, 3, 501 Inverse relation(s), 438, 442 Inverse sine, 874–879 Inverse tangent, 874–879 Inverse variation, 550–557, 603 equations, 551–557 modeling with, 550, 552, 556–557 Investigating Algebra, See Activities Irrational conjugates theorem, 380 Irrational exponent, 425 Irrational number, 2 Iteration, 830, 831, 833 1094 Student Resources n2pe-9060.indd 1094 9/28/05 9:28:10 AM J Joint variation, 553–557, 603 Julia set, 282 Justify results, Throughout. See for example 4, 6, 7, 68, 271, 281, 391, 398, 418, 490, 512, 550, 556, 748, 819, 832 K Kepler’s second law, 904 Key Concept, 2, 3, 10, 12, 18, 42, 51, 52, 53, 72, 73, 74, 80, 82, 83, 84, 89, 90, 91, 92, 98, 107, 115, 123, 126, 133, 154, 160, 161, 168, 174, 179, 187, 195, 203, 204, 205, 206, 210, 212, 218, 236, 237, 245, 246, 248, 253, 266, 275, 276, 279, 284, 292, 294, 330, 339, 353, 354, 363, 364, 370, 379, 380, 381, 388, 394, 414, 415, 420, 421, 428, 438, 440, 446, 452, 478, 481, 486, 492, 493, 494, 499, 502, 507, 508, 515, 517, 551, 553, 558, 559, 565, 573, 575, 576, 582, 583, 584, 614, 615, 621, 626, 634, 642, 650, 653, 655, 682, 685, 690, 692, 693, 698, 699, 700, 707, 717, 718, 724, 725, 744, 745, 751, 752, 757, 763, 764, 768, 794, 796, 797, 802, 804, 810, 812, 821, 827, 852, 853, 859, 860, 861, 866, 867, 868, 875, 882, 883, 885, 889, 891, 908, 909, 911, 915, 924, 949, 955 L number of solutions of, 154–157 solution of, 152, 153, 178 solving algebraically, 160–166, 223 using augmented matrices, 218–219 using Cramer’s rule, 205–206, 208–209, 221, 226 using the elimination method, 161–167, 179–185, 221, 223 by graphing, 153–159, 221, 222 using inverse matrices, 210–217, 221, 226 using the substitution method, 160–167, 181–185, 221 using tables, 152 in three variables, 178–185, 224 Lines classifying by slope, 83 horizontal line test, 440, 443 parallel equations for, 99, 102 slope of, 84–86 perpendicular equations for, 99, 102 slope of, 84–86 of reflection, 988–989 slope of, 83, 100 of symmetry, 652, 655, 990 vertical, slope of, 73–74, 77 List, making to solve problems, 998–999 Local maximum, of a polynomial function, 388 Local minimum, of a polynomial function, 388 Location Principle, 378 Logarithm(s), 499 change-of-base formula and, 508–509, 511 common, 500 natural, 500 properties of, 507, 1034 using, 507–513, 541 Logarithmic equation(s), 517 evaluating, 499–501, 503–505, 541 modeling with, 519, 521–522, 524–525 property of equality for, 517 solving, 517–525, 538, 542 Logarithmic expression(s) condensing, 508, 510, 541 evaluating, 499–501, 503–505, 507–512 expanding, 508, 510, 541 Logarithmic function(s) graphing, 502–505, 514, 541 as inverse of exponential functions, 501 modeling with, 500, 504–505 Index n2pe-9060.indd 1095 INDEX Law of cosines, 889–895, 897, 900 Law of sines, 881–888, 897, 900 Law of universal gravitation, 557 Leading coefficient, of a polynomial function, 337 Least common denominator (LCD), 979 of a rational expression, 583, 986 for solving a rational equation, 590 Least common multiple (LCM), 978–979 for a rational expression, 583 Left distributive property, 197 Legs, of a triangle, 995 Like radicals, 422 adding and subtracting, 422–427 Like terms, 12 combining, 12 Likelihood, of an event, 698 Line graph, 1006–1007 Line plot, 1008–1009 Line of reflection, 988–989 Line symmetry, 990 Line of symmetry for a conic section, 652, 655 for a plane figure, 990 Linear equation(s), See also Linear systems, 18 for best-fitting line, 112–120, 143 direct variation, 107–111 forms of, 140 graphing, 89–97, 177 using slope-intercept form, 90–97, 140, 142 using standard form, 91–96 in three variables, 177 with no solutions, 23 point-slope form of, 98–99, 101, 140 rewriting, 28, 30–32 slope-intercept form of, 90–97, 98, 100, 140, 142 solving, 18–25, 60, 62 using the distributive property, 20–24 standard form of, 91–96 in three variables, 177–185 writing, 19, 20, 23–24, 98–104, 142 Linear function(s), 75–79 graphing, 89–97 inverse, 438–439, 442–444 linear programming and, 174–176 Linear inequalities constraints, 174–176 forms of, 41 graphing in one variable, 41–49, 64, 133 systems of constraints, 174–176, 221, 223 in two variables, 132–138, 144 reading, 41 solution of, 41 solving, 41–49, 64 systems of, 168–173 three or more, 170–173 in two variables, 132–138, 144 Linear programming, 174–176 constraints, 174 feasible region, 174 objective function, 174 Linear-quadratic systems, 658–664 Linear regression, 116 Linear systems, 152, 153 classifying, 154–157 coefficient matrix of, 205–206 with infinitely many solutions, 154, 163, 178, 180 with no solutions, 154, 163, 178, 180 1095 9/28/05 9:28:11 AM Logarithmic inequalities, 527 Logical reasoning, See Reasoning Logistic function, 522 Lower limit of summation, 796 Lower quartile, 1008–1009 INDEX M Major axis, of an ellipse, 634 Make a list, problem solving strategy, 998–999 Make a table, problem solving strategy, 48–49, 272–273, 998–999 Mandelbrot set, 281 Manipulatives, See also Calculator; Graphing calculator algebra tiles, 283 Calculator Based Laboratory, 948 coins, 308, 528 compass, 308 flashlight, 649 index cards, 774 measuring tools, 112, 437, 550 musical instruments, 948 Mapping diagram, 72, 73, 77, 140, 141 Margin of error, 768–771 formula, 768 Mathematical induction, 836–837 Mathematical modeling, formulas from, 1031 Matrix (Matrices), 187 adding and subtracting, 187, 189–193, 194 augmented, 218–219 coefficient, 205–206 Cramer’s rule and, 205–209 describing products, 195 determinant of, 203–209 dimensions of, 187, 195 elements of, 187 equal, 187 equations, 190–192 identity, 210 inverse, 210–217 multiplying, 195–202, 937 order of operations, 188 properties, 188, 197, 1033 row operations, 218–219 scalar multiplication, 188–192 for solving linear systems, 205–209, 210–219 total cost, 198 transition, 201 triangular form, 218–219 Matrix algebra, formulas from, 1026 Maximum value of a polynomial function, 388–392 of a quadratic function, 238–239, 241, 244, 287 of sine and cosine functions, 909 Mean, 744, 746–750, 783, 784, 1005 geometric, 749 transformation and, 751–755 Measurement converting measurements, 5, 7 converting units of, 981 Measures, table of, 1025 Median, 744, 746–750, 783, 784, 1005 transformation and, 751–755 of a triangle, 618 Midline, of a trigonometric graph, 915 Midpoint formula, 615, 669 of a line segment, 615, 617–619 Minimum value of a polynomial function, 388–392 of a quadratic function, 238–239, 241, 244 of sine and cosine functions, 909 Minor axis, of an ellipse, 634 Mixed Review, Throughout. See for example 9, 16, 24, 32, 40, 47, 58, 79, 88, 96, 104, 111, 120, 129, 138, 158 Mixed Review of Problem Solving, 33, 59, 106, 139, 186, 220, 274, 316, 369, 400, 436, 464, 506, 537, 572, 601, 641, 667, 705, 732, 756, 782, 818, 838, 873, 896, 940, 963 Mode, 744, 746–749, 783, 784, 1005 transformation and, 751–755 Modeling, See also Expression(s); Formulas; Graphing calculator; Graphs; Linear equation(s); Polynomial(s); Polynomial function(s); Quadratic function(s) absolute value, 51 using algebraic expressions, 13, 15–16 using area models, 254, 257, 258, 261, 283 using best-fitting quadratic models, 311 choosing a model, 774–781, 786 using circular models, 628, 630–632 circular motion, 916, 921, 922 completing the square, 283 using conic sections, 654, 656–657 direct variation, 107–111, 143 dropped objects, 268–270, 272–273 using eccentricity, 666 using equations of circles, 628, 630–632 using equations of ellipses, 636, 638–639, 640 using equations of hyperbolas, 644, 646–648 using equations of parabolas, 622, 624–625 exercises, 15, 39, 271, 557, 580, 639, 809, 888, 946 exponential decay, 488–491, 776 with exponential equations, 516, 521–523 using exponential functions, 480–481, 483–485, 488–491, 528, 530, 534–536 exponential growth, 480–481, 483–485 factors using a tree, 978 finite differences, 395, 398 fitting a model to data, 774–781 using hundreds squares, 976 infinite geometric series, 819, 822, 824–825 using inverse of a power functions, 441–442, 444–445 using inverse variation, 550, 552, 556–557 launched objects, 295, 298 using linear equations, 19, 20, 23–24, 29, 31–32, 98–105, 775 using linear inequalities, 44, 46–47 using logarithmic equations, 519, 521–522, 524–525 using logarithmic functions, 500, 504–505 using mapping diagrams, 72, 73, 77 using natural base functions, 494, 495, 497–498 normal distribution, 757–762 using a number line, 2, 41–51, 53, 54, 56, 303, 599, 975 operations on sets, 715–716 pendulum periods, 447 with power functions, 532–533, 535, 542 using quadratic equations, 254, 257–258, 261, 262, 264–265 using quadratic functions, 308, 311, 314 using quadratic inequalities, 303, 306–307 using quadratic regression, 308, 311 using quadratic systems, 660, 662–664 using rational equations, 589, 594–597 using rational expressions, 574, 579–580 real numbers, 2 relations, 72 using scatter plots, 112–120 with sine functions, 910, 913–914 1096 Student Resources n2pe-9060.indd 1096 9/28/05 9:28:11 AM inequalities and, 42–47 integer, 975 matrix, 195–202 as opposite of division, 4 of polynomials, 347–352, 403 properties, 3, 18 for matrices, 197 of rational expressions, 575–580, 602, 605 scalar, 188–192 Multiplication property of equality, 18 Multiplicative inverse, 4 of a complex number, 280 Multi-step problems examples, 29, 85, 91, 134, 170, 189, 206, 213, 239, 262, 311, 340, 373, 396, 429, 431, 439, 447, 480, 488, 494, 560, 567, 574, 616, 636, 644, 654, 691, 720, 795, 829, 862, 891, 951 exercises, 8, 23, 32, 33, 39, 47, 57, 59, 78, 88, 95, 103, 106, 137, 139, 184, 186, 200, 209, 216, 220, 250, 257, 274, 298, 307, 314, 316, 335, 344, 351, 358, 369, 376, 400, 418, 433, 436, 444, 458, 464, 484, 491, 497, 505, 506, 535, 537, 556, 570, 572, 580, 601, 619, 625, 631, 641, 647, 663, 667, 688, 696, 705, 712, 732, 748, 756, 761, 782, 800, 818, 838, 864, 872, 873, 888, 893, 896, 921, 940, 954, 961, 963 Mutually exclusive events, 707 N Natural base e, 492 Natural base expression, 492–493, 495–496 Natural base function, 493–498 Natural logarithm, 500 change-of-base formula and, 508–509, 511 Negative angle identities, 924 Negative correlation, 113, 114, 117 Negative exponent property, 330 Negative number, square root of, 275 Newton’s law of cooling, 516 Normal curve, 757 Normal distribution, 757–762, 783, 785 nth root, 414 evaluating, 414–419, 466 Number line, 2 to add integers, 975 for graphing absolute value, 50, 51 for graphing absolute value equations, 50 for graphing absolute value inequalities, 50, 53, 54, 56 for graphing linear inequalities, 41–49 for graphing quadratic inequalities, 303 real numbers on, 2 to show critical x-values, 599 to subtract integers, 975 Numbers absolute value of, 50, 51 classifying, 2 complex, 276 composite, 978 imaginary, 276 integers, 2, 975 irrational, 2 pentagonal, 394 prime, 978–979 pure imaginary, 276 rational, 2 real, 2 triangular, 394 triangular pyramidal, 395 whole, 2 Numerical expression, 10–17, 60 O Objective function, 174 Odd function, 928 Odds, for and against an event, 699–700, 702–703 Ohm’s law, 749 Online Quiz, Throughout. See for example 9, 16, 24, 32, 40, 47, 58, 79, 88, 96, 104, 111, 120, 129, 138 Open-ended problems, 6, 14, 32, 33, 45, 59, 87, 94, 103, 106, 110, 118, 127, 136, 139, 157, 165, 183, 186, 191, 200, 208, 215, 220, 250, 257, 270, 274, 280, 289, 297, 305, 313, 316, 334, 342, 369, 375, 384, 391, 398, 400, 417, 425, 433, 436, 443, 456, 464, 483, 490, 496, 506, 511, 520, 537, 556, 562, 569, 572, 593, 601, 617, 630, 638, 641, 667, 705, 711, 722, 728, 732, 748, 756, 782, 815, 818, 824, 831, 838, 873, 878, 896, 913, 940, 963 Or rule, 1000–1001 Order of operations, 10–17 for matrices, 188 Ordered pair, 987 Ordered triple, 177, 178 Index n2pe-9060.indd 1097 INDEX solutions to absolute value equations, 50 to absolute value inequalities, 50, 53, 54, 56 to linear inequalities, 41, 43, 45, 49 with tangent functions, 918, 921 using tree diagrams, 682, 686, 720, 978 with trigonometric functions, 870, 871–872, 941–948, 967 using Venn diagrams, 2, 430, 698, 706–708, 715–716, 1004 using a verbal model, 13, 19, 20, 29, 34, 35, 36, 42, 54, 63, 66, 100, 101, 134, 155, 162, 181, 239, 254, 261, 262, 356, 373, 389, 560, 589, 829 vertical motion, 295, 298 Monomial, See also Polynomial(s), 252, 985 Moore’s law, 547 Multiples, 978 Multiple choice questions, 228–230, 324–326, 544–546, 608–610, 788–790, 844–846 examples, 3, 19, 36, 82, 132, 155, 162, 268, 286, 332, 339, 355, 365, 430, 453, 508, 518, 575, 590, 614, 627, 708, 717, 745, 769, 805, 821, 853, 877, 933, 956 practice, Throughout. See for example 6, 14, 24, 30, 37, 38, 45, 46, 56, 69, 77, 86, 87, 93, 101, 109 Multiple representations, See also Manipulatives; Modeling examples, 35, 48, 53, 105, 107, 115, 134, 153, 161, 218, 239, 272, 285, 292, 293, 340, 360, 396, 440, 455, 460, 480, 488, 523, 530, 552, 560, 567, 591, 596, 635, 636, 640, 651, 652, 659, 714, 720, 724, 781, 795, 834, 855, 884, 895, 910, 931, 934, 938, 958 exercises, 15, 24, 39, 57, 95, 104, 119, 129, 157, 173, 216, 242, 258, 290, 306, 314, 343, 367, 392, 434, 451, 485, 521, 562, 570, 631, 647, 703, 729, 754, 779, 808, 816, 857, 887, 914, 929, 937 Multiplication of complex numbers, 277–278, 280 cross multiplying, 589–590 of data by a constant, 752–755, 783, 785 of functions, 429–435 1097 9/28/05 9:28:12 AM Ordering, real numbers, 3, 6 Origin, in a coordinate plane, 987 Outcome, 698 Outlier, 746, 747 Overlapping events, 707, 733, 736 INDEX P Parabola, 236 axis of symmetry of, 236, 620 directrix of, 620 eccentricity of, 665–666 equation of, 621–625, 668, 670 translated, 650–657 focal length of, 624 focus of, 620 graph of, 620–625, 668, 670 as graph of a quadratic function, 236–243 latus rectum of, 625 vertex of, 236, 620 Parallel lines equations for, 99, 102 slope of, 84–86 Parent function for absolute value functions, 121, 123 for cosine functions, 908 for cube root functions, 446, 465 for exponential decay functions, 486 for exponential growth functions, 478 for linear functions, 89 for logarithmic functions, 502 for quadratic functions, 236 for simple rational functions, 558 for sine functions, 908 for square root functions, 446, 465 Partial sum, 820 Pascal, Blaise, 692 Pascal’s triangle, 692, 695 binomial expansion and, 693, 695 Pattern(s) exercises, 8, 350, 512, 695 exponential function models and, 529–531 factoring, 354 to make a generalization, 283 Pascal’s triangle, 692, 695 power function models and, 531–533 product patterns for binomials, 347 Pentagonal numbers, 394 Percent calculating with, 977 decrease, 488–489 fractions, decimals, and, 976 increase, 480–481, 483–485 Perfect square, 284 Perfect square trinomial, 253 Perimeter, 991 Period, of a function, 908, 909 Periodic function, See also Cosine function; Sine function, 908 frequency of, 910 Permutation(s), 684–689, 733, 734–735 circular, 689 formula, 685 probability and, 699, 702 with repetition, 685–689 Perpendicular bisector, 615–619 Perpendicular lines equations for, 99, 102 slope of, 84–86 Piecewise function, 130–131 Point discontinuity, 579 Point-slope form, 98–99, 101, 140, 530, 532 Polynomial(s), 337 adding, 346–352, 403 dividing, 362–368 rational expressions, 576–577 factoring, 353–359, 364–368, 404 by grouping, 354, 357 multiplying, 347–352, 403 rational expressions, 575–576 in quadratic form, 355, 357 subtracting, 346–352, 403 theorems involving, 363, 364, 379 Polynomial equation(s) factoring, 353–359, 364–368, 404 solving, 355–359, 404 Polynomial function(s), 337 classifying zeros of, 381–382, 384, 385 degree of, 337, 339 Descartes’ Rule of Signs and, 381–382, 384, 385 end behavior of, 336, 339–344 evaluating, 338–344, 402, 403 by synthetic substitution, 338 finding zeros of, 370–378, 379–386, 401, 405 fundamental theorem of algebra and, 379–386, 405 graphing, 336, 340, 342–344, 387–392, 401, 403, 406 leading coefficient of, 337 maximum of, 388–392 minimum of, 388–392 standard form of, 337 turning points of, 388, 390 types of, 337 writing, 381, 384, 386, 393–399, 406 Polynomial long division, 362–368 Population, 766 Positive correlation, 113, 114, 117 Power function(s), 428–435 inverse, 440–445 modeling with, 532–533, 535, 538, 542 writing, 531–535 Power of a power property, 330, 414, 420 Power of a product property, 330, 420 Power property of logarithms, 507 Power of quotients property, 330, 420 Power regression, 533, 535 Powers, See also; Exponent(s); Exponential function(s) of a binomial difference, 693, 695 of a binomial sum, 693–695 coefficient of, 12 evaluating, 10–17 Practice, See Reviews Precision, significant digits and, 983 Prediction using direct variation, 108, 110 exercises, 15, 104, 110, 112, 158, 308, 344, 437, 535, 550, 631, 774, 791, 896, 973 using exponential decay models, 488–491 using exponential growth models, 480, 483–485 using exponential regression, 530 using an inverse function, 442 using line of fit, 116–120, 146–147 using rate of change, 85 using regression, 396 Premise, 1000–1001 Prerequisite Skills, xxiv, 70, 150, 234, 328, 412, 476, 548, 612, 680, 742, 792, 850, 906 Prime factorization, 978–979 Prime number, 978–979 Probability binomial, 763–765 binomial distribution and, 724–731, 733, 736 combinations and, 699, 702, 733, 735 of the complement of an event, 709–713 of compound events, 707–713 conditional, 718–723 of dependent events, 718–723, 733, 736 of disjoint events, 707 event, 698 experimental, 700, 702 formulas, 698, 1028–1029 fundamental counting principle, 682, 684–689, 734–735 geometric, 701, 703, 704, 738–739 of independent events, 717–719, 721–723, 733, 736 1098 Student Resources n2pe-9060.indd 1098 9/28/05 9:28:13 AM of square roots, 266 of subtraction, 18, 188 zero product, 253 Proportion, 980 Pure imaginary number, 276 Pythagorean identities, 924 Pythagorean theorem, 995 using, 852–858, 866, 895, 898 Q Quadrant(s), of a coordinate plane, 987 Quadrantal angle, 867 Quadratic equation(s), See also Polynomial(s) with complex solutions, 276–282 discriminant in, 294 to model dropped objects, 268–270, 272–273 to model launched objects, 295, 298 to model vertical motion, 295, 298 roots of, 253 solving by completing the square, 284–291, 317, 321 by factoring, 252–265, 317, 319, 320 by finding square roots, 266–271, 284, 317, 320 using the quadratic formula, 292–299, 317, 321 standard form of, 253 systems of, 658–664 Quadratic expression(s) completing the square for, 283 factoring, 252–253, 255–256, 259–260, 263 Quadratic form, of a polynomial, 355, 357 Quadratic formula, 292–299, 317, 321, 933 Quadratic function(s) best-fitting quadratic model and, 311 graphing in intercept form, 246–251, 317, 319 in standard form, 236–243, 317, 318 in vertex form, 245–246, 248–251, 287, 317, 319 maximum value, 238–239, 241, 244, 287 minimum value, 238–239, 241, 244 parent function, 236 writing in intercept form, 309, 312–315, 322 in standard form, 310, 312–315, 322 in vertex form, 309, 312–315, 322 zeros of, 254–256 Quadratic inequality (inequalities) critical x-values of, 303 graphing, 300–307, 322 in one variable, forms, 302–305 solving, 302–307, 322 system of, 301, 304, 305 Quadratic regression, 308, 311 Quadratic system, 658–664, 668, 672 Quartic function, 337 Quartile, 1008–1009 Quotient of powers property, 330, 420 Quotient property of logarithms, 507 of square roots, 266 Quotient of powers property, 495–500, 542, 544 Quotient property of radicals, 720–726, 753, 755 Quotient rule, for fractions, 915 R Radian, 860 converting between degrees and, 860–864, 899 Radian measure, 860–865 Radical(s), 266 index of, 414 like, 422 nth root, 414–419 properties of, 421–427, 507, 1034 simplest form, 422 Radical equation solving, 452–461, 465, 468 with two radicals, 455, 457 Radical function, graphing, 446–451, 465, 468 Radical inequalities, 462–463 Radical sign, 266 Radicand, 266 Radius, of a circle, 626, 992 Random sample, 766 Random variable, 724 Range as an absolute value inequality, 54 data, 745–750, 783, 784, 1005 of a function, 73, 446–447, 479, 487 of a relation, 72 of a sequence, 794 transformation and, 751–755 Rate, 85 Rate of change average, 85, 86 slope and, 82–88, 142 Index n2pe-9060.indd 1099 INDEX of mutually exclusive events, 707 normal distribution and, 757–762 odds and, 699–700, 702–703 outcome, 698 of overlapping events, 707, 733, 736 permutation and, 699, 702, 733, 734–735 standard normal table and, 759–762 theoretical, 698 Venn diagrams and, 698, 706–708 Probability distribution, 724 binomial, 725–731 skewed, 727, 728 symmetric, 727, 728 Problem solving strategies, See also Eliminate choices; Problem Solving Workshop draw a diagram, 35, 37–39 draw a graph, 49, 273 use a formula, 34, 37–39 guess, check, and revise, 998–999 interpret a diagram, 608, 609, 610 look for a pattern, 35, 37–39 make a list, 998–999 make a table, 48–49, 272–273, 998–999 solve a simpler problem, 998–999 use a verbal model, 36–39 write an equation, 36–39 Problem Solving Workshop, 48–49, 105, 218–219, 272–273, 360–361, 460–461, 523–525, 596–597, 640, 714, 781, 834–835, 895, 938–939 Product of powers property, 330, 420 Product property of logarithms, 507 of square roots, 266 Proof, See also Reasoning using mathematical induction, 836–837 of properties of logarithms, 511 Properties of addition, 3, 18, 188 of division, 18 for exponential equations, 515 of exponents, 330, 420, 1033, 1034 of finite differences, 394 of functions, 1034 inverse, 3, 501 for logarithmic equations, 517 of logarithms, 507, 1034 proofs of, 511 of matrix operations, 188, 197, 1033 of multiplication, 3, 18, 197 of radicals, 1034 of rational exponents, 420, 465, 1034 of real numbers, 2–9, 1033 1099 9/28/05 9:28:13 AM INDEX Ratio(s) golden, 594 to identify direct variation, 108, 109 percent as, 976 proportions and, 980 simplest form, 980 trigonometric, 852–858 Rational equation, 589–597, 602, 606 Rational exponent(s), 413–419, 465, 466 equations with, 453, 456, 458–459, 468 properties of, 420–427, 465, 467, 1034 Rational expression(s), 986 adding, 582–588, 602, 605 dividing, 576–580, 602, 605 least common denominator of, 583, 986 multiplying, 575–580, 602, 605 point discontinuity and, 579 simplified form of, 573 simplifying, 573–574, 577–580, 602 subtracting, 582–588, 602, 605 verifying operations with, 581 Rational function(s) graphing general, 565–571, 602, 604 simple, 558–564, 604 inverse variation, 550–557 joint variation, 553–557 parent function for, 558 Rational inequalities, 598–600 Rational numbers, 2 Rational zero theorem, 370 Rationalizing the denominator, 267 Readiness Prerequisite Skills, xxiv, 70, 150, 234, 328, 412, 476, 548, 612, 680, 742, 792, 850, 906 Skills Review Handbook, 975–1009 Reading function notation, 75, 430, 438 graphs, 74 linear inequalities, 41 subscripts, 26 summation notation, 796 Reading math, 54, 83, 174, 277, 339, 830, 854, 861, 868 Real numbers ordering, 3, 6 properties of, 2–9, 61, 1033 subsets of, 2 Reasoning and rule, 1000–1001 biconditional statement, 1002–1003 chain rule, 1000–1001 compound statement, 1001 conclusion, 1000–1003 valid, 1000–1001 conditional statement, 1002–1003 converse, 1002–1003 if-then form, 1002–1003 counterexample, 1003 derivations, 930, 957 direct argument, 1000–1001 exercises, 7, 25, 31, 49, 56, 87, 88, 94, 103, 105, 110, 117, 118, 128, 183, 219, 256, 273, 289, 297, 334, 345, 384, 386, 391, 443, 450, 485, 491, 497, 505, 522, 531, 557, 577, 618, 630, 639, 640, 646, 647, 656, 662, 666, 695, 702, 714, 716, 722, 723, 748, 759, 761, 770, 771, 807, 823, 831, 835, 837, 920, 923, 936, 945, 946, 957, 960 hypothesis, 1002–1003 hypothesis testing, 764–765 indirect argument, 1000–1001 inductive reasoning, 836–837 or rule, 1000–1001 using a pattern to make a generalization, 283 premises, 1000–1001 proof using mathematical induction, 836–837 for properties of logarithms, 511 Venn diagrams, 1004 Reciprocal, multiplying by, 4 Reciprocal identities, 924 Rectangle, area and perimeter of, 991 Rectangular prism surface area of, 993 volume of, 68, 334, 350, 993 Recursive rule, 826, 827 for a sequence, 826–835, 839, 842 Reference angle, 868, 871 Reflection on the coordinate plane, 988–989 of the graph of a parent function absolute value function, 124–129 cosine function, 917, 920 logarithmic function, 502 sine function, 917, 920 line of, 988–989 Refraction, index of, 879, 930, 963 Regression cubic, 396 exponential, 528, 530 linear, 116 power, 533, 535 quadratic, 308, 311 Relation, 72, 140, 141 inverse, 438, 442 Remainder theorem, 363, 404 Repeated solution, 379 Repeating decimal, 822 Reviews, See Big Ideas; Chapter Review; Chapter Summary; Cumulative Review; Mixed Review; Mixed Review of Problem Solving; Prerequisite Skills; Skills Review Handbook Right distributive property, 197 Right triangle trigonometry, 852–858, 897, 898 Root(s), See also Radical(s) evaluating nth roots, 414–419 of a quadratic equation, 253 square, 266, 275 Rotation center of, 988 on the coordinate plane, 988–989 Rounding function, 131 Row equivalent matrices, 218 Row operations, 218–219 Rubric for scoring extended response questions, 146, 470, 738, 970 for scoring short response questions, 66, 408, 674, 902 S Sample(s) biased, 767 classifying, 766, 769, 783 selecting, 766 size of, 768 Sampling, 766–771, 783, 786 SAT, See Standardized Test Preparation Scalar, 188 Scalar multiplication, 188–192 Scale factor, 989 Scatter plot, 113–120, 143 Science, See Applications Scientific notation, 331, 982 properties of exponents and, 331, 333, 334 Secant function, See also Trigonometric function(s) evaluating for any angle, 866–872 evaluating for right triangles, 852–858 Second-order differences, 394 Sector, 861 arc length, 861–865 area, 861–865 central angle of, 861 Self-selected sample, 766 Sequence(s), 794, 839 arithmetic, 802–809, 839, 841 finite, 794 formulas from, 1029–1030 geometric, 810–816, 839, 841 graphing, 795, 798, 800, 801 1100 Student Resources n2pe-9060.indd 1100 9/28/05 9:28:14 AM binomial products, 985 expressions, 984 geometry angle relationships, 994 congruent figures, 996–997 coordinate plane, 987 line symmetry, 990 similar figures, 996–997 transformations, 988–989 triangle relationships, 995 logical reasoning conditional statements, 1002–1003 counterexamples, 1003 logical argument, 1000–1001 Venn diagrams, 1004 measurement area, 991 area of a circle, 992 circumference of a circle, 992 converting units of, 981 perimeter, 991 surface area, 993 volume, 993 number sense factors and multiples, 978–979 fractions, decimals, and percents, 976 integers, operations with, 975 least common denominators, 986 percent, calculating with, 977 ratios and proportions, 980 scientific notation, 982 significant digits, 983 problem solving strategies, 998–999 statistical data graphing, 1006–1007 mean, median, mode, and range, 1005 organizing, 1008–1009 Slope, 82 of best-fitting lines, 114–120 classifying lines by, 83 rate of change and, 82–88, 142 Slope-intercept form, 90–97, 98, 100, 140, 142 Snell’s law, 930 Solution(s) of an absolute value equation, 50 of an absolute value inequality, 50 of an equation, 18 extraneous, 52, 454, 518, 591, 934 of a linear inequality in two variables, 132 of a polynomial function, 379, 387 of a quadratic equation, 253, 294 of a system of linear equations, 152, 153 in three variables, 178 of a system of linear inequalities, 168 Solve a simpler problem, problem solving strategy, 998–999 Special angle, 963 Special patterns, factoring with, 253, 256, 260 Spreadsheet to evaluate a recursive rule, 826 use the Location Principle, 378 Square(s) of a binomial, 347 difference of two, 353 Square root(s), 266 imaginary unit i and, 275, 317 of a negative number, 275 properties of, 266 as solutions to quadratic equations, 267–271, 317, 320 Square root function graphing, 446–451, 465, 468 parent, 446, 465 Standard deviation, 745–750, 783, 784 transformation and, 751–755 Standard form of a complex number, 276 of a linear equation, 91–96, 140 of a number, 982 of a polynomial function, 337 of a quadratic equation, 253 of a quadratic function, 236–243, 317, 318 Standard normal distribution, 758–762 formula, 758 Standard normal table, 759 Standard position, for an angle, 859 Standardized Test Practice, See also Eliminate choices, 68–69, 148–149, 230–231, 326–327, 410–411, 472–473, 546–547, 610–611, 676–677, 740–741, 790–791, 846–847, 904–905, 972–973 examples, 3, 19, 36, 82, 132, 155, 162, 254, 268, 286, 332, 339, 355, 365, 453, 508, 575, 590, 614, 627, 708, 717, 769, 877, 933, 956 exercises, Throughout. See for example 6, 8, 14, 15, 23, 32, 37, 47, 55 Standardized Test Preparation, See also Gridded-answer questions; Multi-step problems; Open-ended problems Index n2pe-9060.indd 1101 INDEX infinite, 794 patterns and, 794–795, 797–801 recursive rules and, 826–835, 839, 842 terms of, 794 writing rules for, 795, 798, 799–800 Series arithmetic, 804–809, 839, 841 finite, 796 formulas from, 797, 1029–1030 geometric, 812–817, 839, 841 infinite, 796 infinite geometric, 819–825, 839, 842 summation notation and, 796–800 Set theory, 715–716 Short response questions, 66–68, 408–410, 674–676, 902–904 practice, Throughout. See for example 9, 15, 23, 31, 33, 37, 38, 55, 59, 77, 78, 87, 94, 95, 103, 106 Shrink, of the graph of the parent absolute value function, 124–129 Sierpinski carpet, 816 Sierpinski triangle, 825 Sigma notation, 796 Significant digits, 348, 983 Similar figures, 996–997 Simplest form radical, 422 Simulation, 714 using random numbers, 714 Sine function, See also Trigonometric equation(s); Trigonometric function(s) difference formula for, 949 using, 949–954 double-angle formula for, 955 using, 955–962 evaluating for any angle, 866–872 evaluating for right triangles, 852–858 graphing, 874, 908–914, 964, 965 reflections, 917, 920 translations, 915–917, 919–922, 966 half-angle formula for, 955 using, 955–962 inverse of, 874–879, 897, 899 sinusoids, and, 941–948 sum formula for, 949 Sines, law of, 881–888, 897, 900 Sinusoidal regression, 943 Sinusoids, 941–948, 967 Sixty-degree angle, trigonometric values for, 853 Skewed distribution, 727, 728 Skills Review Handbook, 975–1009 algebra review 1101 9/28/05 9:28:15 AM INDEX Standardized Test Preparation context-based multiple choice questions, 324–326, 608–610, 844–846 extended response questions, 146–148, 470–472, 738–740, 970–972 multiple choice questions, 228–230, 544–546, 788–790 short response questions, 66–68, 408–410, 674–676, 902–904 Standing wave, 953 State Test Practice, 33, 59, 69, 139, 149, 186, 220, 231, 274, 316, 327, 369, 400, 411, 436, 464, 473, 506, 537, 547, 572, 601, 641, 667, 677, 732, 741, 782, 818, 838, 847, 873, 896, 940, 963, 973 Statistics, See also Data; Graphs; Probability best-fitting line, 114–120 bias in sampling, 767, 769, 771 biased question, 772–773 binomial distribution, 763–765 control group, 773 convenience sample, 766 direct variation, 107–111, 140, 143 experimental group, 773 experiments, designing, 772–773 formulas from, 1029 geometric mean, 749 margin of error, 768–771 measures of central tendency, 744–750, 783, 784, 1005 measures of dispersion, 744–750, 783, 784, 1005 negative correlation, 113, 114, 117 normal distribution, 757–762, 783, 785 outlier, 746, 747 population, 766 positive correlation, 113, 114, 117 random sample, 766 sampling, 766–771, 783, 786 self-selected sample, 766 standard deviation, 745–750, 783, 784 surveys designing, 772–773 sampling, 766–771 systematic sample, 766 tolerance, 54 unbiased sample, 767 Stem-and-leaf plot, 1008–1009 Step function, 131 Stretch, of the graph of the parent absolute value function, 124–129 Subscripts, reading, 26 Subset, 716 Substitution, for checking solutions, 18, 19, 20, 36, 52, 91, 133, 153, 160, 179, 205, 267, 285, 381, 452, 454, 455, 468, 517, 518, 591, 934 Substitution method for evaluating polynomial functions, 338 for solving linear-quadratic systems, 659, 661, 662–664 for solving linear systems, 160–167, 181–185, 221 for solving quadratic systems, 660–664, 668, 672 Subtraction of complex numbers, 276, 279 counting problems and, 691 with fractions, 979 of functions, 428–435 inequalities and, 42–47 integer, 975 matrix, 187, 189–192, 194 as opposite of addition, 4 of polynomials, 346–352, 403 properties, 18 for matrices, 188 with rational expressions, 582–588, 602, 605 Subtraction property of equality, 18 Sum formulas, 949, 964 for special series, 797 using, 949–954, 968 Sum of two cubes, 354 Summation notation, 796–800 Summing rectangles, 640 Supplementary angles, 994 Surface area, 993 Survey designing, 772–773, 786 probability and, 726, 729, 731, 732, 740, 741 sampling and, 766–771, 786 Symbols approximately equal to, 2 empty set, 715 factorial, 684 imaginary unit, 275 inequality, 50, 51 infinity, 336, 339 mean, 744 percent, 976 radical sign, 266 standard deviation, 745 subset, 716 summation, 796 table of, 1024 theta, 852 theta prime, 868 universal set, 715 Symmetric distribution, 727, 728 Symmetry line of for a conic section, 652, 655 for a plane figure, 990 Synthetic division, 363–368 Synthetic substitution, 363 for evaluating polynomial functions, 338 System of linear equations, See Linear systems System of linear inequalities, 168–173 with no solution, 169 three or more inequalities, 170–173 System of quadratic inequalities, 301, 304, 305 Systematic sample, 766 T Table(s) to display data, 8, 9, 47, 57, 59, 69, 108, 110, 111, 112, 115, 117, 118, 119, 120, 206, 400, 421, 426, 472, 552, 553, 570, 580, 777, 779, 780, 781, 782, 787, 824, 904, 946, 969, 973, 1007 to graph cube root functions, 447 to graph equations of parabolas, 621 to graph exponential decay functions, 486 to graph exponential growth functions, 478 to graph linear functions, 75, 80 to graph polynomial functions, 340, 342–344 to graph quadratic functions, 236, 237, 240 to graph square root functions, 446 interpreting, 609, 610 for natural base e, 492 for recording experimental data, 308, 819 to represent relations, 72 to solve linear equations, 25 to solve linear systems, 152 to solve problems, exercises, 15, 24, 39, 95, 104, 129, 290, 306, 314, 343, 451, 570, 647, 729, 808, 914, 929, 937 to solve quadratic inequalities, 302 to solve radical inequalities, 462 to solve rational inequalities, 598 spreadsheet, 826 standard normal, 759 Tables of reference Formulas from algebra, 1027–1028 1102 Student Resources n2pe-9060.indd 1102 9/28/05 9:28:16 AM complex conjugates theorem, 380 factor theorem, 364 fundamental theorem of algebra, 379–386, 405 irrational conjugates theorem, 380 Pythagorean theorem, 995 rational zero theorem, 370 remainder theorem, 363 Theoretical probability, 698 Third-order differences, 395 Thirty-degree angle, trigonometric values for, 853 Tolerance, 54 Total cost matrix, 198 Tower of Hanoi, 800 Transformation, 123 on the coordinate plane, 988–989 data and, 751–755 of exponential data, 529 of general graphs, 126 of the graph of a parent function absolute value, 121–129, 144 exponential, 479, 487 radical, 448 rational, 558, 559 multiple, 125–129 of power data, 532 producing equivalent inequalities, 42 vertical shrinking of a graph, 479, 487 vertical stretching of a graph, 479, 487 Transition matrix, 201 Translation of conic sections, 650–657, 672 on the coordinate plane, 988–989 exercises, 39 of the graph of a parent function absolute value, 121, 122, 123–129 cosine, 915–917, 919–922, 966 exponential, 487 exponential growth, 479 logarithmic, 503 radical, 448 rational, 559 sine, 915–917, 919–922, 966 tangent, 918, 920, 921 horizontal, 916 vertical, 916 Transverse axis, of a hyperbola, 642 Tree diagram for counting possibilities, 682, 686 for factoring numbers, 978 for finding probability, 720 Triangle(s) AAS, 882 ambiguous case, 883–884 area of, 885, 887, 888, 991 ASA, 882 classifying, using the distance formula, 614–615, 617 hypotenuse of, 995 legs of, 995 median of, 618 perimeter of, 991 right, trigonometry and, 852–858, 898 SAS, 889 SSA, 883–884 SSS, 890–891 sum of angle measures of, 995 Triangular numbers, 394 Triangular pyramidal numbers, 395 Trigonometric equation(s), 876–880 identities and, 923–930 solving, 931–939, 964, 967 using double-angle and halfangle formulas, 958, 960–962 in an interval, 932, 935–937 using sum and difference formulas, 949–954 Trigonometric expressions, 925–926, 928, 955–956, 959, 960 Trigonometric formulas double-angle and half-angle, 955–962, 964, 968 sum and difference, 949–954, 964, 968 Trigonometric function(s) of any angle, 866–872, 899 difference formulas for, 949 using, 949–954 double-angle formulas for, 955 using, 955–962 graphing, 874, 908–922, 964, 965 half-angle formulas for, 955 using, 955–962 inverse, 874–880, 897, 899 modeling with, 870, 871–872, 941–948, 967 right angle, 852–858, 898 sum formulas for, 949 using, 949–954 Trigonometric identities, 923–930, 966 verifying, 923, 926–930, 958, 966 Trigonometric ratios, 852–858 Trigonometry formulas from, 1030–1031 identities from, 1030–1031 Trinomial(s), 252 factoring, 252–265 Turning points, of a polynomial function, 388, 390 U Unbiased sample, 767 Unbounded region, 174 Index n2pe-9060.indd 1103 INDEX from combinatorics, 1028 from coordinate geometry, 1026 from geometry, 1032 from mathematical modeling, 1031 from matrix algebra, 1026 from probability, 1028–1029 from sequences and series, 1029–1030 from statistics, 1029 from trigonometry, 1030–1031 Identities, from trigonometry, 1030–1031 Measures, 1025 Properties of exponents, 1033 of functions, 1034 of logarithms, 1034 of matrices, 1033 of radicals, 1034 of rational exponents, 1034 of real numbers, 1033 Symbols, 1024 Theorems, from algebra, 1027–1028 Tangent function, See also Trigonometric equation(s); Trigonometric function(s) difference formula for, 949 using, 949–954 double-angle formula for, 955 using, 955–962 evaluating for any angle, 866–872 evaluating for right triangles, 852–858 graphing, 911–914, 965 translations, 918, 920, 921 half-angle formula for, 955 using, 955–962 inverse, 875–879, 897, 899 sum formula for, 949 using, 949–954 Tangent identities, 924 Technology, See Calculator; Graphing calculator Technology support, See Animated Algebra; @Home Tutor; Online Quiz; State Test Practice Term(s) constant, 12 of an expression, 12 like, 12 of a sequence, 794 variable, 12 Terminal side, of an angle, 859 Test-taking strategies, eliminate choices, 3, 228, 229, 286, 544, 545, 590, 627, 788, 789, 933 Theorems binomial theorem, 693 1103 9/28/05 9:28:16 AM Undefined slope, 83 Union, of sets, 707–713, 715–716 Unit analysis, 5 for checking solutions, 7, 34 with conversions, 5 with operations, 5 Unit circle, 867 Universal gravitational constant, 557 Universal set, 715 Upper limit of summation, 796 Upper quartile, 1008–1009 INDEX V Variable(s), 11 dependent, 74 independent, 74 random, 724 solving for, 26 Variable term, 12 Variation constant of, 107, 551 direct, 107–111, 140, 143 inverse, 550–557, 603 joint, 553–557, 603 Venn diagram, 1004 classifying numbers, 2 to show composition of functions, 430 to show operations on sets, 715–716 to show probability, 698, 706–708 Verbal model examples, 13, 19, 20, 29, 34, 35, 36, 42, 54, 63, 66, 100, 101, 134, 155, 162, 181, 239, 254, 261, 262, 356, 373, 389, 560, 589, 829 exercises, 15, 30, 242, 257, 594, 601 writing and evaluating, 11 Verification, of trigonometric identities, 923, 926–930, 958, 966 Vertex (Vertices) of an absolute value graph, 123 of an angle, 859 of an ellipse, 634, 650 of a feasible region, 174 of a hyperbola, 642, 650 of a parabola, 236, 620, 650 Vertex form, of quadratic a function, 245–246, 248–251, 287, 317, 319 Vertical asymptote, 558 Vertical line, graph of, 92 Vertical line test, 73–74, 77 Vertical motion problem, 295, 298 Vertical shrinking, of a graph, 479, 487 Vertical stretching, of a graph, 479, 487 Vertical translation, graphing, 916 Visual thinking, See also Graphs; Manipulatives; Modeling; Multiple representations; Transformation exercises, 15, 102, 185, 281, 297, 342, 624, 862 Vocabulary overview, 1, 71, 151, 235, 329, 413, 477, 549, 613, 681, 743, 793, 851, 907 prerequisite, xxiv, 70, 150, 234, 328, 412, 476, 548, 612, 680, 742, 792, 850, 906 review, 61, 141, 222, 318, 402, 466, 539, 603, 669, 734, 784, 840, 898, 965 Volume, See Formulas W What If? questions, 11, 13, 21, 29, 36, 44, 74, 83, 85, 91, 101, 109, 126, 135, 155, 163, 170, 181, 198, 219, 239, 246, 247, 254, 262, 269, 278, 287, 295, 341, 356, 361, 365, 373, 383, 389, 416, 431, 447, 453, 461, 480, 488, 494, 501, 509, 519, 525, 531, 552, 561, 567, 574, 592, 597, 628, 636, 644, 661, 683, 684, 691, 692, 699, 700, 701, 708, 709, 717, 718, 719, 721, 746, 752, 759, 767, 795, 797, 805, 822, 829, 835, 855, 870, 877, 891, 911, 939, 942, 957 Whole numbers, 2 Work rate problems, 20, 24 Writing, See also Communication; Verbal model absolute value functions, 125 algebraic expressions, 984 direct variation equations, 107–111 equations of circles, 627–632 equations of ellipses, 635–639 equations of hyperbolas, 643–648 equations of parabolas, 621–625 exponential functions, 529–531, 533–536, 542 linear equations, 19, 20, 23–24, 98–104, 142 linear systems as matrix equations, 212–213, 215–217 piecewise functions, 131 polynomial functions, 381, 384, 386, 392–399 power functions, 531–535 quadratic functions, 309–315, 322 rational equations, 589, 594–595 rules for nth term of a sequence, 803–809, 810–816 rules for sequences, 795, 798, 799–800 systems of equations, 155, 157–158, 162, 165–166, 181, 184–185 systems of linear inequalities, 170, 172–173 trigonometric equations, 877, 879–880 trigonometric functions, 941–948 X x-axis, 987 x-coordinate, 987 x-intercept, 91 of the graph of a polynomial function, 387 x-values, critical, 303, 599 Y y-axis, 987 y-coordinate, 987 as local maximum of a function, 388 as local minimum of a function, 388 y-intercept, 89 Z z-intercept, 177 z-score, 758 standard normal table and, 759–762 Zero exponent property, 330 Zero product property, 253 Zero slope, 83 Zeros of a polynomial function, 364, 365, 366, 367, 370–378, 387, 405 approximating real, 382–383, 384 Descartes’ Rule of Signs and, 381–382, 384, 385 fundamental theorem of algebra and, 379–386, 405 of a quadratic function, 254–256 average of, 262 of a rational function, 566 1104 Student Resources n2pe-9060.indd 1104 9/28/05 9:28:17 AM Credits Photographs Credits n2pe-9070.indd 1105 CREDITS Cover Michael Wong/Corbis; viii PhotoDisc/Getty Images; ix Alex Rosenfeld/Science Photo Library; x Jonathan Nourok/PhotoEdit; xi Mandy Collins/Alamy; xii Dave Bjorn/Photo Resource Hawaii; xiii Joe McBride/Getty Images; xiv Ron Sanford/Corbis; xv Ralph Wetmore/Getty Images; xvi NASA; xvii Jerry Wachter/ Sportschrome, Inc.; xviii AP/Wide World Photos; xix Ted Kinsman/Photo Researchers, Inc.; xx David Madison/Getty Images; xxi Thorney Lieberman/Getty Images; xxiv–1 Jennifer Graylock/AP/Wide World Photos; 2 Sylvain Grandadam/Getty Images; 6 Jason Hawkes/Getty Images; 8 all Courtesy NASA/JPLCaltech; 9 Joel Sartore/Getty Images; 10 Rubberball Productions/ Getty Images; 13 Brand X Pictures/Getty Images; 18 PhotoDisc/ Getty Images; 20 Creatas/PunchStock; 23 Walter Hodges/Getty Images; 24 both Digital Stock/Corbis; 26 Peter Adams/Index Stock Imagery; 29 Rubberball Productions; 33 Gary I. Rothstein/ AP/Wide World Photos; 34 PhotoDisc/Getty Images; 35 Jason Reed/Reuters; 38 Eugene Hoshiko/AP/Wide World Photos; 41 Royalty-Free/Corbis; 42 Barbara Leslie/Getty Images; 44 David A. Northcott/Corbis; 50 all Jay Penni Photography/McDougal Littell; 51 Digital Vision Ltd./SuperStock; 54 Jim Cummins/Getty Images; 59 Dean Hoffmeyer/AP/Wide World Photos; 70–71 George D. Lepp/Corbis; 72 Tom Stock/Getty Images; 74 Paul Battaglia/AP/Wide World Photos; 76 Stephen Frink/Corbis; 79 D.C. Lowe/SuperStock; 82 Ernest Manewal/Index Stock Imagery; 85 both Susan Ragan/AP/Wide World Photos; 89 Angela Wyant/ Getty Images; 91 Paul Nicklen/National Geographic/Getty Images; 98 Nancy Richmond/The Image Works; 100 Steve Skjold/PhotoEdit; 103 David Young-Wolff/PhotoEdit; 107 Dave G. Houser/Corbis; 108 Amos Nachoum/Corbis; 112 all McDougal Littell; 113 Douglas C. Pizac/AP/Wide World Photos; 115 Sandy Huffaker/Getty Images; 123 top right John Coletti/ Index Stock Imagery; 132 Digital Vision/Getty Images; 134 all Stockbyte/Getty Images; 150–151 Myrleen Ferguson Cate/ PhotoEdit; 153 Duomo/Corbis; 155 David Frazier/The Image Works; 160 Rubberball/PictureQuest; 166 Petros Giannakouris/ AP/Wide World Photos; 168 Duncan Smith/Getty Images; 170 Viviane Moos/Corbis; 173 Jim Cummins/Getty Images; 174 Brand X/SuperStock; 176 Kelly-Mooney Photography/Corbis; 178 Chris Donahue/AP/Wide World Photos; 185 Manchan/Getty Images; 186 Comstock Images/Alamy; 187 Stephen J. Carrera/ AP/Wide World Photos; 192 Brand X Pictures/Alamy; 195 Mike Powell/Getty Images; 203 Tim Wakefield/SuperStock; 210 Lee Strickland/Getty Images; 215 Walter Meayers Edwards/Getty Images; 216 Gianni Cigolini/Getty Images; 220 Tom Bean/ Corbis; 234–235 Getty Images; 236 C. B. Knight/Getty Images; 239 Rick Friedman/Corbis; 245 Medford Taylor/Getty Images; 251 David Hall/Nature Picture Library; 252 Kayte M. Deioma/ PhotoEdit; 257 Image Source Limited/Index Stock Imagery; 259 Greg Huglin/SuperStock; 261 John Warden/SuperStock; 262 PhotoDisc/Getty Images; 264 bottom right Seth Thompson/ Getty Images; 266 Hubble Heritage Team/NASA/AP/Wide World Photos; 269 Mike Mergen/AP/Wide World Photos; 270 Steve Allen/Brand X Pictures/PictureQuest; 271 Duomo/Corbis; 274 Cheryl Hatch/AP/Wide World Photos; 275 Dr. Fred Espenak/ SPL/Photo Researchers, Inc.; 281 Imagebroker/Alamy; 282 Alfred Pasieka/SPL/Photo Researchers, Inc.; 283 all McDougal Littell; 284 Jim Cummins/Getty Images; 287 David Madison/ Getty Images; 289 Ron Watts/Corbis; 291 Tom Stewart/Corbis; 292 Mike Yamashita/Corbis; 295 left David Madison/Getty Images; 295 center Roy Morsch/Corbis; 295 right Amwell/Getty Images; 299 Ben Mangor/SuperStock; 300 PhotoDisc/Getty Images; 303 Marcio Jose Sanchez/AP/Wide World Photos; 306 Stephen Frisch/Stock Boston; 309 Photolibrary.com/Index Stock Imagery; 311 Kevin Fleming/Corbis; 314 Michael Wong/Corbis; 316 Stockbyte; 328–329 Reuters/Corbis; 330 NASA/Corbis; 331 right Reuters/Corbis; 331 left Dex Image/Getty Images; 332 Comstock Images; 337 David Young-Wolff/PhotoEdit; 343 bottom ThinkStock LLC/Index Stock Imagery; 344 M. Philip Kahl/Bruce Coleman, Inc.; 346 Tim Larsen/AP/Wide World Photos; 348 David Frazier/Getty Images; 353 Richard T. Nowitz/ Corbis; 362 Orlin Wagner/AP/Wide World Photos; 365 RoyaltyFree/Corbis; 367 Joe Cavarotta/AP/Wide World Photos; 368 Neil Rabinowitz/Corbis; 370 top right Volker Steger/Siemens/SPL/ Photo Researchers, Inc.; 376 age fotostock/SuperStock; 379 Peter Correz/Getty Images; 383 PhotoDisc/Getty Images; 385 Peter Yates/SPL/Photo Researchers, Inc.; 387 top right Turner & de Vries/Getty Images; 393 Terry Renna/AP/Wide World Photos; 398 Tom Stewart/Corbis; 400 Joe McDonald/Corbis; 412–413 David Bergman/Corbis; 414 Javier Soriano/AFP/Getty Images; 416 Azure Computer & Photo Services/Animals Animals; 419 McDougal Littell/Houghton Mifflin Co.; 420 Cameron Heryet/ Getty Images; 426 Lester Lefkowitz/Corbis; 428 Darrell Gulin/ Corbis; 431 StreetStock Images/Brand X Pictures/PictureArts; 434 Courtesy of Professor Tim Pennings, Hope College; 436 Royalty-Free/Corbis; 437 both RMIP/Richard Haynes/McDougal Littell; 438 Paul A. Souders/Corbis; 441 Gail Burton/AP/Wide World Photos; 444 Rod Taylor/AP/Wide World Photos; 446 Duomo/Corbis; 447 Navaswan/Getty Images; 451 Brian Erler/ Getty Images; 452–453 Cathrine Wessel/Corbis; 453 Philippe Giraud/Corbis Sygma; 457 Casey Riffe/Marshfield News-Herald/ AP/Wide World Photos; 458 OSF/Colbeck, M./Animals Animals; 476 background B.A.E. Inc./Alamy; 478 Greer & Associates, Inc./ SuperStock; 484 Bob Daemmrich/Corbis Sygma; 485 Lynda Richardson/Corbis; 486 age fotostock/SuperStock; 488 PhotoDisc/Getty Images; 491 Richard A. Cooke/Corbis; 492 Cousteau Society/Getty Images; 498 Tim Hursley/SuperStock; 499 Chuck Carlton/Index Stock Imagery; 506 Jim McNee/Index Stock Imagery; 507 age fotostock/SuperStock; 509 Royalty-Free/ Corbis; 511 center left image100/Alamy; 511 center Pat LaCroix/ Getty Images; 511 center right Tony Arruza/Corbis; 512 center right Rubberball Productions; 512 center left Cathy Melloan/ PhotoEdit; 512 bottom Rubberball Productions; 515 Roger Ressmeyer/Corbis; 516 Jeff Sherman/Getty Images; 522 both AM Corporation/Alamy; 528 both Jay Penni Photography/McDougal Littell; 529 Mark Chappell/Animals Animals; 530 Nick Dolding/ Getty Images; 532 Gerard Lacz/Animals Animals; 534 Bryn Colton/Assignments Photographers/Corbis; 536 Bettmann/ Corbis; 548–549 Jeff Hunter/Getty Images; 550 both RMIP/ Richard Haynes/McDougal Littell; 551 Lawrence Manning/ Corbis; 552 age fotostock/SuperStock; 553 Rick Bowmer/AP/ Wide World Photos; 558 Ken Reid/Getty Images; 560 both Courtesy of Z Corporation; 565 Bobby Model/National Geographic/Getty Images; 567 Donald C. Johnson/Corbis; 569 The Photolibrary Wales/Alamy; 570 Scaled Composites/SPL/ Photo Researchers, Inc.; 572 Michael Newman/PhotoEdit; 573 Mary Ann Chastain/AP/Wide World Photos; 579 Tony McConnell/SPL/Photo Researchers, Inc.; 582 Luca DiCecco/ Alamy; 589 Denis Boissavy/Getty Images; 592 don jon red/ Alamy; 594 Fédération Internationale de Volleyball/AP/Wide World Photos; 601 PhotoDisc/Getty Images; 612–613 Phototake/Getty Images; 614 Nick Vedros & Assoc./Getty Images; 618 Reuters/Corbis; 620 Stephen Frink/Getty Images; 624 Hank Morgan/Time Life Pictures/Getty Images; 625 Roger Ressmeyer/ Corbis; 626 Royalty-Free/Corbis; 634 Ralph Wetmore/Getty Images; 638 bottom left NASA/ARC; 638 bottom right Detlev Van Ravenswaay/SPL/Photo Researchers, Inc.; 1105 9/28/05 10:32:31 AM CREDITS 641 Tom Uhlman/Visuals Unlimited; 642 Paul A. Souders/ Corbis; 648 Mike Cartwright/AP/Wide World Photos; 649 RMIP/ Richard Haynes/McDougal Littell; 650 Courtesy of Superdairyboy; 656 Erik S. Lesser/AP/Wide World Photos; 658 Yellow Dog Productions/Getty Images; 676 Fermilab Photo; 680–681 Alexander Walter/Getty Images; 682 Douglas C. Pizac/ AP/Wide World Photos; 684 Oliver Morin/AFP/Getty Images; 690 Robbie Jack/Corbis; 696 Marc Lester/AP/Wide World Photos; 698 PhotoDisc/Getty Images; 699 Ellen Senisi/The Image Works; 705 bottom left Big Cheese Photo/FotoSearch; 707 Jeff Greenberg/The Image Works; 717 Dennis MacDonald/PhotoEdit; 719 NOAA/AP/Wide World Photos; 724 John Russell/AP/Wide World Photos; 742–743 Mark E. Gibson/Getty Images; 744 Omar Torres/AFP/Getty Images; 746 Banana Stock/Alamy; 748 Alan Diaz/AP/Wide World Photos; 749 Andy Lyons/Getty Images; 751 NASA-HQ-GRIN; 754 bottom Ric Francis/AP/Wide World Photos; 754 top Phil Cantor/Index Stock Imagery; 756 top age fotostock/SuperStock; 756 bottom Don Heupel/AP/Wide World Photos; 756 bottom center ThinkStock/SuperStock; 757 PhotoDisc/Getty Images; 759 Kennan Ward/Corbis; 762 Barbara Novovitch/Reuters; 765 Royalty-Free/Corbis; 766 David YoungWolff/PhotoEdit; 772 Spencer Grant/PhotoEdit; 774 Jay Penni Photography/McDougal Littell; 775 Ben Margot/AP/Wide World Photos; 780 Jay Penni Photography/McDougal Littell; 782 Dynamic Graphics/PictureQuest; 792–793 Steve Gschmeissner/ SPL/Photo Researchers, Inc.; 794 Roger Wood/Corbis; 799 Frank Chmura/PictureQuest; 802 Richard Cummins/Corbis; 805 Stockbyte/PictureQuest; 808 © 2007 Sol LeWitt/Artist Rights Society (ARS), New York. Photo Credit: Mary Ann Sullivan, Bluffton University; 810 PhotoStockFile/Alamy; 813 GDT/Getty Images; 815 Agence Vandystadt/Photo Researchers, Inc.; 819 all Jay Penni Photography/McDougal Littell; 820 Courtesy of Gayla Chandler; 827 Popperfoto/Alamy; 838 Gary S. Settles/Photo Researchers, Inc.; 850–851 João Paulo/Getty Images; 852 Hugh Sitton/Getty Images; 859 M. Spencer Green/AP/Wide World Photos; 864 top Royalty-Free/Corbis; 864 center Courtesy NASA, Life Sciences Division; 865 both Royalty-Free/Corbis; 866 age fotostock/SuperStock; 869 Courtesy NASA/JPL-Caltech; 873 bottom right Brand X Pictures/Getty Images; 873 right Richard Cummins/SuperStock; 873 bottom right Yoshio Tomii/ SuperStock; 875 Greg Ebersole/AP/Wide World Photos; 879 Paolo Curto/The Image Bank; 882 Richard Berenholtz/Corbis; 885 Steve Bein/Corbis; 889 David Madison/Getty Images; 906– 907 Royalty-Free/Corbis; 908 Richard Olseius/National Geographic/Getty Images; 910 Annabella Bluesky/SPL/Photo Researchers, Inc.; 913 Peter Arnold, Inc./Alamy; 915 both Craig T. Lorenz/Photo Researchers, Inc.; 921 Image Source/PunchStock; 924 Lowell Observatory/NOAO/AURA/NSF; 931 Scott Camazine/Photo Researchers, Inc.; 932 both Christopher Mackay/Tantramar Interactive; 940 PhotoDisc/Getty Images; 941 Lee Cohen/Corbis; 945 Steve Chenn/Corbis; 948 RMIP/ Richard Haynes/McDougal Littell; 949 Howard Kingsnorth/Getty Images; 951 left Larry Dunmire/SuperStock; 951 right Ken Graham/Getty Images; 955 Pascal Rondeau/Getty Images; 963 Bill Ross/Corbis. Illustrations and Maps Argosy 1, 51, 70, 151, 235, 329, 413, 477, 549, 557 top, 613, 622 top, 646, 654, 662, 667 top left, 681, 743, 793, 818 bottom left; 851, 907, 921, 942, 954, 957; Kenneth Batelman 504, 636, 752, 766 , 871, 879, 930, 946; Steve Cowden 15, 32, 39, 95, 241, 243 top center, 247, 268, 298 center, 434, 439, 457, 464, 535; Stephen Durke 290 bottom, 552, 570, 622 center, 660, 689 top, 800, 808 top, 832, 858, 872 center right, 896, 918, 929 center, 929 top right, 940; John Francis 580, 761; Patrick Gnan/Deborah Wolfe, Ltd. 5, 129, 254, 343 top, 351, 356, 360, 377, 392, 585, 641, 644, 657, 676, 818 top right; 822, 838, 880, 890, 902, 905, 927, 953, 972; Chris Lyons 139, 158, 647 center; Steve McEntee 480, 563, 870, 873 bottom left, 877, 887, 893 bottom, 893 center, 904, 914, 916; Paul Mirocha 497, 574; Laurie O’Keefe 88, 494, 729; Steve Stankiewicz 258, 505; Doug Stevens 125; Dan Stuckenschneider 16, 46, 87, 108, 119, 137, 185, 189, 250, 257, 265 top right, 369, 373, 426, 450, 519, 524, 556, 557 center, 688, 795, 824, 936; Matt Zang/American Artists 500; Carol Zuber-Mallison 111, 204, 208, 209, 358, 521, 587, 616, 618, 619, 663, 704, 855, 885, 961 top right, 961 bottom. All other illustrations © McDougal Littell/Houghton Mifflin Company. 1106 Student Resources n2pe-9070.indd 1106 9/28/05 10:32:33 AM Sk ills Re Rev v iew Handboo Handbook k To add positive and negative numbers, you can use a number line. To subtract any number, add its opposite. EXAMPLE To add a positive number, move to the right. To add a negative number, move to the left. 26 25 24 22 0 21 1 2 3 4 5 6 Add or subtract. a. 1 1 (25) End 23 SKILLS REVIEW HANDBOOK Operations with Positive and Negative Numbers b. 22 2 (25) 5 22 1 5 Move 5 units to the left. 25 24 23 22 21 0 Start Start 1 Move 5 units to the right. 23 22 21 2 c 1 1 (25) 5 24 The opposite of 25 is 5. 0 1 2 End 3 4 c 22 2 (25) 5 3 To multiply or divide positive and negative numbers, use the following rules. • The product or quotient of two numbers with the same sign is positive. • The product or quotient of two numbers with different signs is negative. EXAMPLE Multiply or divide. a. 3 p 7 5 21 b. 23(27) 5 21 c. 18 4 2 5 9 d. 218 4 (22) 5 9 e. 23(7) 5 221 f. 3(27) 5 221 g. 218 4 2 5 29 h. 18 4 (22) 5 29 PRACTICE Perform the indicated operation. 1. 2 1 (28) 2. 5 2 12 3. 26(10) 4. 230 4 (22) 6. 7(25) 7. 18 2 10 8. 27 1 (212) 9. 11(4) 5. 24 1 6 10. 81 4 (29) 11. 212 4 3 12. 29(28) 13. 21 1 13 14. 45 4 (29) 15. 26(12) 16. 14 2 (29) 17. 232 4 16 18. 223 1 (25) 19. 28 2 (25) 20. 17 2 (218) 21. 29(21) 22. 23 2 (211) 23. 218 4 (23) 24. 14 1 (27) 25. 5(23) 26. 21 1 (28) 27. 22 2 10 28. 29 1 26 29. 220 4 (24) 30. 22 4 (22) 31. 27(26) 32. 1 2 24 33. 215 2 2 34. 0 1 (24) 35. 16 4 8 Skills Review Handbook n2pe-9020.indd 975 975 11/21/05 10:26:41 AM SKILLS REVIEW HANDBOOK Fractions, Decimals, and Percents A percent is a ratio with a denominator of 100. The word percent means “per hundred,” or “out of one hundred.” The symbol for percent is %. In the model at the right, 71 of the 100 squares are shaded. You can write the shaded part of the model as a fraction, a decimal, or a percent. 71 Fraction: seventy-one divided by one hundred, or }} 100 Decimal: seventy-one hundredths, or 0.71 Percent: seventy-one percent, or 71% EXAMPLE Write as a fraction. 94 5 47 a. 94% 5 }} }} 100 50 EXAMPLE 3 c. 0.3 5 three tenths 5 }} 10 Write as a decimal. 15 5 0.15 a. 15% 5 }} 100 EXAMPLE 20 5 1 b. 20% 5 }} } 5 100 106 5 1.06 b. 106% 5 }} 100 5 5 5 4 8 5 0.625 c. } 8 Write as a percent. 41 5 41% a. 0.41 5 }} 100 8 5 80 5 80% b. 0.8 5 }} }} 10 100 5 5 5 p 25 5 125 5 125% c. } }}} }} 4 4 p 25 100 PRACTICE Write as a fraction. 1. 0.65 2. 0.08 3. 1.5 4. 0.13 5. 0.7 6. 50% 7. 26% 8. 3% 9. 95% 10. 110% Write as a decimal. 1 11. } 4 9 12. } 10 30 13. } 25 2 14. } 5 3 15. } 8 16. 16% 17. 142% 18. 1% 19. 30% 20. 6.5% 21. 0.6 22. 0.24 23. 1.3 24. 0.07 25. 0.45 1 26. } 10 4 27. } 5 17 28. } 20 5 29. } 2 3 30. } 16 Write as a percent. 976 n2pe-9020.indd 976 Student Resources 11/21/05 10:26:45 AM Calculating with Percents EXAMPLE Word what of is n 3 5 Symbol Answer the question. a. What is 15% of 20? b. What percent of 8 is 6? n 5 0.15 3 20 n53 c. 80% of what number is 4? n3856 0.8 3 n 5 4 n 5 6 4 8 5 0.75 5 75% 3 is 15% of 20. n 5 4 4 0.8 5 5 75% of 8 is 6. SKILLS REVIEW HANDBOOK You can use equations to calculate with percents. Replace words with symbols as shown in the table at the right. Below are three types of questions you can answer with percents. 80% of 5 is 4. Amount of increase or decrease . To find a percent of change, calculate }}}}}}}}}}}}}} Original amount EXAMPLE Find the percent of change. a. A class increases from 21 students to 25 students. 25 2 21 4 }}}} 5 }} ø 0.19 5 19% increase 21 21 b. A price decreases from $12 to $9. 12 2 9 3 }}} 5 }} 5 0.25 5 25% decrease 12 12 PRACTICE Answer the question. 1. What is 98% of 200? 2. What is 25% of 8? 3. What is 30% of 128? 4. What is 5% of 700? 5. What is 100% of 17? 6. What is 150% of 14? 7. What is 0.2% of 500? 8. What is 6.5% of 3000? 9. What percent of 100 is 54? 10. What percent of 18 is 9? 11. What percent of 80 is 8? 12. What percent of 15 is 20? 13. What percent of 30 is 6? 14. What percent of 5 is 8? 15. What percent of 50 is 1? 16. 50% of what number is 6? 17. 55% of what number is 44? 18. 10% of what number is 6? 19. 75% of what number is 45? 20. 1% of what number is 2? 21. 90% of what number is 63? 22. 12% of what number is 60? 23. 200% of what number is 16? Find the percent of change. Round to the nearest percent if necessary. 24. A class increases from 20 to 28 students. 25. Time decreases from 60 to 45 minutes. 26. A price is reduced from $200 to $180. 27. Votes increase from 200 to 300. 28. A test is shortened from 40 to 32 items. 29. Membership increases from 820 to 1605. 30. A wage rises from $8.75 to $10.00. 31. The temperature drops from 248F to 58F. Skills Review Handbook n2pe-9020.indd 977 977 11/21/05 10:26:46 AM SKILLS REVIEW HANDBOOK Factors and Multiples Factors are numbers or expressions that are multiplied together. A prime number is a whole number greater than 1 that has exactly two whole number factors, 1 and itself. The table shows all the prime numbers less than 100. A composite number is a whole number greater than 1 that has more than two whole number factors. Prime Numbers Less Than 100 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 When you write a composite number as a product of prime numbers, you are writing its prime factorization. EXAMPLE Write the prime factorization of 60. Use a factor tree. Write 60 at the top. Then draw two branches and write 60 as the product of two factors. Continue to draw branches until all the factors are prime numbers. Two factor trees for 60 are given at the right. Both show 60 5 2 p 2 p 3 p 5. 60 60 2 p 30 3 p 20 2 p 15 2 c The prime factorization of 60 is 2 p 3 p 5. 3 p 5 4 p 5 2 p 2 A whole number that is a factor of two or more nonzero whole numbers is a common factor of the numbers. The largest of the common factors is the greatest common factor (GCF). EXAMPLE Find the greatest common factor (GCF) of 18 and 45. Method 1 List factors. Method 2 Use prime factorization. Factors of 18: 1, 2, 3, 6, 9, 18 Prime factorization of 18: 2 p 3 p 3 Factors of 45: 1, 3, 5, 9, 15, 45 Prime factorization of 45: 3 p 3 p 5 The GCF is 9, the greatest of the common factors. The GCF is the product of the common prime factors: 3 p 3 5 9. A multiple of a whole number is the product of the number and any nonzero whole number. A common multiple of two or more numbers is a multiple of all of the numbers. The least common multiple (LCM) is the smallest of the common multiples. EXAMPLE 978 n2pe-9020.indd 978 Find the least common multiple (LCM) of 12 and 15. Method 1 List multiples. Method 2 Use prime factorization. Multiples of 12: 12, 24, 36, 48, 60, . . . Prime factorization of 12: 22 p 3 Multiples of 15: 15, 30, 45, 60, . . . Prime factorization of 15: 3 p 5 The LCM is 60, the least of the common multiples. Form the LCM of the numbers by writing each prime factor to the highest power it occurs in either number: 22 p 3 p 5 5 60. Student Resources 11/21/05 10:26:48 AM EXAMPLE 3 10 SKILLS REVIEW HANDBOOK The least common denominator (LCD) of two fractions is the least common multiple of the denominators. Use the LCD to add or subtract fractions with different denominators. 5 8 Add: }} 1 } The least common multiple of the denominators, 10 and 8, is 40. So, the least common denominator (LCD) of the fractions is 40. 3 5 3 p 4 5 12 and 5 5 5 p 5 5 25 Rewrite the fractions using the LCD of 40: }} }}} }} } }}} }} 10 10 p 4 40 8 8p5 40 3 1 5 5 12 1 25 5 37 Add the numerators and keep the same denominator: }} } }} }} }} 10 8 40 40 40 PRACTICE Write the prime factorization of the number. If the number is prime, write prime. 1. 42 2. 104 3. 75 4. 23 5. 70 6. 27 7. 72 8. 180 9. 47 10. 100 11. 88 12. 49 14. 142 15. 32 13. 83 Find the greatest common factor (GCF) of the numbers. 16. 4, 6 17. 24, 40 18. 10, 25 19. 55, 44 20. 28, 35 21. 8, 20 22. 5, 8 23. 15, 12 24. 16, 32 25. 70, 90 26. 2, 18 27. 9, 21 28. 36, 42, 54 29. 7, 12, 17 30. 45, 63, 81 Find the least common multiple (LCM) of the numbers. 31. 4, 16 32. 2, 14 33. 5, 6 34. 16, 24 35. 6, 8 36. 12, 20 37. 3, 6 38. 18, 8 39. 9, 12 40. 9, 5 41. 10, 15 42. 7, 9 43. 40, 4, 5 44. 25, 30, 3 45. 27, 81, 33 Perform the indicated operation(s). Simplify the result. 113 46. } } 2 8 32 5 47. } }} 4 16 7 23 48. }} } 5 10 111 49. } } 2 3 5 11 50. }} } 3 12 411 51. } } 5 8 1 13 52. }} } 4 10 521 53. } } 6 2 7 2 11 54. } }} 8 16 9 21 55. }} } 3 10 221 56. } } 3 6 2 1 1} 57. } 5 4 41 1 25 58. } }} } 5 12 6 32 3 23 59. } }} } 4 2 10 9 2121 60. }} } } 5 10 2 71 3 21 61. } }} } 4 16 8 8122 7 62. } } }} 9 3 12 4 11 1 1 }} 63. } } 15 3 6 11211 64. } } } 4 2 3 15 2 7 1 1 65. }} }} } 16 10 2 5 211 7 66. }} } }} 24 6 12 321 1 1} 67. } } 5 4 2 5232 2 68. } } }} 5 6 15 4132 7 69. } } }} 4 9 12 Skills Review Handbook n2pe-9020.indd 979 979 11/21/05 10:26:48 AM SKILLS REVIEW HANDBOOK Ratios and Proportions A ratio uses division to compare two quantities. Three Ways to Write the Ratio of a to b You can write a ratio of two quantities a and b, where b is not equal to 0, in three ways. a to b a b a:b } You should write ratios in simplest form. EXAMPLE Write the ratio of 12 boys to 16 girls in three ways. Boys Girls 12 5 12 4 4 5 3 First write the ratio as a fraction in simplest form: }}} 5 }} }}} } 16 16 4 4 4 3. c Three ways to write the ratio of boys to girls are 3 to 4, 3 : 4, and } 4 A proportion is an equation stating that two ratios are equal. You can use cross multiplication to solve a proportion. EXAMPLE 9 a b c d b. }} 5 } If } 5 }, where b ? 0 and d ? 0, then ad 5 bc. Solve the proportion. n 54 5 } 5 }} a. Using Cross Multiplication to Solve Proportions 5 p 54 5 9 p n 270 5 9n 30 5 n Cross multiply. Simplify. Solve for n. x 40 3 8 x p 8 5 40 p 3 8x 5 120 x 5 15 Cross multiply. Simplify. Solve for x. PRACTICE Write the ratio in simplest form. Express the answer in three ways. 1. 3 to 9 2. 16 to 24 3. 10 to 8 4. 6 to 2 5. 25 to 30 6. 60 to 10 7. 4 to 4 8. 8 to 20 9. 32 to 72 10. 42 to 15 11. 14 to 2 12. 12 to 15 x 5 12 13. }} }} 14 24 8 5 d 14. }} }} 24 36 15 5 3 15. }} } 4 n 9 5 5 16. }} } 45 h a5 4 17. } }} 6 12 13 5 91 18. }} }} t 7 75 5 r 19. }} } 120 8 b 52 20. }} } 90 3 4 5 n 21. }} }} 11 110 5 5 150 22. } }} 90 z 95x 23. } } 8 6 72 5 24 24. }} }} 105 m 17 5 51 25. }} }} 33 a 20 5 24 26. }} }} 125 n 16 5 8 27. }} } 144 x 96 5 t 28. }} } 6 3 Solve the proportion. 980 n2pe-9020.indd 980 Student Resources 11/21/05 10:26:50 AM Converting Units of Measurement SKILLS REVIEW HANDBOOK The table of measures on page 1025 gives many statements of equivalent measures. Using each statement, you can write two different conversion factors. Statement of Equivalent Measures Conversion Factors 100 cm 5 1 m }}}} 5 1 and }}}} 5 1 100 cm 1m 1m 100 cm To convert from one unit of measurement to another, multiply by a conversion factor. Use the one that will eliminate the starting unit and keep the desired unit. EXAMPLE Copy and complete. a. 3.5 m 5 ? cm b. 620 cm 5 ? m 100 cm 5 (3.5 3 100) cm 5 350 cm 3.5 m 3 }}}} 1 m 5 620 m 5 6.2 m 620 cm 3 }}}} }} c So, 3.5 m 5 350 cm. c So, 620 cm 5 6.2 m. 1m 100 cm 100 Sometimes you need to use more than one conversion factor. EXAMPLE Copy and complete: 7 days 5 ? sec Find the appropriate statements of equivalent measures. 24 h 5 1 day, 60 min 5 1 h, and 60 sec 5 1 min 24 h , 60 min , and 60 sec Write conversion factors: }}} }}}} }}} 1 day 1h 1 min Multiply by conversion factors to eliminate days and keep seconds. 24 h 3 60 min 3 60 sec 5 (7 3 24 3 60 3 60) sec 5 604,800 sec 7 days 3 }}} }}}} }}} 1 day 1 min 1h c So, 7 days 5 604,800 sec. PRACTICE Copy and complete. 1. 6 L 5 ? mL 2. 2 mi 5 ? ft 3. 80 oz 5 ? lb 4. 4 days 5 ? h 5. 77 mm 5 ? cm 6. 5 gal 5 ? qt 7. 48 ft 5 ? yd 8. 1500 mL 5 ? L 10. 125 lb 5 ? oz 11. 800 g 5 ? kg 12. 900 sec 5 ? min 13. 72 in. 5 ? ft 14. 2.5 ton 5 ? lb 15. 90 min 5 ? h 16. 65,000 mg 5 ? g 17. 100 yd 5 ? in. 18. 3.5 kg 5 ? g 19. 6 pt 5 ? qt 20. 1 week 5 ? min 21. 2 oz 5 ? lb 22. 1 km 5 ? mm 23. 1 mi 5 ? in. 24. 5 gal 5 ? c 25. 288 in.2 5 ? ft 2 26. 24 pt 5 ? gal 27. 4 kg 5 ? g 28. 7 hr 5 ? sec 9. 40 m 5 ? cm Skills Review Handbook n2pe-9020.indd 981 981 11/21/05 10:26:52 AM SKILLS REVIEW HANDBOOK Scientific Notation Scientific notation is a way to write numbers using powers of 10. A number is written in scientific notation if it has the form c 3 10n where 1 ≤ c < 10 and n is an integer. The table shows some powers of ten in order from least to greatest. Power of Ten 1023 1022 1021 100 101 102 103 Value 0.001 0.01 0.1 1 10 100 1000 EXAMPLE a. 12,800,000 Write the number in scientific notation. Standard form b. 0.0000039 Standard form 12,800,000 Move the decimal point 7 places to the left. 0.0000039 Move the decimal point 6 places to the right. 1.28 3 107 Use 7 as an exponent of 10. 3.9 3 1026 Use 26 as an exponent of 10. EXAMPLE a. 6.1 3 104 Write the number in standard form. Scientific notation b. 5.74 3 1025 Scientific notation 6.1 3 104 The exponent of 10 is 4. 5.74 3 1025 The exponent of 10 is 25. 61,000 Move the decimal point 4 places to the right. 0.0000574 Move the decimal point 5 places to the left. 61,000 Standard form 0.0000574 Standard form PRACTICE Write the number in scientific notation. 1. 0.6 2. 25,000,000 3. 0.08 4. 0.00542 5. 40.8 6. 7 7. 0.000385 8. 8,145,000 9. 41,236 10. 0.0000016 11. 486,000 12. 0.000000009 13. 0.01002 14. 1,000,000,000 15. 7050.5 16. 0.37 17. 9850 18. 0.0000206 19. 805 20. 0.0005 Write the number in standard form. 982 n2pe-9020.indd 982 21. 5 3 103 22. 4 3 1022 23. 8.2 3 1021 24. 6.93 3 102 25. 3.2 3 1023 26. 9.01 3 1025 27. 7.345 3 105 28. 2.38 3 1022 29. 1.814 3 100 30. 2.7 3 108 31. 1 3 106 32. 4.9 3 1024 33. 8 3 1026 34. 5.6 3 104 35. 1.87 3 109 36. 7 3 1024 37. 6.08 3 106 38. 9.009 3 1023 39. 3.401 3 107 40. 5.32 3 101 Student Resources 11/21/05 10:26:53 AM Significant Digits SKILLS REVIEW HANDBOOK Significant digits indicate how precisely a number is known. Use the following guidelines to determine the number of significant digits. • All nonzero digits are significant. • All zeros that appear between two nonzero digits are significant. • For a decimal, all zeros that appear after the last nonzero digit are significant. For a whole number, you cannot tell whether any zeros after the last nonzero digit are significant, so you should assume that they are not significant. Sometimes calculations involve measurements that have various numbers of significant digits. In this case, a general rule is to carry all digits through the calculation and then round the result to the same number of significant digits as the measurement with the fewest significant digits. When you calculate with units that cannot be divided into fractional parts, such as number of people, consider only the significant digits of the other number(s). EXAMPLE a. 12.6 3 0.05 0.63 0.6 Perform the calculation. Write your answer with the appropriate number of significant digits. 3 significant digits 1 significant digit b. 840 2 significant digits 1 702 3 significant digits The product has 2 significant digits. 1542 The sum has 4 significant digits. Round to 1 significant digit. 1500 Round to 2 significant digits. c. $61.20 restaurant bill 4 6 people The number of people is exact, so consider only the 4 significant digits of the bill, $61.20. The answer should have 4 significant digits. $61.20 4 6 5 $10.20 c Each person pays $10.20. PRACTICE Perform the calculation. Write your answer with the appropriate number of significant digits. 1. 600 1 30 2. 5 2 2.6 3. 12 p 6.75 4. 0.098 1 0.14 1 0.369 5. 3.6053 2 1.720 6. 40 4 3.5 7. 8.0 2 3.1 8. 31.7 p 6.8 p 0.435 9. 30.5 p 6.40 13. 4016 2 3007 10. 3.18 1 2.0005 11. 0.088 4 2.44 12. 8650 1 380 2 49 14. 1.35 1 14.8 15. 320 4 18 16. 38.1 p 3.04 4 0.024 17. $1.45 per notebook p 12 notebooks 18. 10.0 liters of water 2 4.5 liters of water 19. 260 pints of milk 4 106 students 20. 0.5 yard of fabric 1 0.87 yard of fabric 21. 27,973 books 4 11 libraries 22. 12.76 gallons of gas 1 6.08 gallons of gas 23. $6.95 per ticket p 180 tickets 24. 1540 pounds 2 160 pounds 2 85 pounds Skills Review Handbook n2pe-9020.indd 983 983 11/21/05 10:26:54 AM SKILLS REVIEW HANDBOOK Writing Algebraic Expressions To solve a problem using algebra, you often need to write a phrase as an algebraic expression. EXAMPLE Write the phrase as an algebraic expression. a. 6 less than a number b. The cube of a number c. Double a number “Less than” indicates subtraction. “Cube” indicates raising to the third power. “Double” indicates multiplication by 2. cn26 c n3 c 2n EXAMPLE Write an algebraic expression to answer the question. a. Rebecca walks three times as far to school as Meghan does. If Meghan walks m blocks to school, how many blocks to school does Rebecca walk? c 3m b. Kate is 8 inches taller than Noah. If Noah is n inches tall, how tall is Kate? cn18 PRACTICE Write the phrase as an algebraic expression. 1. 8 more than a number 2. 10 times a number 3. Twice a number 4. 6 less than a number 5. One fifth of a number 6. 4 greater than a number 7. 5 times a number 8. A number squared 9. 25% of a number 10. Half a number 11. 2 less than a number 12. The square root of a number Write an algebraic expression to answer the question. 13. Allison is 4 years younger than her sister Camille. If Camille is c years old, how old is Allison? 14. Ryan bought a movie ticket for x dollars. He paid with a $20 bill. How much change should Ryan get? 15. Bridget spent $5 more than Tom spent at the mall. If Tom spent x dollars, how much did Bridget spend? 16. Marc has twice as many baseball cards as hockey cards. If Marc has h hockey cards, how many baseball cards does he have? 17. Elizabeth’s ballet class is 45 minutes long. If Elizabeth is m minutes late for ballet class, how many minutes will she spend in class? 18. Steve drove x miles per hour for 5 hours. How many miles did Steve drive? 19. Wendy bought 10 pens priced at x dollars each. How much did she spend? 984 n2pe-9020.indd 984 Student Resources 11/21/05 10:26:55 AM Binomial Products EXAMPLE Simplify (2x 1 1)(x 1 3). Draw a rectangle with dimensions 2x 1 1 and x 1 3. Use the dimensions to divide the rectangle into parts. Then find the area of each part. The binomial product (2x 1 1)(x 1 3) is the sum of the areas of all the parts. 2 There are 2 blue parts with area x , 7 green parts with area x, and 3 yellow parts with area 1. x 2x 1 1 x 1 x x2 x2 x 1 1 1 x x x x x x 1 1 1 x13 (2x 1 1)(x 1 3) 5 2x2 1 7x 1 3 SKILLS REVIEW HANDBOOK A monomial is a number, a variable, or the product of a number and one or more variables. A binomial is the sum of two monomials. In other words, a binomial is a polynomial with two terms. You can use a geometric model to find the product of two binomials. Another way to find the product of two binomials is to use the distributive property systematically. Multiply the first terms, the outer terms, the inner terms, and the last terms of the binomials. This is called FOIL for the words First, Outer, Inner, and Last. EXAMPLE Simplify (x 1 2)(4x 2 5). First Outer Inner Last (x 1 2)(4x 2 5) 5 x(4x) 1 x(25) 1 2(4x) 1 2(25) 2 Use FOIL. 5 4x 2 5x 1 8x 2 10 Multiply. 5 4x2 1 3x 2 10 Combine like terms. PRACTICE Simplify. 1. (a 1 5)(a 1 3) 2. (m 1 4)(m 1 11) 3. (t 1 8)(t 1 7) 4. (z 1 1)(z 1 6) 5. (y 1 4)(y 1 2) 6. (x 1 9)(x 1 9) 7. (y 2 2) 2 8. (n 1 6) 2 9. (4 2 z)2 10. (a 1 10)(a 2 10) 11. (y 1 3)(y 2 7) 12. (k 1 1)2 13. (5x 2 4)(5x 1 4) 14. (3 1 n)2 15. (c 1 5)(2c 2 7) 16. (a 1 5)(a 1 5) 17. (7 2 z)(7 1 z) 18. (3x 2 8)(x 2 6) 20. (3 2 g)(2g 1 3) 21. (4 2 x)(8 1 x) 22. (3n 2 1)(n 2 4) 23. (2a 1 9)(a 2 9) 24. (8x 1 1)(x 1 1) 25. (5x 1 2)(2x 2 5) 26. (2d 2 5)(3d 2 1) 27. (24z 1 3)(6z 2 1) 19. (4a 1 3) 2 Skills Review Handbook n2pe-9020.indd 985 985 11/21/05 10:26:56 AM SKILLS REVIEW HANDBOOK LCDs of Rational Expressions A rational expression is a fraction whose numerator and denominator are nonzero polynomials. The least common denominator (LCD) of two rational expressions is the least common multiple of the denominators. To find the LCD, follow these three steps: STEP 1 Write each denominator as the product of its factors. STEP 2 Write the product consisting of the highest power of each factor that appears in either denominator. STEP 3 Simplify the product from Step 2 to write the LCD. EXAMPLE Find the least common denominator of the rational expressions. 3 and 1 b. }} }} 12x 8x 2 2 and 2 a. }} }}3 5xy y STEP 1 Factors: Factors: 5xy 5 5 p x p y 8x2 5 23 p x2 3 y 5y 3 Factors: 2 STEP 2 Product: 5 p x p y 3 STEP 3 LCD: 5xy x 21 and c. }}} }}}}}} 3x 1 6 x2 2 3x 2 10 3 3x 1 6 5 3 p (x 1 2) 2 12x 5 2 p 3 p x x 2 3x 2 10 5 (x 1 2) p (x 2 5) Product: 23 p 3 p x2 Product: 3 p (x 1 2) p (x 2 5) LCD: 24x 2 LCD: 3(x 1 2)(x 2 5) PRACTICE Find the least common denominator of the rational expressions. 986 n2pe-9020.indd 986 1 and 4 1. }} }} 2ab a2 5 and 6 2. }} }} 6k 2 7k 2 2 and 2 3. }} }} z3 z2 4 and 23 4. }} }} 5x 10x m and 1 5. }} }}} 14 18m 19 and 3 6. }}} }}} 20xy 16xy 1 and 1 7. }} }} 3y 3y 2 24 and 2 8. }}} }}} 9ab2 21a2b n and n2 9. }}} }}} n12 n22 21 and 3 10. }}} }}} x21 x13 28 and 4 11. }}} }}} 5n 1 5 n11 y 1 12. } and }}} 8 2y 1 8 1 2 13. }}}} and }}}} 2m 2 6 3m 2 9 a and 2a 14. }} }}}} n2 n2 2 6n 1 and 1 15. }}} }}}} x24 (x 2 4)2 3 4 16. }}}} and }}}} 4x 1 12 6x 1 18 29 1 and 17. }} }}}}} 2n3 10n2 1 8n 10 17b 18. }}}} and }}}} 15b 2 30 9b 2 18 25 and 3 19. }}}} }}}} (k 1 3)4 (k 1 3)2 8 1 and 20. }}} }}}} y25 3y 2 15 n2 n 21. }}}}} and }}}} 10n 1 20 7n 1 14 20 1 22. }}}} and }}}} 5z 2 40 9z 2 56 2a 2 23. }}}}}} and }}} a12 a2 1 4a 1 4 1 21 24. }}} and }}}}} 2z 2 6 z2 2 z 2 6 3k and 2k 25. }}} }}}}}} k23 k 2 2 5k 1 6 x 2x 26. }}} and }}}}}} x2 2 9 x 2 1 3x 2 18 m2 25 27. }}}}}}} and }}}}}}} 2 m 2 11m 1 28 m2 1 5m 2 45 Student Resources 11/21/05 10:26:58 AM The Coordinate Plane Each point in a coordinate plane is represented by an ordered pair. The first number is the x-coordinate, and the second number is the y-coordinate. The ordered pair (3, 1) is graphed at the right. The x-coordinate is 3, and the y-coordinate is 1. So, the point is right 3 units and up 1 unit from the origin. EXAMPLE y-axis Quadrant II 4 (2, 1) 3 origin 2 (0, 0) y Quadrant I (1, 1) (3, 1) 1 2 3 4 5 6x 262524232221 21 22 23 (2, 2) Quadrant III 24 x-axis (1, 2) Quadrant IV SKILLS REVIEW HANDBOOK A coordinate plane is formed by the intersection of a horizontal number line called the x-axis and a vertical number line called the y-axis. The axes meet at a point called the origin and divide the coordinate plane into four quadrants, numbered I, II, III, and IV. Graph the points A(2, 21) and B(24, 0) in a coordinate plane. A(2, 21) Start at the origin. The x-coordinate is 2, so move right 2 units. The y-coordinate is 21, so move down 1 unit. Draw a point at (2, 21) and label it A. B(24, 0) Start at the origin. The x-coordinate is 24, so move left 4 units. The y-coordinate is 0, so move up 0 units. Draw a point at (24, 0) and label it B. 4 3 2 1 B(24, 0) y 3 4 5 6x 1 262524232221 21 22 23 24 A(2, 21) PRACTICE Graph the points in a coordinate plane. 1. A(7, 2) 2. B(6, 27) 3. C(2, 23) 4. D(28, 0) 5. E(24, 28) 6. F(1, 3) 7. G(3, 0) 8. H(1, 25) 9. I(0, 22) 10. J(26, 5) 11. K(5, 8) 12. L(8, 22) 13. M(23, 24) 14. N(27, 8) 15. P(25, 1) 16. Q(22, 26) 17. R(0, 6) 18. S(24, 21) 19. T(4, 4) 20. V(23, 7) Give the coordinates and the quadrant or axis of the point. 21. A 24. D 22. B 25. E 23. C 5 26. F 27. G 28. H 29. J 30. K 31. L 32. M 33. N 34. O 35. P 36. Q 37. R 38. S 39. T 40. U 41. V 42. W 43. X 44. Y T 4 F 3 N 2 1 D y A M G S W U E L 26 24 22 O R V H 3 4 5 6x 1 K J B23 X Œ 24 Y P C Skills Review Handbook n2pe-9020.indd 987 987 11/21/05 10:26:59 AM SKILLS REVIEW HANDBOOK Transformations A transformation is a change made to the position or to the size of a figure. Each point (x, y) of the figure is mapped to a new point, and the new figure is called an image. A translation is a transformation in which each point of a figure moves the same distance in the same direction. A figure and its translated image are congruent. EXAMPLE Translation a Units Horizontally and b Units Vertically (x, y) → (x 1 a, y 1 b) Translate } FG right 3 units and down 1 unit. y 13 F 21 F9 To move right 3 units, use a 5 3. To move down 1 unit, use b 5 21. So, use (x, y) → (x 1 3, y 1 (21)) with each endpoint. 1 F(2, 4) → F9(2 1 3, 4 1 (21)) 5 F9(5, 3) G(1, 1) → G9(1 1 3, 1 1 (21)) 5 G9(4, 0) G G9 1 x Graph the endpoints (5, 3) and (4, 0). Then draw the image. A reflection is a transformation in which a figure is reflected, or flipped, in a line, called the line of reflection. A figure and its reflected image are congruent. EXAMPLE Reflection in x-axis Reflection in y-axis (x, y) → (x, 2y) (x, y) → (2x, y) Reflect n ABC in the y-axis. y A9 A B9 B Use (x, y) → (2x, y) with each vertex. A(4, 3) → A9(24, 3) B(1, 2) → B9(21, 2) C(3, 1) → C9(23, 1) 1 C9 Change each x-coordinate to its opposite. C x 1 Graph the new vertices. Then draw the image. A rotation is a transformation in which a figure is turned about a fixed point, called the center of rotation. The direction can be clockwise or counterclockwise. A figure and its rotated image are congruent. EXAMPLE Rotation About the Origin 1808 either direction (x, y) → (2x, 2y) 908 clockwise (x, y) → (y, 2x) 908 counterclockwise (x, y) → (2y, x) Rotate RSTV 1808 about the origin. y R(2, 2) → R9(22, 22) S(4, 2) → S9(24, 22) T(4, 1) → T9(24, 21) V(1, 0) → V9(21, 0) Change every coordinate to its opposite. R 2 Use (x, y) → (2x, 2y) with each vertex. S V9 S9 T V T9 2 x R9 Graph the new vertices. Then draw the image. 988 n2pe-9020.indd 988 Student Resources 11/21/05 10:27:01 AM EXAMPLE Dilation with Scale Factor k with Respect to the Origin SKILLS REVIEW HANDBOOK A dilation is a transformation in which a figure stretches or shrinks depending on the dilation’s scale factor. A figure stretches if k > 1 and shrinks if 0 < k < 1. A figure and its dilated image are similar. (x, y) → (kx, ky) Dilate JKLM using a scale factor of 0.5. The scale factor is k 5 0.5, so multiply every coordinate by 0.5. Use (x, y) → (0.5x, 0.5y) with each vertex. J(4, 4) → J9(0.5 p 4, 0.5 p 4) 5 J9(2, 2) K(6, 4) → K9(0.5 p 6, 0.5 p 4) 5 K9(3, 2) L(6, 21) → L9(0.5 p 6, 0.5 p (21)) 5 L9(3, 20.5) M(4, 21) → M9(0.5 p 4, 0.5 p (21)) 5 M9(2, 20.5) y J J9 K K9 1 M9 L9 5 M x L Graph the new vertices. Then draw the image. PRACTICE Find the coordinates of N(23, 8) after the given transformation. For rotations, rotate about the origin. 1. Rotate 1808. 2. Reflect in x-axis. 3. Translate up 3 units. 4. Reflect in y-axis. 5. Rotate 908 clockwise. 6. Translate left 5 units. 7. Rotate 908 counterclockwise. 8. Translate right 2 units and down 9 units. Transform n PST. Graph the result. For rotations, rotate about the origin. 9. Reflect in x-axis. y T 10. Rotate 908 counterclockwise. 11. Rotate 908 clockwise. 12. Translate down 7 units. 13. Reflect in y-axis. 14. Translate left 4 units. 15. Rotate 1808. 16. Translate right 2 units. x 1 22 P S 17. Translate right 1 unit and up 4 units. 18. Translate left 6 units and up 2 units. The coordinates of the vertices of a polygon are given. Draw the polygon. Then find the coordinates of the vertices of the image after the specified dilation, and draw the image. 19. (1, 3), (3, 2), (2, 5); dilate using a scale factor of 3 3 20. (2, 8), (2, 4), (6, 8), (6, 4); dilate using a scale factor of } 2 1 21. (3, 3), (6, 3), (3, 23), (6, 23); dilate using a scale factor of } 3 22. (2, 2), (2, 7), (5, 7); dilate using a scale factor of 2 1 23. (2, 22), (6, 22), (4, 26), (0, 26); dilate using a scale factor of } 2 Skills Review Handbook n2pe-9020.indd 989 989 11/21/05 10:27:02 AM SKILLS REVIEW HANDBOOK Line Symmetry A figure has line symmetry if a line, called a line of symmetry, divides the figure into two parts that are mirror images of each other. Below are four figures with their lines of symmetry shown in red. Trapezoid No lines of symmetry EXAMPLE Isosceles Triangle 1 line of symmetry Rectangle 2 lines of symmetry Regular Hexagon 6 lines of symmetry A line of symmetry for the figure is shown in red. Find the coordinates of point A. Point A is the mirror image of the point (3, 26) with respect to the line of symmetry y 5 22. The x-coordinate of A is 3, the same as the x-coordinate of (3, 26). Because 26 is the y-coordinate of (3, 26), and 22 2 (26) 5 4, the point (3, 26) is down 4 units from the line of symmetry. Therefore, point A must be up 4 units from the line of symmetry. So, the y-coordinate of A is 22 1 4 5 2. The coordinates of point A are (3, 2). y A 1 x 2 y 5 22 C B(3, 26) PRACTICE Tell how many lines of symmetry the figure has. 1. 2. 3. 4. 5. A parallelogram 6. A square 7. A rhombus 8. An equilateral triangle A line of symmetry for the figure is shown in red. Find the coordinates of point A. 9. 4 (24, 3) 10. y 11. y (0, 4) A 1x 990 n2pe-9020.indd 990 y5x 1 y51 A y A x52 1 1 1 x x (2, 22) Student Resources 11/21/05 10:27:03 AM Perimeter and Area SKILLS REVIEW HANDBOOK The perimeter P of a figure is the distance around it. To find the perimeter of a figure, add the side lengths. EXAMPLE Find the perimeter of the figure. a. b. 13 in. 5 in. 18 m 4m 4m 18 m 12 in. P 5 5 1 12 1 13 5 30 in. P 5 2(4) 1 2(18) 5 8 1 36 5 44 m The area A of a figure is the number of square units enclosed by the figure. Area of a Triangle Area of a Rectangle Area of a Parallelogram Area of a Trapezoid b1 h w h l b 1 2 b A 5 lw A 5 }bh EXAMPLE h b2 1 2 A 5 }(b1 1 b2)h A 5 bh Find the area of the figure. a. b. c. 7 in. 6m 5 ft 3m 15 in. A 5 (15)(7) 5 105 in.2 A 5 (5)(5) 5 25 ft 2 1 (6)(3) 5 9 m 2 A5} 2 PRACTICE Find the perimeter and area of the figure. 1. 2. 8 ft 3 cm 17 ft 3. 3 in. 15 ft 4. 4 in. 5 in. 5m 12 in. 6m 2 cm 5. 6. 10 yd 8 yd 7. 8. 12 mm 17 yd 8 in. 2.7 m 3m 9 mm 9 mm 21 yd 4m 12 mm Skills Review Handbook n2pe-9020.indd 991 991 11/21/05 10:27:04 AM SKILLS REVIEW HANDBOOK Circumference and Area of a Circle A circle consists of all points in a plane that are the same distance from a fixed point called the center. The distance between the center and any point on the circle is the radius. The distance across the circle through the center is the diameter. The diameter is twice the radius. circle radius diameter center The circumference of a circle is the distance around the circle. For any circle, the ratio of the circumference to the diameter is π (pi), an irrational number 22 . that is approximately 3.14 or }} 7 To find the circumference C of a circle with radius r, use the formula C 5 2πr. To find the area A of a circle with radius r, use the formula A 5 πr 2. EXAMPLE Find the circumference and area of a circle with radius 6 cm. Give an exact answer and an approximate answer for each. Circumference Area C 5 2πr A 5 πr 2 5 2π(6) 5 π(6)2 5 12π 5 36π < 12(3.14) < 36(3.14) < 37.7 < 113 c The circumference is 12π centimeters, or about 37.7 centimeters. 6 cm c The area is 36π square centimeters, or about 113 square centimeters. PRACTICE Find the circumference and area of the circle. Give an exact answer and an approximate answer for each. 1. 2. 3. 4. 5 in. 10 m 2 cm 5. 6. 4 in. 7. 8. 6 ft 12 ft 16 m 9 cm 9. 10. 2 cm 992 n2pe-9020.indd 992 11. 14 ft 12. 22 in. 36 cm Student Resources 11/21/05 10:27:05 AM Surface Area and Volume SKILLS REVIEW HANDBOOK A solid is a three-dimensional figure that encloses part of space. The surface area S of a solid is the area of the solid’s outer surface(s). The volume V of a solid is the amount of space that the solid occupies. Cylinder Rectangular Prism h S 5 2lw 1 2lh 1 2wh V 5 lwh w l EXAMPLE r 2 S 5 2πr 1 2πrh h V 5 πr 2h Find the surface area and volume of the rectangular prism. Surface area Volume S 5 2lw 1 2lh 1 2wh V 5 lwh 7 ft 5 2(5)(3) 1 2(5)(7) 1 2(3)(7) 5 (5)(3)(7) 5 30 1 70 1 42 5 105 ft 3 5 ft 3 ft 5 142 ft 2 EXAMPLE Find the surface area and volume of the cylinder. Surface area 3m Volume 2 12 m 2 V 5 πr h S 5 2πr 1 2πrh 2 5 2π(3) 1 2π(3)(12) 5 π(3)2 (12) 5 90π m 2 Exact answer 5 108π m3 Exact answer < 283 m 2 Approximate answer < 339 m3 Approximate answer PRACTICE Find the surface area and volume of the solid. 1. 2. 6.5 mm 3. 3 in. 3 cm 12 mm 5 in. 3 cm 3 cm 8 in. 4. 2m 4m 5. 6. 14 yd 10 ft 4 yd 10 m 15 ft Skills Review Handbook n2pe-9020.indd 993 993 11/21/05 10:27:07 AM SKILLS REVIEW HANDBOOK Angle Relationships An angle bisector is a ray that divides an angle into two congruent angles. Two angles are complementary angles if the sum of their measures is 908. Two angles are supplementary angles if the sum of their measures is 1808. EXAMPLE Find the value of x. ]› a. BD bisects ∠ ABC and m ∠ ABC 5 648. C b. ∠ GFJ and ∠ HFJ are J D x8 648 ]› Because BD bisects ∠ ABC, the value of x is half m ∠ ABC. 64 5 32 x 5 }} 2 D 4x 8 (3x 2 1)8 F G A supplementary. H (2x 2 6)8 B c. ∠ CBD and ∠ ABD are complementary. E C Because ∠ GFJ and ∠ HFJ are complementary angles, their sum is 908. (x 2 3)8 B A Because ∠ CBD and ∠ ABD are supplementary angles, their sum is 1808. (2x 2 6) 1 4x 5 90 (3x 2 1) 1 (x 2 3) 5 180 6x 2 6 5 90 4x 2 4 5 180 x 5 16 x 5 46 PRACTICE ]› BD is the angle bisector of ∠ ABC. Find the value of x. 1. 2. A D 3. A D 248 788 x8 C B A (11x 2 19)8 (8x 1 5)8 B (2x 2 4)8 B D C C ∠ ABD and ∠ DBC are complementary. Find the value of x. 4. 5. A 6. B D A A (3x 2 18)8 (3x 2 4)8 (7x 1 5)8 D (4x 1 10)8 B D (5x 2 20)8 C (5x 1 1)8 C C B ∠ ABD and ∠ DBC are supplementary. Find the value of x. 7. 8. D (x 2 28)8 (4x 1 17)8 (3x 1 2)8 A 994 n2pe-9020.indd 994 B A C 9. D D 3x 8 B (4x 1 2)8 (2x 1 4)8 C A B C Student Resources 11/21/05 10:27:08 AM Triangle Relationships SKILLS REVIEW HANDBOOK The sum of the angle measures of any triangle is 1808. EXAMPLE Find the value of x. 60 1 35 1 x 5 180 x8 95 1 x 5 180 608 358 x 5 85 The sum of the angle measures is 1808. Simplify. Solve for x. In a right triangle, the hypotenuse is the side opposite the right angle. The legs are the sides that form the right angle. The Pythagorean theorem states that the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse. Pythagorean Theorem a2 1 b2 5 c 2 c a b EXAMPLE Find the value of x. a. b. 12 cm x 6 ft x 13 cm 8 ft 6 2 1 82 5 x 2 2 Simplify. 100 5 x2 Simplify. 36 1 64 5 x x 5 10 ft x2 1 122 5 132 Pythagorean theorem Pythagorean theorem 2 x 1 144 5 169 Simplify. x2 5 25 Solve for x2 . x 5 5 cm Solve for x. Solve for x. PRACTICE Find the value of x. 1. 2. 688 378 3. x8 4. x8 348 728 x8 x8 5. 6. 40 cm x 568 x8 7. 8 in. 8 in. x x8 37 ft 35 ft 30 cm 8. 5m 6m x x 9. A triangle with angles that measure x8, x8, and 708 Skills Review Handbook n2pe-9020.indd 995 995 11/21/05 10:27:09 AM SKILLS REVIEW HANDBOOK Congruent and Similar Figures Two figures are congruent if they have the same shape and the same size. If two figures are congruent, then corresponding angles are congruent and corresponding sides are congruent. The triangles at the right are congruent. Matching arcs show congruent angles, and matching tick marks show congruent sides. Two figures are similar if they have the same shape but not necessarily the same size. If two figures are similar, then corresponding angles are congruent and the ratios of the lengths of corresponding sides are equal. EXAMPLE Tell whether the figures are congruent, similar, or neither. a. 7 3 3 3 3 7 7 b. 10 11 As shown, corresponding angles are congruent, but corresponding sides have different lengths. So, the figures are not congruent, but they may be similar. F 3.75 E A 3 B D As shown, corresponding angles are congruent and corresponding sides are congruent. So, the figures are congruent. 7 6 7.5 C G 12.5 13.75 H The figures are similar if the ratios of the lengths of corresponding sides are equal. BC FG 3 3.75 AB EF }} 5 }} 5 0.8 6 7.5 }} 5 }} 5 0.8 CD GH 11 13.75 }} 5 }}} 5 0.8 AD EH 10 12.5 }} 5 }} 5 0.8 c Because corresponding angles are congruent and the ratios of the lengths of corresponding sides are equal, ABCD is similar to EFGH. EXAMPLE The two polygons are similar. Find the value of x. a. 608 x8 The angle with measure x° corresponds to the angle with measure 60°, so x 5 60. 308 b. 8 12 x 9 The side with length 12 corresponds to the side with length 8, and the side with length 9 corresponds to the side with length x. 12 8 9 x }} 5 } 12x 5 72 x56 996 n2pe-9020.indd 996 Write a proportion. Cross multiply. Solve for x. Student Resources 11/21/05 10:27:10 AM PRACTICE SKILLS REVIEW HANDBOOK Tell whether the figures are congruent, similar, or neither. Explain. 1. 2. 4 2 6 14 11 3.5 2 3. 1 10 7 14 3 4. 5. 7 6. 8 12 9 9 7 7 12 8 8. 6 4.5 9 1.5 12 1.6 3.2 5 1.5 3 9. 2.4 4.8 4 3 3 8 8 5 7. 12 6 4.8 7 3 3.2 7 1.6 3 3 2.4 The two polygons are similar. Find the value of x. 10. 11. 438 7 x21 12. 12 14 20 18 8 4x 1 3 x8 13. 14. 1138 34.5 27 15. 568 568 678 (5x 2 3)8 16. 18 17. 538 568 568 1248 5x 2 7 538 18. 1198 618 1198 36 15 30 x15 (3x 1 4)8 (11x 2 5)8 1198 618 (7x 1 4)8 538 Skills Review Handbook n2pe-9020.indd 997 997 11/21/05 10:27:11 AM SKILLS REVIEW HANDBOOK More Problem Solving Strategies Problem solving strategies can help you solve mathematical and real-life problems. Lesson 1.5 shows how to apply the strategies use a formula, look for a pattern, draw a diagram, and use a verbal model. Below are four more strategies. Strategy When to Use How to Use Make a list or table Make a list or table when a problem requires you to record, generate, or organize information. Make a table with columns, rows, and any given information. Generate a systematic list that can help you solve the problem. Work backward Work backward when a problem gives you an end result and you need to find beginning conditions. Work backward from the given information until you solve the problem. Work forward through the problem to check your answer. Guess, check, and revise Guess, check, and revise when you need a place to start or you want to see how the problem works. Make a reasonable guess. Check to see if your guess solves the problem. If it does not, revise your guess and check again. Solve a simpler problem Solve a simpler problem when a problem can be made easier by using simpler numbers. Think of a way to make the problem simpler. Solve the simpler problem, then use what you learned to solve the original problem. EXAMPLE Lee works as a cashier. In how many different ways can Lee make $.50 in change using quarters, dimes, and nickels? Use the strategy make a list or table. Then count the number of different ways. Quarters Dimes Nickels 2 0 0 1 2 1 1 1 3 1 0 5 0 5 0 0 4 2 0 3 4 0 2 6 0 1 8 0 0 10 Start with the greatest number of quarters. Then list all the possibilities with 1 quarter, starting with the greatest number of dimes. Then list all the possibilities with 0 quarters, starting with the greatest number of dimes. c Lee can make $.50 in quarters, dimes, and nickels in 10 different ways. EXAMPLE In a cafeteria, 3 cookies cost $.50 less than a sandwich. If a sandwich costs $4.25, how much does one cookie cost? Use the strategy work backward. 4.25 2 0.50 5 3.75 Cost of 3 cookies 3.75 4 3 5 1.25 Cost of 1 cookie CHECK 1.25 3 3 5 3.75 Cost of 3 cookies 3.75 1 0.50 5 4.25 Cost of sandwich c One cookie costs $1.25. 998 n2pe-9020.indd 998 Student Resources 11/21/05 10:27:12 AM Nolan’s class has 6 more boys than girls. There are 28 students altogether. How many girls are in Nolan’s class? Use the strategy guess, check, and revise. Guess a number of girls that is less than half of 28. First guess: 12 girls, 12 1 6 5 18 boys, 12 1 18 5 30 students Too high ✗ Second guess: 10 girls, 10 1 6 5 16 boys, 10 1 16 5 26 students Too low ✗ Third guess: Correct ✓ 11 girls, 11 1 6 5 17 boys, 11 1 17 5 28 students c There are 11 girls in Nolan’s class. EXAMPLE SKILLS REVIEW HANDBOOK EXAMPLE How many diagonals does a regular decagon have? Use the strategy solve a simpler problem. A decagon has 10 sides, so find the number of diagonals of polygons with fewer sides and look for a pattern. 3 sides 0 diagonals 4 sides 2 diagonals 5 sides 5 diagonals 6 sides 9 diagonals 7 sides 14 diagonals Notice that the difference of the numbers of diagonals for consecutive figures keeps increasing by 1: 22052 52253 92554 14 2 9 5 5 So, an 8-sided polygon has 14 1 6 5 20 diagonals, a 9-sided polygon has 20 1 7 5 27 diagonals, and a 10-sided polygon has 27 1 8 5 35 diagonals. c A regular decagon (a 10-sided polygon) has 35 diagonals. PRACTICE 1. Ben has a concert at 7:30 P.M. First he must do 2 hours of homework. Then, dinner and a shower will take about 45 minutes. Ben wants to allow a half hour to get to the concert. What time should Ben start his homework? 2. Quinn and Kyle collected 87 aluminum cans to recycle. Quinn collected twice as many cans as Kyle. How many cans did each person collect? 3. In how many different ways can three sisters form a line at a ticket booth? 4. The 8 3 8 grid at the right has some 1 3 1 squares, some 2 3 2 squares, some 3 3 3 squares, and so on. How many total squares does the grid have? 5. If Kaleigh draws 20 different diameters in a circle, into how many parts will the circle be divided? 6. Six friends form a tennis league. Each friend will play a match with every other friend. How many matches will be played? 7. Susan has 13 coins in her pocket with a total value of $1.05. She has only dimes and nickels. How many of each type of coin does Susan have? Skills Review Handbook n2pe-9020.indd 999 999 11/21/05 10:27:13 AM ALGEBRA 2 n2pe-0010.indd i 10/6/05 2:32:32 PM About Algebra 2 The content of Algebra 2 is organized around families of functions, including linear, quadratic, exponential, logarithmic, radical, and rational functions. As you study each family of functions, you will learn to represent them in multiple ways—as verbal descriptions, equations, tables, and graphs. You will also learn to model real-world situations using functions in order to solve problems arising from those situations. In addition to its algebra content, Algebra 2 includes lessons on probability and data analysis as well as numerous examples and exercises involving geometry and trigonometry. These math topics often appear on standardized tests, so maintaining your familiarity with them is important. To help you prepare for standardized tests, Algebra 2 provides instruction and practice on standardized test questions in a variety of formats—multiple choice, short response, extended response, and so on. Technology support for both learning algebra and preparing for standardized tests is available at classzone.com. n2pe-0010.indd ii 10/6/05 2:32:34 PM ALGEBRA 2 Ron Larson Laurie Boswell Timothy D. Kanold Lee Stiff n2pe-0010.indd iii 10/6/05 2:32:36 PM Copyright © 2007 McDougal Littell, a division of Houghton Mifflin Company. All rights reserved. Warning: No part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying and recording, or by any information storage or retrieval system without the prior written permission of McDougal Littell unless such copying is expressly permitted by federal copyright law. Address inquiries to Supervisor, Rights and Permissions, McDougal Littell, P.O. Box 1667, Evanston, IL 60204. ISBN-13: 978-0-6185-9541-9 ISBN-10: 0-618-59541-4 123456789—DWO—09 08 07 06 05 Internet Web Site: http://www.mcdougallittell.com iv n2pe-0010.indd iv 10/6/05 2:32:40 PM CHAPTER 2 Linear Functions, p. 76 P(d) 5 1 1 0.03d Unit 1 Linear Equations, Inequalities, Functions, and Systems Linear Equations and Functions Prerequisite Skills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 2.1 Represent Relations and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 2.2 Find Slope and Rate of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 2.3 Graph Equations of Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Graphing Calculator Activity Graph Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 2.4 Write Equations of Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Problem Solving Workshop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Mixed Review of Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 2.5 Model Direct Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 2.6 Draw Scatter Plots and Best-Fitting Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Investigating Algebra: Fitting a Line to Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 2.7 Use Absolute Value Functions and Transformations . . . . . . . . . . . . . . . . . . . . . . 123 Investigating Algebra: Exploring Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 2.8 Graph Linear Inequalities in Two Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Mixed Review of Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 ASSESSMENT Quizzes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96, 120, 138 Chapter Summary and Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 ★ Standardized Test Preparation and Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 "MHFCSB Activities . . . . 71, 73, 86, 90, 95, 98, 102, 107, 115, 133 DMBTT[POFDPN Chapter 2 Highlights PROBLEM SOLVING ★ ASSESSMENT • Mixed Review of Problem Solving, 106, 139 • Multiple Representations, 95, 104, 105, 119, 129 • Multi-Step Problems, 78, 88, 95, 103, 106, 137, 139 • Using Alternative Methods, 105 • Real-World Problem Solving Examples, 74, 76, 85, 91, 100, 108, 115, 125, 134 • Standardized Test Practice Examples, 82, 132 • Multiple Choice, 77, 86, 87, 93, 102, 109, 110, 118, 127, 128, 136 • Short Response/Extended Response, 77, 78, 79, 87, 88, 94, 95, 103, 106, 111, 119, 128, 129, 136, 137, 139, 146 • Writing/Open-Ended, 76, 86, 87, 93, 94, 101, 103, 106, 109, 110, 117, 118, 127, 128, 135, 136, 139 TECHNOLOGY At classzone.com: • Animated Algebra, 71, 73, 86, 90, 95, 98, 102, 107, 115, 133 • @Home Tutor, 70, 78, 87, 94, 97, 103, 110, 119, 121, 128, 137, 141 • Online Quiz, 79, 88, 96, 104, 111, 120, 129, 138 • Electronic Function Library, 140 • State Test Practice, 106, 139, 149 Contents n2pe-0050.indd ix ix 9/23/05 2:02:50 PM Sele elecc ted Answers Chapter 1 a b c d a d b c ad 5} bc Definition of division ad 5} Commutative property of multiplication 55. } 4 } 5 } p } 1.1 Skill Practice (pp. 6–7) 1. reciprocal 3. 23 21 1 3 5 22 21 0 1 2 5. cb Definition of multiplication of fractions 11. Associative property of addition 13. Commutative property of multiplication 15. Distributive property a d 5} p} 1 17. 6 p (a 4 3) 5 6 p a p } 3 1 56p }pa 3 Definition of multiplication of fractions a b 5} 4} Definition of division 1 1 2 2 1 5 6p} pa 3 1 2 Definition of division Commutative property of multiplication Associative property of multiplication 5 2a 1 23. Sample answer: a 5 22, b 5 } 25. $8.50 per h 4 2 27. $36.25 29. 195 mi 31. 116 } yd 33. 2200 g 3 35. 1.75 gal 37. 0.00175 ton 39. The unit multiplier 0.82 euro 0.82 euro should be } ; 25 dollars p } 5 20.5 euros. 1 dollar 1 dollar 41. 29.3 ft/sec 43. 31.1 mi/h 45. 0.04 oz/sec 47. 1800 mi/h 49. Always; this represents the associative property of addition, which is true for all real numbers. 51. Sometimes; it is true for b ≤ 0 and c ≤ 0. 53. Always; this represents the distributive property, which is true for all real numbers. c b d 1.1 Problem Solving (pp. 8–9) 57. a. Lance: 6, Darcy: 2, Javier: 3, Sandra: 22 b. Sandra, Darcy, Javier, Lance 59. a. Pluto, Neptune, Uranus, Saturn, Jupiter, Mars, Earth, Mercury, Venus b. Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, Pluto c. Sample answer: The planets are in opposite orders in parts (a) and (b) with the exception of Mercury and Venus. d. Mercury or Venus 61. a. cheetah: 102.67; three-toed sloth: 0.15; squirrel: 17.6; grizzly bear: 30 b. Sample answer: The cheetah is about 467 times faster than the three-toed sloth. 1.2 Skill Practice (pp. 13–15) 1. base: 12, exponent: 7 3. The negative sign should be applied after evaluating the power, 234 5 281. 5. 81 7. 49 9. 232 11. 210,000 13. 264 15. 64 17. 25 19. 2100 21. 75 23. 6 25. 5x 1 5 27. 13z 2 2 2z 1 10 29. 11m 2 1 SELECTED ANSWERS Multiplication Definition of subtraction 5 c 1 ((23) 1 3) Associative property of addition 5c10 Inverse property of addition 5c Identity property of addition 21. 7a 1 (4 1 5a) 5 7a 1 (5a 1 4) Commutative property of addition 5 (7a 1 5a) 1 4 Associative property of addition 5 12a 1 4 Combine like terms. 19. (c 2 3) 1 3 5 (c 1 (23)) 1 3 c 1 31. 25p 2 1 21 35. 10n 1 24; 44 37. 26 39. 49 41. } 9 43. 27d 1 11c 45. 2m 2 1 n 2 2 8m 47. 13m 2 2 5 49. 28s 1 8t 51. Sample answer: 3k 1 4k 1 (28) 2 2j ; 7k 2 8 2 2j 53. (4 1 3) p (5 2 2) 5 21 55. (3 p 4)2 2 (23 1 3)2 5 23 1.2 Problem Solving (pp. 15–16) 57. 0, 10, 20, 30; $1.89, $3.20, $4.51, $5.82 59. 270 2 4.5x; no; when x > 60 there will be a negative balance on the card, which means you will have spent more than what you had on the card. 63. 26.5x 2 6y 1 200; $88 1.3 Skill Practice (pp. 21–23) 1. solution 3. 3 5. 12 2 7. 6 9. 2 } 11. 4 13. 21 15. 18 17. 29 21. 1 23. 4 9 1 2 25. 27 27. 21 } 29. 4 31. 22 } 33. 4 35. 27 37. 22 3 3 39. 28 41. Both sides of the equations should be 3 3 divided by } instead of subtracting } from each side; 7 7 3 3 } x 5 15, x 5 15 4 }, x 5 35. 43. 12 45. 60 47. 223 7 7 Selected Answers n2pe-9090.indd SA1 SA1 12/1/05 12:19:41 PM 2 49. 1} 51. 6; 15, 8 53. 2; 6, 6, 3 55. 4 57. 2 59. 4 3 29. 3x 1 18 5 72, 18 in., 24 in., 30 in. 61. 2.9 63. no solution 65. all real numbers d2b a2c 67. x 5 }; a 5 c and b ? d or a ? c and b 5 d; x 1 12 a 5 c and b 5 d x16 1.3 Problem Solving (pp. 23–24) 69. 3 h 71. 9 h 5 1 1 1 73. a. 3c 1 2g 5 8 b. 2 } ; }; 2 } ; } 75. 18 min 4 2 12 4 A 1.4 Skill Practice (pp. 30–31) 1. formula 3. l 5 }; w 2A 5 mm 5. h 5 } ; 6 cm 7. y 5 26 2 3x; 5 b1 1 b2 6 31 3 21 9. y 5 2 } x 1 }; 11 11. y 5 } x 2 } ; 23 5 5 2 2 7 11 13. y 5 } x 2 } ; 6 17. The variable y should only 4 4 appear on one side of the equation, not both; SELECTED ANSWERS S 9 4y 2 xy 5 9, y(4 2 x) 5 9, y 5 } . 19. h 5 } 2 k; 42x pr 40 1 3x 16x 1 28 about 4.96 cm 21. y 5 } ; 11 23. y 5 } ; x 3x 15 5 2 7} 25. y 5 }; 5 27. Method 1: y 5 } x 2 3, 3 1 2 2x 3 5 1 y5} p 2 2 3, y 5 } ; Method 2: 15 p 2 2 9y 5 27, 3 3 1 30 2 9y 5 27, 29y 5 23, y 5 } ; Sample answer: 3 Method 1 is more efficient because it is already x1y xy 2 1 xy xy 2 y 2 x solved for y. 29. z 5 } 31. z 5 } C 1.4 Problem Solving (pp. 31–32) 33. d 5 }; about 36 in. p 5 35. C 5 } (F 2 32); 108C 37. R 5 80c 1 150d; 9 R 2 80c d5} ; 80 designer tuxedos; 160 designer 150 tuxedos; 240 designer tuxedos b. Sample answer: r 5 1.5, R 5 2.25; r 5 1.15, R 5 3.83; r 5 0.849, 2 2 4p 4p lw w l R 5 7.03 39. V 5 } ;V5} x 31. 40x 1 7(20 2 x ) 5 404; 8 boxes of books, 12 boxes of clothes 33. about 4.07 in. 1.6 Skill Practice (pp. 44–45) 1. solution 3. 0 22 2 4 6 5. 26 24 0 22 2 11. 23 ≤ x ≤ 1 13. x < 22 or x > 4 17. 24 0 22 2 4 19. 22 23. x ≤ 22 25. x < 4 0 2 4 23 22 0 22 6 0 21 2 1 4 6 35. The inequality symbol should not be reversed when subtracting; 10 > 2x, 5 > x. 37. 26 < x < 3 39. 1 < x ≤ 7 28 24 0 4 2 4 6 8 0 43. x < 24 or x > 2 1 45. x ≤ 2 } or x ≥ 1 2 24 22 0 22 21 8 0 2 1 4 2 49. no solution 51. no solution 1.6 Problem Solving (pp. 46–47) 53. 45x 1 35 ≤ 250, 7 x ≤ 4} days 55. a. 0 ≤ e < 500 b. 1400 ≤ e < 2429 9 pattern shows the output is decreased by 10 each time; an equation that represents the table is y 5 75 2 10x. 23. y 5 7x 2 16 c. 0 ≤ e < 500 or 1400 ≤ e < 2429 57. 50 ≤ F ≤ 95; 10 ≤ C ≤ 35 59. a. Amy: 0.65(84) 1 0.15(80) 1 0.2w ≥ 85, Brian: 0.65(80) 1 0.15(100) 1 0.2x ≥ 85, Clara: 0.65(75) 1 0.15(95) 1 0.2y ≥ 85, Dan: 0.65(80) 1 0.15(90) 1 0.2z ≥ 85 b. w ≥ 92; x ≥ 90; y ≥ 110; z ≥ 97.5 c. Amy, Brian, and Dan. Sample answer: It is impossible to score over 100 points on a test, so Clara will not be able to achieve a grade of 85 or better. 1.5 Problem Solving (pp. 38–39) 25. 3.75 km/min 1.6 Problem Solving Workshop (p. 49) 27. y 5 1.5x 1 15; no; the bamboo shoot will 1. y 5 235x 1 200; x < 20 3. x ≥ $7000 1.5 Skill Practice (pp. 37–38) 1. verbal model 3. 0.5 h 5. 90 mi 7. 54 ft 9. 20 m 11. y 5 4x 1 11 13. y 5 46 2 10x 17. 4x 1 9 5 12, 0.75 ft 19. The eventually slow it’s growth rate and stop growing. 1.7 Skill Practice (pp. 55–56) 1. An apparent solution that must be rejected because it does not satisfy the original equation. 3. solution 5. not a solution 7. solution SA2 n2pe-9090.indd SA2 Selected Answers 12/1/05 12:19:44 PM 9. 29, 9 11. 0 212 0 26 6 5. domain: 22, 1, 6, range: 23, 21, 5, 8 12 y 24 0 22 2 4 6 7 4 21. 24, 9 23. }, 2 25. 27, 4 27. 27, 2 29. 1 } ;3 9 Input Output 22 23 21 5 8 1 2 31. 220, 4 33. No; the equation has no solutions 6 x 22 because an absolute value will never be negative. 3 4 1 1 1 35. 23 37. 21 } , 2} , 39. 2 } , 23 } 41. When writing 2 2 6 the second equation, the right side of the equation should be 2x 2 3; 5x 2 9 5 2x 2 3, 6x 2 9 5 2 3, 6x 5 6, x 5 1, the solutions are 3 and 1. 25 5 43. 25 ≤ x ≤ 5 26 45. 25 < m < 9 2 22 26 6 2 22 10 6 10 65. c > 0, c 5 0, c < 0 67. no solution 69. x < 9 or x > 9 0 3 6 9 11. Yes; each input has exactly one output. 13. Yes; each input has exactly one output. 15. x is the input and y is the output, so there should be one value of y for each value of x; the relation given by the table is not a function because the inputs 1 and 0 each have more than one output. 17. No; the input 22 has more than one output. 19. No; the input 21 has more than one output. 21. function 23. not a function 25. Chapter Review (pp. 61–64) 1. exponent, base 3. extraneous solution 5. Sample answer: 3(x 2 4) and 3x 2 12 7. Inverse property of multiplication 9. Distributive property 11. 3x 2 6y 13. 18b 233 1 15. 22t 4 1 5t 2 17. 2 } 19. 9 21. 21 23. $74.99 6 2.1 Problem Solving (pp. 78–79) 43. Yes; each input has exactly one output. 45. About 905; V(6) represents the volume of a sphere with radius 6. 47. a. h(l) domain: 74 15 ≤ l ≤ 24, 72 70 range: 68 57.95 ≤ h(l) ≤ 75.5 25. y 5 210x 1 7; 223 27. y 5 }; 15 29. y 5 } x 2 5; S 2 2p r 220 31. h 5 } ; about 7.73 cm 33. 602 mi 2p r 0 1 37. x ≤ 2 } 2 39. 23 ≤ x ≤ 3 2 4 6 8 22 21 0 1 24 0 22 0 24 0 15 17 19 21 Length (inches) 23 l 11,350,000, 12,280,000, 12,420,000, 15,980,000, 18,980,000, 20,850,000, 33,870,000, range: 20, 21, 27, 31, 34, 55 b. Yes; each input p has exactly one output. c. No; the input 21 has more than one output. 4 2 41. 23, 1 } 43. no solution 3 45. y < 21 or y > 6 66 64 62 60 58 0 b. 59 in. or 4 ft 11 in. c. 21.7 in. 49. a. domain: 2 2 x 35. not linear; 10 37. linear; 6 39. linear; 23 5 2 215 x26 21 SELECTED ANSWERS 81. e 2 6008 ≤ 5992, m 2 46,000 ≤ 45,000 1 x 21 Height (inches) 75. p 2 6.5 ≤ 1 77. b 2 21 > 1 79. x 2 45 ≤ 15 35. x ≤ 6 y 1 12 2c 2 b c2b b2c c1b 71. x ≤ } or x ≥ } 73. x < } or x > } a a a a 2 27. y 4 8 12 47. v 2 26 ≤ 0.5, 25.5 in. ≤ v ≤ 26.5 in. Extension (p. 81) 1. discrete; 21, 1, 3, 5, 7 y Chapter 2 2.1 Skill Practice (pp. 76–78) 1. independent, dependent 3. domain: 24, 22, 1, 3, range: 23, 21, 2, 3 y 1 21 x Input Output 24 22 1 3 23 21 2 3 1 21 x Selected Answers n2pe-9090.indd SA3 SA3 12/1/05 12:19:46 PM 3. continuous; y > 26 y 1 2.3 Skill Practice (pp. 93–94) 1. slope-intercept 3. x 21 Both graphs have a y-intercept of 0, but the graph of y 5 3x has a slope of 3 instead of 1. y 1 x 21 5. 5. d (x ) 5 3.5x Both graphs have a slope of 1, but the graph of y 5 x 1 5 has a y-intercept of 5 instead of 0. y Distance (miles) d(x) 14 10 1 6 2 0 x 21 9. 0 1 2 3 Hours 11. y 1 x 4 y x 21 domain: x ≥ 0, range: d (x ) ≥ 0; continuous 7. m (x ) 5 3x m(x) Gallons of milk SELECTED ANSWERS 30 27 24 21 18 1 15 12 9 6 3 0 x 21 21. The slope and y-intercept were switched around. y 1 0 1 2 3 4 5 6 7 8 9 10 x Weeks 1 x domain: whole numbers, range: multiples of 3; discrete 25. x-intercept: 215, y-intercept: 23 27. x-intercept: 5, y-intercept: 210 29. x-intercept: 6, y-intercept: 24.5 3 2.2 Skill Practice (pp. 86–87) 1. slope 3. }; rises 2 5 7 5. 2 }; falls 7. 24; falls 9. }; rises 11. undefined; 4 3 31. is vertical 13. 0; is horizontal 15. The x and y coordinates were not subtracted in the correct 21 2 (23) 2 2 (24) 1 order; } 5 } . 19. neither 21. perpendicular 33. y 1 1 (8, 0) x 22 43. 3 2 6 21 (4, 0) x 21 45. y y 1 x 21 1 23. parallel 25. 13 mi/gal 27. 2 m/sec 29. 2 31. } 3 2 y (0, 2) x 33. 2 } 35. No; no. Sample answer: The slope of 120 221 1 PQ 5 } 5 2} . The slope of QR 5 } 5 23 2 (21) 2 21 2 (22) 325 21 2 1 1 1 2} , the slope of ST 5 } 5 2 } . 2 2 7 2.2 Problem Solving (pp. 87–88) 41. } 43. 6.5% 12 3 1 47. a. } b. yes c. } 8 8 SA4 n2pe-9090.indd SA4 A 55. Sample answer: x 5 3, y 5 22 57. slope: 2 } , B C y-intercept: } B Selected Answers 12/1/05 12:19:48 PM 19 4 2.3 Problem Solving (pp. 94–96) 1 27. y 5 22x 1 6 29. y 5 2 } x 1 } 31. y 5 23x 1 11 59. 2 33. y 5 2 } x 1 7 35. y 5 5x 1 23 37. y 5 23x 1 17.5 $480 y 600 Cost 4 3 525 450 375 41. 24x 1 y 5 23 43. 4x 2 5y 5 27 45. 4x 1 3y 5 32 1 47. Sample answer: y 5 2 } x18 2 300 225 150 75 0 2.4 Problem Solving (pp. 103–104) 51. n 5 15t 1 50 53. 15x 1 9y 5 4500 0 1 2 3 4 5 6 7 8 x y 500 Months Cost 61. 400 $1.50; $3 C(g) 36 300 32 28 24 200 20 16 100 12 8 4 0 0 0 1 2 3 4 5 6 7 8 g Number of games 63. 30; fall; the value of the card will decrease after Walking time (hours) b. 8 0 4 8 12 16 c. Sample answer: r x 12 0 2 200 16 1 1 150 w 4 0 100 Find the point on the line where x is 200 then the corresponding y-coordinate is how many student tickets were sold. 55. y 5 1.661x 1 21.62; $48.20 57. a. 2l 1 2w 5 24 2 0 50 SELECTED ANSWERS you buy each smoothie, so the line will fall from left to right. 65. w Sample answer: r 5 0 and w 5 4, 4 r 5 1.75 and w 5 1, 3 r 5 0.875 and w 5 2.5 0 l l w 6 6 7 5 8 4 9 3 10 2 Running time (hours) t (minutes) h (feet) 0 200 1 350 2 500 3 650 4 800 5 950 b. h 900 800 Height (feet) 67. a. 700 600 500 400 300 200 100 0 0 1 2 3 4 m Minutes c. h (t ) 5 150t 1 200 2.4 Skill Practice (pp. 101–103) 1. standard 3. y 5 2 5 5. y 5 6x 7. y 5 2 } x 1 7 9. y 5 4x 2 2 11. y 5 2x 1 11 4 4 1 13. y 5 29x 1 85 15. y 5 2 } x 1 1 17. y 5 2 } x22 7 3 2.4 Problem Solving Workshop (p. 105) 1. y 5 4x 1 7 1 3. y 5 2 } x 1 16 5. y 5 32.14x 1 1764.36; 2 about 2825 years old 2.5 Skill Practice (pp. 109–110) 1. Sample answer: If y 5 ax, then a is the constant of variation. a is a constant ratio of y to x for all ordered pairs (x, y ). y 3. y 5 3x 1 19. The x- and y-coordinates were transposed; y 2 1 5 22(x 2 5), y 2 1 5 22x 1 10, y 5 22x 1 11. 21 x 1 1 21. y 5 2x 1 8 23. y 5 23x 1 13 25. y 5 2 } x2} 4 4 Selected Answers n2pe-9090.indd SA5 SA5 12/1/05 12:19:49 PM 5. y 5 23.5x 19. y 5 0.05x 1 1.14 y 3 y 6 4 x 21 2 1 11. y 5 2x; 24 13. y 5 20.2x; 22.4 15. y 5 } x; 4 0 3 20 40 60 x 80 19. not direct variation 21. direct variation; 2.5 21. a. Sample answer: Measuring the depth of water 1 4 4 23. direct variation; } 25. y 5 2 } x; 3 27. y 5 27x; } 7 6 3 5 1 29. y 5 27.2x; } 31. direct variation; y 5 2 } x 9 3 at different times while filling a swimming pool. The number of gallons of milk you buy and the total cost. b. Sample answer: The age of a car and its current value. The number of miles you have driven since you last put gas in the tank and the amount of gas left in the tank. c. Sample answer: The height of a person and the month they were born. The age of a person and the number of vehicles they own. to be compared to each other, not the products; 24 1 8 3 12 2 6 4 } 5 24, } 5 6, } ø 2.7, } 5 1.5, because the ratios are not equal, the data does not show direct variation. 2.5 Problem Solving (pp. 110–111) 39. w 5 3600d; 6300 lb 41. direct variation; t 5 5.1s 43. a. direct variation; P 5 4s b. Not a direct variation; the ratios of A to s are not equal. c. Not a direct variation; the ratios of A to P are not equal. 2.6 Skill Practice (pp. 117–118) 1. best-fitting line 2.6 Problem Solving (pp. 119–120) 25. Sample answer: y 5 101.3x 1 2236.6 27. a. (0, 37), (4, 49), (8, 57), (12, 64), (14, 67), (18, 72), (22, 77) y b. c. Sample answer: 80 y 5 1.8x 1 40.7; 102 countries 60 Countries 33. direct variation; y 5 24x 35. The quotients need SELECTED ANSWERS 0 3. negative correlation 5. no correlation 7. 0 9. 21 11. a. y b. Sample answer: 120 20 y 5 220x 1 141 c. about 2259 96 40 0 0 6 12 18 x 24 Years since 1980 72 2.7 Skill Practice (pp. 127–128) 1. vertex 48 3. translated down 7 units y 2 24 x 22 0 13. a. 0 1 2 3 5 x 4 b. Sample answer: y 120 y 5 6.7x 1 1 c. about 135 90 5. translated left 4 units and down 2 units y 1 x 21 60 30 0 0 4 8 12 16 1 1 15. y 5 23x 17. y 5 } x 19. y 5 } x 1 2 2 1 3 x 21. 17. The line should go through the middle of the data points. Sample answer: (24, 0) y 60 1 1 x y 1 2 (22, 1 ) 2 (21, 21) (0, 23) 40 20 0 2 23. y x 21 29. The graph should be 0 2 4 6 8 x (1, 1) (2, 0) y translated left 3 units. 1 21 x 33. No. Sample answer: It does not pass the vertical line test. SA6 n2pe-9090.indd SA6 Selected Answers 12/1/05 12:19:51 PM 23. 2.7 Problem Solving (pp. 128–129) 37. y 1 15,000 pairs of shoes s 50 Sales (thousands) 25. y 1 40 30 1 20 x 21 10 0 0 10 20 30 40 29. solution, not a solution 31. solution, not a solution y y 35. 33. t Weeks 1 140 39. y 5 2 }x 2 69 1 140 69 41. a. b. 1 0 0.5 1 1.5 2 2.5 3 d 90 60 30 0 30 60 90 2 3 3 5 39. Sample answer: y > x 1 3 41. y > 2 } x 1 3; pick 60 40 20 1 2 3 4 t 5 Extension (p. 131) 1. 21 3. } 2 2.8 Problem Solving (pp. 137–138) 43. 0.03x 1 0.06y ≤ 20 7. y 45. 1.5x 1 2.5y ≤ 75 y 1 1 x 21 y 30 Linen 5. 20 10 x 21 0 9. y 0 10 20 30 40 50 x Cotton y ≤ 15.6 yd 47. a. 11x 1 26y ≤ 120 1 x y 10 Bike 22 2.8 Skill Practice (pp. 135–136) 1. half-plane 3. solution, not a solution 5. solution, solution 7. 9. y 1 21 8 6 4 2 0 0 2 4 6 8 10 12 14 16 18 x Canoe b. Sample answer: 2 days canoeing and 5 days biking, y 1 3 days canoeing and 2 days biking, 2 days canoeing and 2 days biking y c. 11x 1 26y ≤ 96 5 Sample answer: 1 day 4 canoeing and 3 days biking, 3 4 days canoeing and 2 days 2 1 biking, 2 days canoeing and 0 0 1 2 3 4 5 6 7 8 9 x 2 days biking x 21 x 19. The boundary line should be a dashed line. y 1 21 x Selected Answers n2pe-9090.indd SA7 SELECTED ANSWERS two points on the boundary line to find the slope and then use the point-slope form of an equation to find the equation. The boundary line is dotted, so the inequality dos not include points on the boundary. Then choose a point to determine which inequality sign to use. Sample answer: You and your sister want to spend at least $15 on your little brother’s birthday. You want to buy him some racecars that cost $3 each and some building block sets that cost $5 each. 1 3 } ≤ t ≤ 2} 0 x 21 c. d 5 60t 2 1.5; 80 x 22 t d 0 x SA7 12/1/05 12:19:52 PM Chapter Review (pp. 141–144) 1. standard 3. direct variation 5. domain: 22, 21, 2, 3, range: 22, 0, 6, 8; function 7. linear function; 51 9. undefined 11. 0 13. 15. y 1 x 21 y 1 x 21 3 4 17. y 5 2 } x 1 2 19. y 5 28x; 224 21. y 5 20.8x; 22.4 23. Sample answer: y 5 2x 1 2.3 y 25. shrunk vertically by 1 1 1 2 21. (28, 6) 23. no solution 25. (7, 3) 27. Failed to multiply the constant by 22. 26x 2 4y 5 214 5x 1 4y 5 15 2x 5 1 x 5 21 29. (25, 26) 31. infinitely many solutions 33. (28, 0) 3 2 1 2 1 3 4 2 2 45. about (2.90, 22.16) 47. (21, 2) 49. (7, 1) 1 51. 2 } , 6 53. (5, 4) 1 x 21 2 2 1 35. (7, 26) 37. 2 }, 4 39. 2 }, } 41. (2, 3) 43. (3, 2) 3 a factor of } 4 1 3.2 Skill Practice (pp. 164–165) 1. substitution 3. (6, 21) 4 5. no solution 7. } , 2 9. (0, 3) 11. (23, 8) 3 1 1 1 13. (44, 217) 15. 7, } 17. (26, 22) 19. 2 } ,} 2 2 6 2 9 3.2 Problem Solving (pp. 165–166) 55. 5 acoustic, 27. $1.75, $1.25 y 4 electric 57. The company can fill its orders by operating Factory A for 5 weeks and Factory B for 3 weeks. 59. 12 doubles games, 14 singles games 61. 80 pounds of peanuts, 20 pounds of cashews 1 x SELECTED ANSWERS 21 29. solution 31. 3.3 Skill Practice (pp. 171–172) 1. The ordered pair 33. y 1 21 x must satisfy each inequality of the system. y 5. 7. no solution y 1 21 x 1 x 21 9. 17. y y Chapter 3 3.1 Skill Practice (pp. 156–157) 1. independent 1 3. (1, 21) 5. (4, 21) 7. (5, 0) 9. (22, 4) 11. (6, 0) 1 81 51 23 23 x 21 1 2 13. }, } 17. (2, 21); consistent and independent 19. no solution; inconsistent 21. infinitely many solutions; consistent and dependent 23. (2, 0); consistent and independent 25. (3, 21); consistent and independent 27. infinitely many solutions; consistent and dependent 31. no solution 33. (24, 2) and (24, 2) 3.1 Problem Solving (pp. 157–158) 35. lifeguard: 6 h, cashier: 8 h 37. 11 days; the number of days will decrease; the number of days will be divided by a larger number, which will decrease the quotient, which is the number of days. 39. a. m 5 20.096x 1 50.8 b. w 5 20.12x 1 57.1 c. in the year 2192 d. No. Sample answer: It is not likely that women’s times will ever catch up to men’s times or that the times will continue to decrease infinitely. SA8 n2pe-9090.indd SA8 x 21 19. y 1 x 23 3 27. Sample answer: y < x 2 1, y < 2 }x 1 4 4 y 29. 1 22 x Selected Answers 12/1/05 12:19:54 PM Chapter 8 Lesson 7.6 (pp. 519–522) 15. 5x 5 33 Lesson 8.1 (pp. 554–557) 5x 5 log11 33 15. y 5 } → 2 5 } → 14 5 a 7 11 log11 11 a x log 33 log 11 5x 5 log11 33 5 } → x ø 0.2916 35. 5.2 log4 2x 5 16 14 log4 2x ø 3.0769 21. x p y: 12(132) 5 1584 y/x: 132/12 5 11 18(198) 5 3564 198/18 5 11 23(253) 5 5819 253/23 5 11 29(319) 5 9251 319/29 5 11 34(374) 5 12,716 374/34 5 11 4(log4 2x) ø 43.0769 WORKED-OUT SOLUTIONS 14 y5} →y5} x 3 Check: 2x ø 71.2020 a Intersection X=35.601007 Y=16 x ø 35.6010 57. R 5 100e20.00043t → 5 5 100e20.00043t x and y show direct variation because the ratios y/x are equal. 0.05 5 e20.00043t ln 0.05 5 ln e20.00043t a A 22.9957 ø 20.00043t 172 An equation is P 5 } . A t ø 6967 years 172 Boots: P 5 } → P ø 2.87 lb/in.2 60 Lesson 7.7 (pp. 533–536) 5.66 2 2.89 521 ln y 11. m 5 } ø 0.69 (5, 5.66) (3, 4.28) (4, 5.00) (2, 3.58) (1, 2.89) ln y 2 2.89 5 0.69(x 2 1) ln y 5 0.69x 1 2.2 Lesson 8.2 (pp. 561–563) 5. 1 The graph of y 5 } x 5 2 (22, ) y 5 2x lies farther from the axes than the graph x y 5 e 2.2(e 0.69)x ø 9(2) x 1 (25, 1) 1 1 ln x 0 0.693 1.099 1.386 1.609 ln y 20.511 1.411 2.518 3.296 3.902 y5 (5, 21) 1 x 5 2 (2, 2 ) (1, 25) 3.902 2 (20.511) m 5 }} ø 2.743 1.609 2 0 ln y 21. ln y 2 (20.511) 5 2.743(ln x 2 0) ln y 5 ln x2.743 2 0.511 y 5 eln x 25 y (21, 5) 5 1 y 5 e 0.69x 1 2.2 23. a 400 39. Snow shoes: P 5 } → 0.43 5 } → 172 5 a 1 (1, 0) 2.743 x 5 1 ln x 2.743 2 0.511 y 5 e20.511 p eln x y (3, 2) 1 x of y 5 }x , and it lies in Quadrants II and IV instead of Quadrants I and III. The domain is all real numbers except 4, and the range is all real numbers except 21. (7, 22) ø 0.6x2.743 x (5, 24) y5 23 x24 21 33. a. A model is y 5 0.48(2.08) . b. Linear if a graph of (x, y) appears linear; exponential if a graph of (x, ln y) appears linear; power if a graph of (ln x, ln y) appears linear. The graph of (x, y) appears linear, so a model is y 5 33.8x 1 28. 1000 0.6T 1 331 1000 0.6(25) 1 331 39. a. t 5 } 5 }} ø 2.89 2.89 seconds to travel 1 kilometer; 2.89(5) 5 14.45 seconds to travel 5 kilometers WS14 Worked-Out Solutions n2pe-9080.indd WS14 10/13/05 11:40:57 AM Time (seconds) 39. b. From the graph, you can estimate the temperature to be 3.98C. t 3.02 3.01 3.00 2.99 2.98 2.97 0 0 1 2 3 4 5 6 T Air temperature (8C) Lesson 8.5 (pp. 586–588) 2x 9 2 2x 9 5. } 2} 5} x11 x11 x11 5 32 15x Pi Pi 43. a. M 5 }} 5 }} 12t 1 1 12 } 11i 1 (1 1 i ) 5 (x 1 1)(x 2 1) (2 2x x2 2 1 1 4 , 2 3 (2, ) 4 3 b. P 5 15,500; i 5 0.06; t 5 4 (3, ) (0, 0) (23, 2 ) (22, 2 ) 3 4 (1 1 0.06) ( 1 , 2 4 23 3x 21 4 9 0} Check: } (6) 1 2 3(6) 4 9 0} } 8 18 1 1 }5}✓ 2 2 4(3x) 5 9(x 1 2) ) 4 3 x56 Depth Temp. b. 1000 4.7634 1050 4.5796 1100 4.4094 1150 4.2515 1200 4.1044 1250 3.9672 1300 3.8389 Temperature (8C) 33. a. 4 x12 9 }5} 5. x 2 22 15,500(0.06)(1 1 0.06)48 m 5 }} ø $990.41 48 Lesson 8.6 (pp. 593–595) 3 4 ) 21 (1 1 i ) WORKED-OUT SOLUTIONS y5 Pi(1 1 i )12t (1 1 i ) 2 1 }} 12t No x-intercept; x 5 21 and x 5 1 are vertical asymptotes. y (1 1 i ) 5} 5 }} 12t 12t 7. y 5 } → y 5 }} 2 15. 2 12} 12t Pi Lesson 8.3 (pp. 568–571) 5 x 21 32 2 15x 8 17. }2 2 } 5 }2 2 }2 5 } 4x 12x 12x 12x 2 3x 15. T 4.8 4.6 4.4 3x 2 6x } 1 3x 1 4.2 4.0 3.8 3.6 1 4 6 3x 1 4 } 5 6x } 6 3x 1 2 }1}5} 2 1 4 2 }1}0} 1 2 12 2(2) 1 1(x) 5 4(2) → x 5 4 0 0 1000 1100 1200 1300 Depth (meters) d 4 2 0} Check: } 1} 6 3(4) 3(4) 6 12 4 4 }5}✓ 12 12 635t 2 2 7350t 1 27,200 n 5 }} 2 35. t 2 11.5t 1 39.4 635t 2 2 7350t 1 27,200 The mean temperature is 48C at about 1238 meters. 720 5 }} 2 t 2 11.5t 1 39.4 2 720t 2 8280t 1 28,368 5 635t 2 2 7350t 1 27,200 Lesson 8.4 (pp. 577–580) (x 2 5)(x 1 4) 7. }} Cannot be simplified (x 1 5)(x 2 3) 48x7y4 6x y 6 p 8 p x3 p x4 p y4 8x4 y } 25. } 3 6 5 }} 3 4 2 5 2 6px py py 2407t 1 7220 5.92t 2 131t 1 1000 S 4 A 5 }} 4 }} 2 2 6.02t 2 125t 1 1000 26420t 1 292,000 }} 930 6 Ï(930)2 2 4(1168)(85) t 5 }}} ø 1.45, 9.45 2(85) Because 9.45 is not in the domain (0 ≤ t ≤ 9), t ø 1.45 → 1995. 49. 26420t 1 292,000 85t 2 2 930 1 1168 5 0 2 5.92t 2 131t 1 1000 5 }} p }} 2 2407t 1 7220 6.02t 2 125t 1 1000 247,060 373.08 For 1999, t 5 7: S 4 A 5 } p} ø $50.21 419.98 4371 Chapter 9 Lesson 9.1 (pp. 617–619) }}} 7. d 5 } Ï(6 2 2)2 1 (25 2 (21))2 5 4Ï2 12 1 6 21 1 (25) 2 2 Midpoint 5 } , } 5 (4, 23) 2 Worked-Out Solutions n2pe-9080.indd WS15 WS15 10/13/05 11:40:59 AM 27. A(24, 1), B(22, 6), C(0, 21) Lesson 9.3 (pp. 629–632) }}} 2 } }}} 2 } }}} } AB 5 Ï(22 2 (24)) 1 (6 2 1) 5 Ï 29 17. 15x 2 1 15y 2 5 60 BC 5 Ï (0 2 (22)) 1 (21 2 6) 5 Ï 53 2 2 2 x 1y 54 } AC 5 Ï(0 2 (24)) 1 (21 2 1) 5 Ï20 5 2Ï 5 2 2 y 2) x 2 1 y 2 5 4 (0, 2 1 } (22, 0) r 5 Ï4 5 2 1 2 1 }} 2 } x 2 1 y 2 5 (Ï260 )2 → x 2 1 y 2 5 260 } b. VS 5 Ï (26 2 0) 1 (5 2 0) 5 Ï 61 Î1 2 65. WORKED-OUT SOLUTIONS }}} SM 5 } 2 Ï117 3 2 } 2} 2 (26) 1 (8 2 5) 5 2 2 2 y x 2 1 y 2 5 225 x 2 1 y 2 5 25 } Ï117 } VS 1 SM 5 Ï 61 1 } 2 20 x A 0.1 mi ø 13.2 units p } 5 1.32 mi 1 unit } Ï117 c. MP 5 } 2 }} } Ï117 E F x 2 1 y 2 5 100 A: x 2 1 (24)2 5 225 → x ø 614.5 → (214.5, 24) B: x 2 1 (24)2 5 100 → x ø 69.2 → (29.2, 24) } C: x 2 1 (24)2 5 25 → x 5 63 → (23, 24) MP 1 PV 5 } 1 Ï130 2 0.1 mi 1 unit ø 16.8 units p } 5 1.68 mi Lesson 9.2 (pp. 623–625) D: x 2 1 (24)2 5 25 → x 5 63 → (3, 24) E: x 2 1 (24)2 5 100 → x ø 69.2 → (9.2, 24) F: x 2 1 (24)2 5 225 → x ø 614.5 → (14.5, 24) 15. 5x2 5 215y → x2 5 23y a. AF ø ⏐14.5 2 (214.5)⏐ 5 29 mi y 2 3 4p 5 23 → p 5 2} 4 y5 2 Focus: 1 0, 2} 42 3 3 4 x b. BE ø ⏐9.2 2 (29.2)⏐ 5 18.4 mi c. CD ø ⏐3 2 (23)⏐ 5 6 mi (0, 2 ) 3 4 3 4 C D 220 } 2 B y 5 24 PV 5 Ï (0 2 3) 1 (0 2 11) 5 Ï 130 2 Directrix: y 5 } } 39. r 5 Ï (28 2 0)2 1 (14 2 0)2 5 Ï260 }} 2 x (0, 22) AB Þ BC Þ AC, so nABC is scalene. 26 1 3 5 1 11 3 53. a. M 5 }, } 5 2}, 8 2 2 2 (2, 0) 1 Lesson 9.4 (pp. 637–639) Axis of symmetry: x 5 0 11. 16x 2 1 9y 2 5 144 (0, 4) y 2 27. Focus: (25, 0) → p 5 25 → y 5 220x 57. a. y2 x2 } 1 } 5 1; a 5 4, b 5 3 16 9 y y 48 in. (0, 248) Vertices: (0, 64); 48 in. 146 in. x 146 in. (248, 0) b. x 2 5 4(48)y → x 2 5 192y x y 2 5 4(248)x → y 2 5 2192x 2 c. Using x 5 192y and x 5 73, y ø 27.8 Using y 2 5 2192x and y 5 73, x ø 227.8. The dish is about 27.8 inches deep. (0, 1 (23, 0) (3, 0) x 1 Co-vertices: (63, 0); } (0, 24) Foci: (0, 6Ï 7 ) (0, 2 7) } 29. b 5 Ï7 ; c 5 3; a2 5 b2 1 c 2 → a 5 4 2 y y2 x2 x2 }2 1 } } 2 5 1, or } 1 } 5 1 (Ï 7 ) 4 16 7 49. Largest field: 2a 5 185 → a 5 92.5; 2b 5 155 → b 5 77.5 2 y2 2 92.5 6006.25 y2 8556.25 x x }2 1 }2 5 1, or } 1 } 5 1 77.5 A 5 π(92.5)(77.5) ø 22,521 square meters WS16 Worked-Out Solutions 7) Smallest field: 1 6 23 1 1 Center: (h, k) 5 1 6} , }2 2a 5 135 → a 5 67.5; 2b 5 110 → b 5 55 Distance between vertex (6, 23) and (h, k): 2 y2 y2 x2 x2 }2 1 }2 5 1, or } 1 } 5 1 4556.25 3025 67.5 55 2 a 5 23 2 k 5 23 2 (21) 5 2 A 5 π(67.5)(55) ø 11,663 square meters Distance between focus (6, 26) and (h, k): 11,663 ≤ A ≤ 22,521 c 5 26 2 k 5 26 2 (21) 5 5 } b2 5 c 2 2 a2 5 25 2 4 5 21 → b 5 Ï 21 Lesson 9.5 (pp. 645–648) (0, 9) y 49. x 2 2 10x 1 4y 5 0; A 5 1, B 5 0, C 5 0 2 y x2 }2 }51 81 16 B2 2 4AC 5 0 2 4(1)(0) 5 0 → Parabola 4 } a 5 4; b 5 9; c 5 Ï 97 (2 ( 97 ,0) (24, 0) 2 97 , 0) (4, 0) Vertices: (64, 0) x 2 2 10x 1 4y 5 0 x (x 2 2 10x 1 25) 5 24y 1 25 (0, 29) } Foci: (6Ï 97 , 0) 25 (x 2 5)2 5 24 1 y 2 } 42 9 Asymptotes: y 5 6} x 4 25 25 (h, k) 5 1 5, } 5 6.25 feet 2; height 5 } } 23. c 5 4Ï 5 ; a 5 4; b2 5 c 2 2 a2 2 a → b 5 8 y 2 2 x2 8 y 16 4 41. a. A(30.5, 0); B(85, 240) Lesson 9.7 (pp. 661–664) b. Vertices: (630.5, 0); horizontal trans. axis y2 y2 x2 x2 }2 2 }2 5 1 → }2 2 }2 5 1 b b a 30.5 2 2 (240) 85 2 b y2 2 x So, an equation is } 2} 5 1. 236.5 930.25 2 5. Intersection X=.47247477 Y=-2.582576 5 1 → b ø 236.5 }2 2 } 2 30.5 4 When y 5 0, the x-intercepts are 0 and 10, so the distance of the jump is 10 feet. x2 64 }2 2 }2 5 1, or } 2 } 5 1 4 (x 2 6)2 WORKED-OUT SOLUTIONS 13. 81x 2 2 16y 2 5 1296 (y 1 1)2 4 An equation is } 2 } 5 1. 21 Intersection X=3.5275252 Y=6.5825757 The solutions are approximately (0.5, 22.6) and (3.5, 6.6). y2 42 c. x 5 42: } 2} 5 1 → y ø 14.6 236.5 930.25 15. 4x 2 2 5y 2 5 276 2x 1 y 5 26 → y 5 22x 2 6 h 5 40 1 14.6 5 54.6 feet Substitute 22x 2 6 for y in Equation 1. Lesson 9.6 (pp. 655–657) 3. (x 1 4)2 5 28(y 2 2) 4x 2 2 5(22x 2 6)2 5 276 y y54 (24, 2) Parabola; vertical axis; vertex at (h, k) 5 (24, 2). 4x 2 2 20x2 2 120x 2 180 5 276 2 2 x (24, 0) 4p 5 28 → p 5 22 216x 2 2 120x 2 104 5 0 2x 2 1 15x 1 13 5 0 Focus: (h, k 1 p) 5 (24, 0) 13 13 (2x 1 2) 1 x 1 } 5 0 → x 5 21, x 5 2} 22 2 Directrix: y 5 k 2 p → y 5 4 19. Vertices: (6, 23), (6, 1); Focus: (6, 26), (6, 4) (y 2 k)2 (x 2 h)2 a b 2} 51 Vertical transverse axis; } 2 2 When x 5 21: y 5 22(21) 2 6 5 24 13 13 : y 5 22 1 2} 2657 When x 5 2} 2 22 13 The solutions are (21, 24) and 1 2} , 7 2. 2 Worked-Out Solutions n2pe-9080.indd WS17 WS17 10/13/05 11:41:04 AM 1 29. (2s4 1 5)5 5 5C 0(2s4)550 1 5C1(2s4)451 41. a. Oak Lane: m 5 2} , (x1, y1) 5 (22, 1) 7 1 1 1 5C2(2s4)352 1 5C3(2s4)253 1 5C4(2s4)154 5 y 2 1 5 2} (x 1 2) → y 5 2} x1} 7 7 7 1 5C5(2s4)055 5 1(32s20) 1 5(16s16)(5) Circle: x 2 1 y 2 5 1 1 10(8s12)(25) 1 10(4s8)(125) 1 5(2s4)(625) 5 2 1 x 1 2} x1} 51 7 7 1 2 b. 2 10 1 1 1(1)(3125) 5 32s 20 1 400s16 1 2000s12 1 5000s8 1 6250s4 1 3125 25 x2 1 } x2 2 } x1} 51 49 49 49 49. You can choose 3 of the 18 types of flowers. WORKED-OUT SOLUTIONS 49x 2 1 x 2 2 10x 1 25 5 49 18 3 4 5 3 5 1 4 1 3 4 } 1 } 5 }; y 5 2} 2} 1 } 5 } y 5 2} 7 5 7 5 7 5 7 5 1 2 3 4 4 3 The solutions are } , } and 2} ,} . 5 5 5 5 1 Î1 2 1 2 18 p 17 p 16 p 15! 15! p 3! 3 (5x 2 4)(10x 1 6) 5 0 → x 5 } , x 5 2} 5 5 1 2 18! 15!3! C 5 } 5 }} 5 816 50x 2 2 10x 2 24 5 0 2 Lesson 10.3 (pp. 702–704) 7. Factors of 150 from 1 to 50: 1, 2, 3, 5, 6, 10, 15, 25, 30, 50 Factors of 150 10 1 P 5 }} 5 } 5} 5 50 Integers from 1 to 50 }} c. d 5 3 5 } 3 2 4 2 4 1 } 2} 5 Ï2 ø 1.4 mi 5 5 5 }2} 2 1 2 Chapter 10 numbers. Only 1 is the correct combination. 1 Lesson 10.1 (pp. 686–689) 13. a. 26 p 26 p 26 p 26 p 10 p 10 5 45,697,600 b. 26 p 25 p 24 p 23 p 10 p 9 5 32,292,000 362,880 9! 9! 35. 9P 2 5 } 5 } 5 } 5 72 5040 (9 2 2)! 7! 65. Permutations of 9 objects taken 3 at a time: 9! (9 2 3)! 17. There are 48C 6 different combinations of 6 362,880 9! 6! P 5}5}5} 5 504 ways 9 3 720 Area of smallest circle Area of entire target and 4 of the 48 that are not queens. 4! 48! C p 48C4 5 } p } 5 778,320 4 1 3!1! 44!4! No queen: Choose 5 cards from the 48 in a deck that are not queens. 48! 43!5! C 5 } 5 1,712,304 48 5 The total number of possible hands is 778,320 1 1,712,304 5 2,490,624. π p 82 π p 40 1 25 39. P 5 }} 5 }2 5 } 5 0.04 Lesson 10.4 (pp. 710–713) 11. P(A or B) 5 P(A) 1 P(B) 2 P(A and B) 0.71 5 0.28 1 0.64 2 P(A and B) 20.21 5 2P(A and B) → P(A and B) 5 0.21 21. P(K or ♦) 5 P(K) 1 P(♦) 2 P(K and ♦) 13 4 1 4 5 1} 1 1} 2 1} 5} 52 2 52 2 52 2 13 Lesson 10.2 (pp. 694–697) 17. Exactly one queen: Choose 1 of the 4 queens 1 P(correct numbers) 5 } 5 } 12,271,512 C 48 6 45. The number of combinations of 6 food items is 106. The number of combinations of 6 different food items is 10 p 9 p 8 p 7 p 6 p 5. So, the probability that at least 2 bring the same item is P 5 1 2 P(none are the same) 5 10 p 9 p 8 p 7 p 6 p 5 10 1 2 }} 5 0.8488. 6 Lesson 10.5 (pp. 721–723) 13. P 5 P(blue) p P(green) p P(red) 3 60 5 4 p } p } 5} ø 0.015 5 1} 16 2 1 16 2 1 16 2 4096 WS18 Worked-Out Solutions n2pe-9080.indd WS18 10/13/05 11:41:06 AM d. p 5 P(Rh1) 5 P(O1) 1 P(A1) 1 P(B1) 25. Primes from 1 to 20: 2, 3, 5, 7, 11, 13, 17, 19. P(number of odd primes) Event C: player wins 55 39. 0. 5 0. Event A: player 0 wins toss .45 Total # of matches 0.5 Event D: player loses Event C: player wins 0. 47 Event D: player loses P(C) 5 P(A and C) 1 P(B and C) P(k 5 5) 5 10C5(0.85) 5(1 2 0.85)10 2 5 ø 0.008 P(k 5 6) 5 10C6 (0.85) 6 (1 2 0.85)10 2 6 ø 0.040 P(k 5 7) 5 10C7(0.85)7(1 2 0.85)10 2 7 ø 0.130 P(k 5 8) 5 10C 8 (0.85) 8 (1 2 0.85)10 2 8 ø 0.276 P(k 5 9) 5 10C9 (0.85) 9 (1 2 0.85)10 2 9 ø 0.347 P(k 5 10) 5 10C10 (0.85)10 (1 2 0.85)10 2 10 ø 0.197 P(k ≥ 5) 5 P(k 5 5) 1 P(k 5 6) 1 P(k 5 7) 1 P(k 5 8) 1 P(k 5 9) 1 P(k 5 10) ø 0.998 5 P(A) p P(CA) 1 P(B) p P(CB) 5 (0.50) p (0, 55) 1 (0, 50)(0.47) 5 0.51 Lesson 10.6 (pp. 727–730) N Outcomes P(N) 1 10 }5} 10 100 Lesson 11.1 (pp. 747–749) 5. Mean: 1 8 10 1 100 90 1000 9 100 900 1000 9 10 2 90 } 5} 3 900 } 5} Probability 5. Chapter 11 69 1 70 1 75 1 84 1 73 1 78 1 74 1 73 1 78 1 71 }}}}} 10 6 10 WORKED-OUT SOLUTIONS Event B: player loses toss 0.5 3 1 P(AB1) 5 0.85 7 P(oddprime) 5 }}} 5 } 8 P(number of primes) 5 74.5 4 10 Median: 69, 70, 71, 73, 73, 74, 75, 78, 78, 84 2 10 73 1 74 2 } 5 73.5 0 1 2 3 Number of digits 21. Each question has 4 possible answers, so the probability of guessing a correct answer is p 5 0.25. There are 30 questions, so n 5 30. The probability of randomly guessing 11 correct answers is P(k 5 11) 5 C (0.25)11(1 2 0.25)30 2 11 ø 0.055. 30 11 45. a. p 5 0.34; Mode: 73 and 78 15. Range: 158 2 135 5 23 135 1 142 1 148 1 136 1 152 1 140 1 158 1 154 } 5 }}}} x 8 5 145.625 s5 Î }}}}} P(k 5 5) 5 10C5(0.34) 5(1 2 0.34)10 2 5 ø 0.14 b. p 5 P (Rh2) 5 P (O2) 1 P (A2) 1 P (B2)1 P (AB2) 5 0.15 2 10 2 2 P(k 5 2) 5 10C2(0.15) (1 2 0.15) c. p 5 P(O) 5 P (O ø 0.28 ) 1 P(O ) 5 0.43 1 2 P(k 5 0) 5 10C 0 (0.43) 0 (1 2 0.43)10 2 0 ø 0.004 P(k 5 1) 5 10C1(0.43)1(1 2 0.43)10 2 1 ø 0.027 (135 2 145.625)2 1 (142 2 145.625)2 1 . . . 1 (154 2 145.625)2 }}}}} 8 Î } 519.875 5 } ø 8.1 8 29. a. The outlier is 5. b. With outlier: 20 1 23 1 . . . 1 23 Mean: } x 5 }} 5 20.2 10 P(k 5 2) 5 10C2(0.43)2(1 2 0.43)10 2 2 ø 0.093 Median: 22 Mode: 23 Range: 25 2 5 5 20 P(k ≤ 2) 5 P(k 5 0) 1 P(k 5 1) 1 P(k 5 2) Std. Dev.: ø 0.124 (20 2 20.2)2 1 (23 2 20.2)2 1 . . . 1 (23 2 20.2)2 s 5 }}}} 10 ø 5.4 Worked-Out Solutions n2pe-9080.indd WS19 WS19 10/13/05 11:41:08 AM Without outlier: 19.4 2 20 0.25 20.4 2 20 20.4 ounces: z 5 } 5 1.6 0.25 33. a. 19.4 ounces: z 5 } 5 22.4 20 1 23 1 . . . 1 23 Mean: } x 5 }} ø 21.9 9 b. The table shows that P(x ≤ 22.4) 5 0.0082. Median: 23 Mode: 23 Range: 25 2 19 5 6 So, the probability is 0.0082. Std. Dev.: Î c. P(x ≤ 20.4) 5 0.9452; P(x ≤ 19.4) 5 0.0082 }}}} s5 (20 2 20.2)2 1 (23 2 20.2)2 1 . . . 1 (23 2 20.2)2 }}}} 9 P(x ≤ 20.4) 2 P(x ≤ 19.4) 5 0.937 ø 2.1 WORKED-OUT SOLUTIONS c. The outlier causes the mean and median to decrease, and the range and standard deviation to increase. The mode stays the same. Original data set 1 Ïn Adding 17 to data values 60.056 5 6 } } Mean 78 78 1 17 5 95 77 77 1 17 5 94 Mode 77 77 1 17 5 94 Range 9 9 Standard deviation 2.8 2.8 Mean Original data set Multiplying data values by 4 61.9 61.9(4) 5 247.6 Median 62 62(4) 5 248 Mode 58 58(4) 5 232 Range 9 9(4) 5 36 Standard deviation 3.4 3.4(4) 5 13.6 19. The margin of error is about 63.2%. 19. Margin of error 5 6 } } Ïn Median 11. 1 1 7. Margin of error 5 6} } 5 6} } ø 60.032 Ïn Ï1000 1 Lesson 11.2 (pp. 753–755) 5. Lesson 11.4 (pp. 769–771) Heights without stilts Heights with stilts Mean 70.8 70.8 1 28 5 98.8 Median 72 72 1 28 5 100 Mode 72 72 1 28 5 100 Range 8 8 Standard deviation 2.4 2.4 1 0.003136 5 } → n ø 319 n 29. Sample Answer: It is reasonable to assume that Kosta is going to win the election, because the margin of error is 65%. If the margin of error works in favor of Murdock and against Kosta, Kosta will have 49% (54% 2 5%) and Murdock will have 51% (46% 1 5%). Lesson 11.5 (pp. 778–780) 3. Model: y 5 20.38x 2 1 1.1x 1 16 11. A model for the data is y 5 0.55x 1 1.1 Chapter 12 Lesson 12.1 (pp. 798–800) 2 2 2 2 19. } ,} ,} ,} ,. . . 3p1 3p2 3p3 3p4 Lesson 11.3 (pp. 760–762) 2 11. 29 and 37 are one standard deviation on either side of the mean, which accounts for 68% of the data. So, the probability is 0.68. 2 2 Next term: } 5} ; A rule is an 5 } . 15 3n 3. P(x ≤ } x 2 s) 5 0.0015 1 0.0235 1 0.135 5 0.16 3p5 4 47. ∑ n3 5 03 1 13 1 23 1 33 1 43 n50 5 0 1 1 1 8 1 27 1 64 5 100 WS20 Worked-Out Solutions n2pe-9080.indd WS20 10/13/05 11:41:10 AM 65. n 1 2 3 4 5 an 1 3 7 15 31 Lesson 12.4 (pp. 823–825) ` 13. k51 a6 5 26 2 1 5 63 moves for 6 rings 1 a1 5 7; r 5 2} ;s5 } 5} 5} 9 12r 17 8 } a 0.625 625 WORKED-OUT SOLUTIONS n 5 345 1 345(0.783) 1 345(0.783)2 1 345(0.783)3 1 . . . A rule for the nth term is a a20 5 2(20) 2 5 5 35 345 1 5} 5} ø 1590 million 12r 1 2 0.783 an 5 a1 1 (n 2 1)d 5 23 1 (n 2 1)2 5 2n 2 5 Lesson 12.5 (pp. 830–833) 8 ∑ (23 2 2i) 15. Geometric sequence; a1 5 4; r 5 23 i51 an 5 r p an 2 1 5 23an 2 1 a1 5 23 2 2(1) 5 25; a8 5 23 2 2(8) 5 219 25 1 (219) 2 A recursive rule is a1 5 4, an 5 23an 2 1. 2 s8 5 8 } 5 296 1 2 27. f (x) 5 } x 2 3, x0 5 2 65. a. a1 5 1; d 5 8 x1 5 f (x0) an 5 a1 1 (n 2 1)d 5 1 1 (n 2 1)8 5 27 1 8n 3 2 } 3 19. Geometric sequence; a1 5 2; r 5 5 } 4 2 3 an 5 a1r n 2 1 5 2 1 } 2 1 1 5} (2) 2 3 2 5} (22) 2 3 2 5} (24) 2 3 2 5 22 5 24 5 25 a1 5 2000, an 5 1.014an 2 1 2 100. Because a24 5 62.14, the balance at the beginning of the 24th month is $62.14. So, she will be able to pay off the balance at the end of the 24th month. 4 729 5} 5} 4096 2048 8 ∑ 6(4)i 2 1 Chapter 13 i51 a1 5 6(4)1 2 1 5 6; r 5 4 Lesson 13.1 (pp. 856–858) 2 r8 2 48 s8 5 a1 1} 5 6 1 1} 2 5 131,070 1 12r2 5 f (24) 45. Recursive rule: n21 1458 x3 5 f (x2) 5 f (22) 1 Lesson 12.3 (pp. 814–817) } x2 5 f (x1) 5 f (2) b. a12 5 27 1 8(12) 5 89 blocks are visible 721 625(0.001) 39. B; a1 5 23; d 5 21 2 (23) 5 2 4 63 1 5} 5} 5} 5} 12r 1 2 0.001 0.999 999 15. Arithmetic sequence 3 a7 5 2 1 } 2 7 27. 625(0.001) 1 625(0.001)2 1 625(0.001) 3 1 . . . Lesson 12.2 (pp. 806–809) 1 a 8 1 2 1 29 2 a8 5 28 2 1 5 255 moves for 8 rings 49. k21 A formula for the sequence is an 5 2n 2 1. a7 5 27 2 1 5 127 moves for 7 rings 41. ∑ 71 2}89 2 5. 124 x 1 59. a. a1 5 1024 4 2 5 512; r 5 } 2 1 an 5 a1(r) n 2 1 5 512 1 } 2 u Using the Pythagorean theorem: 11 } } x 5 Ï 112 2 82 5 Ï 57 8 n21 8 sin u 5 } 2 11 b. n 5 10; after 10 passes, the number of 10 2 1 1 items remaining is a10 5 512 1 } 2 2 5 1. 11 csc u 5 } 8 } Ï 57 cos u 5 } 11 } 11Ï57 sec u 5 } 57 } 8Ï 57 tan u 5 } 57 } Ï 57 cot u 5 } 8 Worked-Out Solutions n2pe-9080.indd WS21 WS21 10/13/05 11:41:12 AM opp adj 7 3 37. 11. tan u 5 } 5 } } x 7 } x 5 Ï 72 1 32 5 Ï 58 Stop u } } 7Ï 58 sin u 5 } 58 } WORKED-OUT SOLUTIONS 33. 250 ft 3 Start 3 cot u 5 } 7 d tan 158 5 } 1500 15⬚ 10 ft Ground u9 5 2708 2 2558 5 158 h sin 158 5 } 75 d ø 402 1500 ft 19.4 ø h The total depth is 402 1 250 5 652 feet. When the ride stops, you are about 10 1 75 1 19.4 5 104.4 feet above the ground. If the radius is doubled, your height above the ground is doubled only if your starting height above the ground is also doubled. As the angle of the dive increases, the depth increases. Lesson 13.2 (pp. 862–865) 5 p radians 1808 5p 11. } 5} } 5 508 18 1 π radians 2 18 x 75 ft } d 75⬚ 2558 158 7 tan u 5 } 3 Ï58 sec u 5 } 3 15⬚ 75 ft h 3Ï 58 cos u 5 } 58 Ï58 csc u 5 } 7 y Lesson 13.4 (pp. 878–880) y p p 7. When 2} ≤ u ≤ }, or 2908 ≤ u ≤ 908, the 2 2 508 } } p Ï3 Ï3 angle whose sine is } is u 5 sin21 } 5} , 3 x 2 2 } Ï3 or u 5 sin21 } 5 608. 2 p radians 2p 23. 408 5 408 } 5 } radians 1 2 1808 9 23. p 15 rev 1 min 2p rad 51. a. } } } 5 } rad/sec 2 1 min 60 sec 1 rev 21 1 2 tan u 5 3.2; 1808 < u < 2708 y u tan21(3.2) ø 72.68, which is in Quadrant I. To find the angle in Quadrant III (1808 < u < 2708): u ø 180 1 72.6 5 252.68 72.68 x p b. Arc length: s 5 r u 5 29 1 } 2 ø 45.6 feet 2 Lesson 13.3 (pp. 870–872) } 37. }} 11 } 5. r 5 Ï x 2 1 y 2 5 Ï (27)2 1 (224)2 5 Ï 625 5 25 y 224 sin u 5 }r 5 } 25 y 224 r 225 24 tan u 5 }x 5 } 5} 27 7 sec u 5 }x 5 } 7 17. 27 r 225 x 7 csc u 5 }y 5 } 24 cot u 5 }y 5 } 24 u9 5 1808 2 1508 5 308 y u 9 5 308 x f cos u 5 } 5 } 25 u 5 1508 tan u 5 } 17 u 11 ft 11 u 5 tan211 } 2 ø 338 17 ft 17 Because the angle of repose remains the same, you can use u 5 338 to find the radius of the 15 foot high pile: 15 338 15 ft r 15 tan 338 tan 338 5 } →r5} r ø 23 The diameter is about d 5 2r 5 2(23) 5 46 ft. x WS22 Worked-Out Solutions n2pe-9080.indd WS22 10/13/05 11:41:14 AM 1 2 Lesson 13.5 (pp. 886–888) 45. nADB: s 5 }(743 1 1210 1 1480) 5 1716.5 sin 1048 B 13. sin }5 } Area 5 25 16 }}}}} Ï1716.5(1716.5 2 743)(1716.5 2 1210)(1716.5 2 1480 16 sin 1048 sin B 5 } ø 0.6210 → B ø 38.48 25 ø 447,399 A ø 1808 2 1048 2 38.48 5 37.68 25 sin 1048 1 25 sin 37.68 sin 1048 a } 5 } → a 5 } ø 15.7 sin 37.68 nCDB: s 5 } (1000 1 858 1 1480) 5 1669 2 Area 5 }}}}} 45. a. Ï1669(1669 2 1000)(1669 2 858)(1669 2 1480) B ø 413,697 C 1 acre Area 5 (447,399 1 413,697) ft 2 } 2 62 ft 1 43,560 ft 2 A sin 588 sin C b. } 5 } 62 ø 20 acres 54 If you first found the length of } AC , you could repeat the same process using nABC and nADC. 54 sin 588 ø 0.7386 → C ø 47.68 sin C 5 } 62 A ø 1808 2 588 2 47.68 5 74.48 62 sin 588 62 sin 74.48 sin 588 a } 5 } → a 5 } ø 70.4 sin 74.48 1 2 Chapter 14 WORKED-OUT SOLUTIONS 588 54 ft Lesson 14.1 (pp. 912–914) 1 2 c. Area 5 } bc sin A 5 }(62)(54)(sin 70.48) 5. Amplitude: 1; period: 2 ø 1577 17. f (x) 5 4 tan x 1577 ft 2 4 200 ft 2/bag ø 7.9 bags y Period: π; intercept: (0, 0) You will need 8 bags of fertilizer. 4 Lesson 13.6 (pp. 892–894) x 4 p Asymptotes: x 5 6} 2 p p Halfway points: 1 } , 4 2 1 2} , 24 2 4 4 17. a2 5 b2 1 c 2 2 2bc cos A 31. a. Equation has the form y 5 a cos bt. 102 5 32 1 122 2 2(3)(12)cos A 1 a5} (3.5) 5 1.75 2 53 72 } 5 cos A → A ø 438 2p b p Equation is y 5 1.75 cos } t. 3 B ø 128; C ø 180 2 438 2 128 5 1258 b. Choose y 5 a cos bt because at t 5 0, the buoy is at its highest point. In nABC, A ø 438, B ø 128, and C ø 1258. 1 2 1 2 Lesson 14.2 (pp. 919–922) 25. s 5 }(a 1 b 1 c) 5 }(5 1 11 1 10) 5 13 11. y 5 2 cos x 1 1 }} Area 5 Ïs(s 2 a)(sb)(s 2 c) }}} } 5 Ï 13(13 2 5)(13 2 11)(13 2 10) 5 Ï 624 ø 25 square units p Period is 6, so 6 5 } → b 5 } . 3 3(sin 438) 3 10 } 5 } → } 5 sin B sin B 10 sin 438 Amplitude: a 5 2 2p Period: } 5 2p b y (0, 3) (2p, 3) 2 ( , 1) ( 2 4 3 2, ) 1 x (p , 21) Horizontal shift: h 5 0; Vertical shift: k 5 1 Worked-Out Solutions n2pe-9080.indd WS23 WS23 10/13/05 11:41:16 AM 1 23. y 5 2sin } x13 y 2 Closest distance: (3p, 4) (0, 3) (2p, 3) Amplitude: a 5 1 0.5 a.u. p }} 5 46.5 million mi 1 a.u. (p, 2) 2p 2p Period: } 5} 5 4π 1 b 93,000,000 mi (4 p, 3) 2 Farthest distance: x 2 } 2 93,000,000 mi 35.6 a.u. p }} ø 3.31 billion mi 1 a.u. h 5 0; k 5 3; a < 0, so graph is reflected. 200 2 d 300 53. a. } 5 tan u WORKED-OUT SOLUTIONS d 5 2300 tan u 1 200 u 300 ft d b. , 4 (2 Lesson 14.4 (pp. 935–937) d Your friend 200 ft You p 5. 12 sin 2 1 } 2 2 3 0 0 6 2 1 12 1 } 2 2300→32350✓ ) 500 2 } Ï3 3 13. 4 cos2 x 2 3 5 0 → cos2 x 5 } → cos x 5 6} 4 (0, 200) 2200 ( , 4 2 u 3 8 ) 5p p c. 100 5 2300 tan u 1 200 → u ø 18.48 x5} 1 2nπ or x 5 } 1 2nπ 6 6 1 5 5. cos u 5 }, 3π < u < 2π 6 1 b. 5 2 sin 2 u 1 1 } 51 62 } Negative because u is in Quadrant III } sin u Ï 1 sin u 26 Ï 11 cos u sin u 11 tan u 5 } 5 2} cos u 5 5 Ï 11 X 119 120 121 122 123 124 X=122 1 6 sec u 5 } 5} cos u 5 2sin u sin(2u) 11. } 5 } 5 tan u 2cos u cos(2u) 1.069 1.069 41. a. r 5 }} 5 }} π 1 2 0.97 cos u 1 2 0.97 sin1 } 2 u2 2 sin u 2 A value of u ø 54.78 minimizes the surface area. c. Minimum X=54.735619 Y=7.9432427 Lesson 14.5 (pp. 944–947) 5. M 5 6, m 5 2 M1m b. 2 When u ø 1228, S 5 9 in.2 Y1 8.8886 8.9246 8.9619 9.0005 9.0405 9.0819 cot u 5 } 5 2} } csc u 5 } 5 } } } 3 2 cos u 5 6.75 1 0.84375 Ï } sin 2 u 1 cos2 u 5 1 Ï 11 } 3 Ï3 2 cos u 43. a. S 5 6(1.5)(0.75) 1 }(0.75)2 } 2 sin u Lesson 14.3 (pp. 927–930) sin u 5 2} ← 6 5p p In 0 ≤ x < 2π, x 5 } and x 5 } . 6 6 2100 612 Vertical shift: k 5 } 5} 54 2 2 The graph is a cosine curve with h 5 0. 2p b p Period 5 4 5 } → b 5 } 2 c. u 0 } p 4 } p 2 } 3p 4 π r 35.6 3.4 1.1 0.6 0.5 u } 5p 4 } 3p 2 } 7p 4 2π r 0.6 1.1 3.4 35.6 M2m 622 5} 52 a 5 } 2 2 The graph is a reflection, so a 5 22. p x 1 4. The function is y 5 22 cos } 2 WS24 Worked-Out Solutions n2pe-9080.indd WS24 10/13/05 11:41:17 AM 9. M 5 6, m 5 26 43. a. 6 1 (26) M1m WQ NA Vertical shift: k 5 } 5} 50 2 2 f tan(u 2 t) 1 f tan t h tan u } 5 }} f h The graph is a sine curve with h 5 0. 1 5 } (tan(u 2 t) 1 tan t)1 } 2 2p b 1 Period 5 2(3π 2 π) 5 4π 5 } → b 5 } 2 tan u f tan u 2 tan t tan t(1 1 tan u tan t) 1 5 } }} 1 }} 1} 2 1 1 tan u tan t) 1 f u 1 tan u tan2 t 1 5 } tan }} 1 } 2 h The graph is not a reflection, so a 5 6. 1 f tan u(1 1 tan2 t) h 1 1 tan u tan t tan u 1 2 25. When t 5 0, m 5 4; when t 5 1, M 5 9. 2 f h 1 1 tan u tan t sec t 5 } }} 1 M1m 914 13 k5} 5} 5} 2 2 2 2p b t51 M2m 924 5 5} 5} a 5 } 2 2 2 t50 5 ft 5 The graph is a reflection, so a 5 2} . 2 Ground p 1 p 7. cos } 5 cos } 1 } 2 5 8 2 4 Î 3 2 Î p 1 1 cos } 4 } 2 Ï2 } Î 3p 4 } 2 a 2 a 2 } < } < π → } is in Quadrant II. 2 Î } 1 } } 12} Ï3 a 2 cos a 1 sin } 5 1} 5 }3 5 } 5} 3 2 2 2 3 Î } } Î Î } 1 2 1 } } 11} Ï6 a 1 cos a 2 cos } 5 2 1} 5 2 }3 5 2 } 5 2} 3 2 2 2 3 Î } } Î } 3p 3p 3p 23. sin x 2 } 5 sin x cos } 2 cos x sin } 2 f h 1 3p 13. cos a 5 }, } < a < 2π Ï6 2 Ï2 5} 4 1 2 } } 11} Î2 1 Ï}2 2 1 Ï2 5 } 5 2} 5} 2 4 2 Ï3 Ï2 Ï2 1 5} 2} 1} } 2 2 2 2 } 1 } } 3p 3p p p 5 cos } cos } 1 sin } sin } 4 4 3 3 2 f h 110 2 } 4 ft Lesson 14.6 (pp. 952–954) } 1 Lesson 14.7 (pp. 959–962) 5 13 A model is y 5 2} cos π x 1 } . 2 2 5p 3p p 9. cos 2} 5 cos } 2 } 4 12 3 f sec2(0) h 1 1 tan u tan(0) WQ NA 1 } 5 } }} 5 } } 5 } Period 5 2 5 } → b 5 π 1 2 b. When t 5 0: The graph is a cosine curve with h 5 0. 1 WORKED-OUT SOLUTIONS 2 2 2 tan u 1 1 tan u tan t 1 5 } }} 1 } 2 1 The function is y 5 6 sin } x. 1 2 tan u h 1 1 tan u tan t 6 2 (26) M2m 5} 56 a 5 } 2 2 2 2 5 (sin x)(0) 2 (cos x)(21) 5 cos x Ï3 a } } } sin } } Ï3 23 2Ï3 3 2 2Ï 2 a tan } 5 }a 5 } } 5 } p } } 5 } } 5 } 3 2 2 Ï6 Ï6 Ï6 cos } 2} 2 3 1 1 u 53. When M 5 2.5: sin } 5 } 5 } 2.5 M 2 Using the Pythagorean Theorem: } Ï5.25 cos }u 5 } 2.5 2 } Ï5.25 1 sin u 5 2 sin }u cos }u 5 2 1 } ø 0.7332 2} 2 2 2.5 1 2.5 2 u ø 478 Worked-Out Solutions n2pe-9080.indd WS25 WS25 10/13/05 11:41:19 AM n2pe-9080.indd WS26 10/13/05 11:41:22 AM