Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
GUIDED PRACTICE ✓ Concept Check ✓ Skill Check ✓ Vocabulary Check 1. Yes; no; every function is a set of ordered pairs and, so, is a relation. A relation is a function only if for each input there is exactly one output. 3 APPLY 1. Is every function a relation? Is every relation a function? Explain. See margin. 2. Describe a line that cannot be the graph of a linear function. a vertical line BASIC Evaluate the function ƒ(x) = 3x º 10 for the given value of x. 3. x = 0 –10 4. x = 20 50 Day 1: pp. 259–260 Exs. 11–19, 20–28 even, 29–31, 32–48 even Day 2: pp. 260–262 Exs. 33–49 odd, 50–52, 56–58, 65, 70, 72–79, Quiz 3 Exs. 1–19 2 6. x = ᎏᎏ º8 3 5. x = º2 º16 In Exercises 7–9, decide whether the relation is a function. If it is a function, give the domain and the range. 7. Input Output 10. 10 100 20 200 30 300 40 400 50 500 8. Input Output yes; 10, 20, 30, 40, 50; 100, 200, 300, 400, 500 9. 5 3 y AVERAGE 15 no 4 5 3 1 6 3 5 x 2 no Extra Practice to help you master skills is on p. 800. Day 1: pp. 259–260 Exs. 11–19, 20–28 even, 29–31, 32–48 even Day 2: pp. 260–262 Exs. 33–49 odd, 53–55, 59–62, 65, 70, 72–79, Quiz 3 Exs. 1–19 BLOCK SCHEDULE pp. 259–262 Exs. 11–19, 20–28 even, 29–31, 32–49, 50–53, 56–58, 65, 70, 72–79, Quiz 3 Exs. 1–19 GRAPHICAL REASONING Decide whether the graph represents y as a function of x. Explain your reasoning. 11º13. See margin. y 11. Day 1: pp. 259–260 Exs. 11–19, 20–28 even, 29–31, 32–48 even Day 2: pp. 260–262 Exs. 33–49 odd, 50–53, 56–58, 65, 70, 72–79, Quiz 3 Exs. 1–19 ADVANCED CAR TRAVEL Your average speed during a trip is 40 miles per hour. Write a linear function that models the distance you travel d(t) as a function of t, the time spent traveling. d(t) = 40t PRACTICE AND APPLICATIONS STUDENT HELP ASSIGNMENT GUIDE y 12. y 13. x 12. No; there are vertical lines that pass through two points on the graph. 13. No; there are vertical lines that pass through two or more points on the graph. STUDENT HELP HOMEWORK HELP Example 1: Example 2: Example 3: Example 4: Exs. 11–19 Exs. 20–28 Exs. 29–40 Exs. 50–52 EXERCISE LEVELS Level A: Easier 11–13, 29–31, 63–66 Level B: More Difficult 14–28, 32–40, 42–52, 56–58, 67–79 Level C: Most Difficult 41, 53–55, 59–62 x x 11. Yes; no vertical line passes through two points on the graph. RELATIONS AND FUNCTIONS In Exercises 14–19, decide whether the relation is a function. If it is a function, give the domain and the range. 14. Input Output yes; 1, 2, 3, 4; 5, 4, 5 1 3, 2 17. 2 4 3 3 4 2 15. Input Output yes; 1, 2, 3, 4; 0 1 2 0 3 yes; 0, 1, 2, 3; 2, 4, 6, 8 Input Output 0 1 2 3 2 4 6 8 16. Input Output no 7 7 9 9 8 10 4 10 no 18. Input Output 0 2 4 2 1 2 3 4 19. yes; 1, 3, 5, 7; 1, 2, 3 Input Output 1 3 5 7 1 2 3 1 4.8 Functions and Relations HOMEWORK CHECK To quickly check student understanding of key concepts, go over the following exercises: Exs. 12, 16, 18, 22, 32, 29, 42, 50. See also the Daily Homework Quiz: • Blackline Master (Chapter 5 Resource Book, p. 11) • Transparency (p. 37) 259 259 EVALUATING FUNCTIONS Evaluate the function when x = 2, x = 0, and x = º3. STUDENT HELP NOTES Homework Help Students can find help for Exs. 42–49 at www.mcdougallittell.com. The information can be printed out for students who don’t have access to the Internet. 20. ƒ(x) = 10x + 1 21. g(x) = 8x º 2 22. h(x) = 3x + 6 21; 1; º29 14; º2; º26 12; 6; º3 23. g(x) = 1.25x 24. h(x) = 0.75x + 8 25. ƒ(x) = 0.33x º 2 2.5; 0; º3.75 9.5; 8; 5.75 º1.34; º2; º2.99 3 2 2 26. h(x) = x º 4 27. g(x) = x + 7 28. ƒ(x) = x + 4 4 5 74 1 º2.5; º4; º6.25 7.8; 7; 5.8 4 }}; 4; 3 }} 7 7 GRAPHICAL REASONING Match the function with its graph. 1 A. ƒ(x) = 3x º 2 B. ƒ(x) = 2x + 2 C. ƒ(x) = x º 2 2 APPLICATION NOTE EXERCISES 50–52 Although the relationships between years and enrollments represent functions, they cannot be modeled by simple linear functions. Each function could be modeled by the union of a set of linear functions over restricted domains. 5 x 3 41. The points (1, 2) and (3, º1) are on the graph, so the slope –1 – 2 3–1 5 INT 1 1 1 1 34. 3 5 x NE ER T 2 2 2 6 2 6 1 1 1 1 36. h(x) = ºx + 2 3 1 37. ƒ(x) = ºx + 1 2 38. ƒ(x) = 4x + 1 39. h(x) = 5 40. g(x) = º3x º 2 41. Writing Describe how to find the slope of the graph of linear function ƒ given that ƒ(1) = 2 and ƒ(3) = º1. See margin. FINDING SLOPE Find the slope of the graph of the linear function f. 42. ƒ(2) = º3, ƒ(º2) = 5 º2 43. ƒ(0) = 4, ƒ(4) = 0 º1 44. ƒ(º3) = º9, ƒ(3) = 9 3 45. ƒ(6) = º1, ƒ(3) = 8 º3 46. ƒ(9) = º1, ƒ(º1) = 2 º0.3 47. ƒ(º2) = º1, ƒ(2) = 6 1.75 48. ƒ(2) = 2, ƒ(3) = 3 1 49. ƒ(º1) = 2, ƒ(3) = 2 0 number of high school students in year x. Estimate ƒ(2005). about 16 million 52. Let g(x) represent the projected y 5 number of college students in year x. Estimate g(2005). 3 y 1x 2 3 about 17 million 1 1 1 3 x 260 37–40, 59. See Additional Answers beginning on page AA1. 260 Student Enrollment 51. Let ƒ(x) represent the projected y 2x 3 36. B 35. g(x) = 2x º 3 a function of the year? Explain. 3 x 1 x A 50. Is the projected high school 50. Yes; yes; in both cases, each input is paired with enrollment a function of the year? exactly one output. Is the projected college enrollment y 1 1 HIGH SCHOOL AND COLLEGE STUDENTS In Exercises 50–52, use the graph below. It shows the projected number of high school and college students in the United States for different years. Let x be the number of years since 1980. 51–52. Estimates may vary. x y 5x 6 35. HOMEWORK HELP Visit our Web site www.mcdougallittell.com for help with Exs. 42–49. y 3 3 2 STUDENT HELP y x 4 3 x GRAPHING FUNCTIONS In Exercises 32–40, graph the function. 32–40. See margin. 33. g(x) = ºx + 4 34. h(x) = 5x º 6 32. ƒ(x) = º2x + 5 is }} = º}}. y 1 Enrollment (millions) 33. 1 1 1 x C y 2x 5 1 y 1 1 5 1 1 31. y 1 y 3 30. y Chapter 4 Graphing Linear Equations and Functions INT 32. 29. NE ER T 18 16 14 12 High school College 10 0 0 5 10 15 20 Years since 1980 25 x DATA UPDATE of National Center for Education Statistics data at www.mcdougallittell.com 53. ZYDECO MUSIC The graph shows the number of people who attended the Southwest Louisiana Zydeco Music Festival for different years, where k is the number of years since 1980. Is the number of people who attended the festival a function of the year? Explain. Music Festival Attendance FOCUS ON APPLICATIONS APPLICATION NOTE EXERCISE 53 Students should recognize that the data do not represent a linear function, although a linear function can be used for modeling purposes. Additional information about Zydeco Music is available at www.mcdougallittell.com. 9000 6000 3000 0 䉴 Source: Louisiana Zydeco Music Festival 0 5 10 15 k Years since 1980 Yes; each input is paired with exactly one output. 54. MASTERS TOURNAMENT The table shows information about the top RE FE L AL I 7 winners of the 1997 Masters Tournament in Augusta, Georgia. Graph the relation. Is the money earned a function of the score? Explain. 䉴 Source: Golfweb ZYDECO MUSIC blends elements from a variety of cultures. The accordion has German origins. The rub board was invented by Louisiana natives whose roots go back to France and Canada. Test Preparation Check graphs; yes; each input is paired with exactly one output. Score Prize ($) 55. 282 283 284 285 285 486,000 291,600 183,600 129,600 102,600 102,600 286 78,570 SCIENCE CONNECTION It takes 4.25 years for starlight to travel 25 trillion miles. Let t be the number of years and let ƒ(t) be trillions of miles traveled. Write a linear function ƒ(t) that expresses the distance traveled as a function 100 }t, or f(t) ≈ 5.88t of time. f(t) = } 17 QUANTITATIVE COMPARISON In Exercises 56–58, choose the statement that is true about the given quantities. A ¡ B ¡ C ¡ D ¡ 1 60. If f(x) = }}, there is no x output value corresponding to the input value 0; 1 if f(x) = }}, there is x+3 no output value corresponding to the input value –3. 56. 61. Let the domain of f(x) = 1 }} be all real numbers 57. x except 0; let the domain 1 58. of f(x) = }} be all x+3 real numbers except º3. ★ Challenge 270 The number in column A is greater. The number in column B is greater. The two numbers are equal. The relationship cannot be determined from the given information. Column A Column B ƒ(x) = º7x + 4 when x = º4 ƒ(x) = º7x + 4 when x = º5 B 1 ƒ(x) = ᎏᎏx º 3 when x = 7 2 1 ƒ(x) = ᎏᎏx º 5 when x = 11 2 C 1 7 2 7 g(x) = 6x º ᎏᎏ when x = ᎏᎏ 1 7 2 7 h(t) = 6t º ᎏᎏ when t = ᎏᎏ C RESTRICTING DOMAINS In Exercises 59–62, use the equations 1 1 y = }} and y = }}. 60–62. See margin. x+3 x 62. For x < 0 but close to 0, the graph falls sharply. For x > 0 but close to 0, the graph rises sharply. EXTRA CHALLENGE www.mcdougallittell.com ADDITIONAL PRACTICE AND RETEACHING 59. Use a graphing calculator or computer to graph both equations. See margin for graph. 60. Explain why each equation is not a function for all real values of x. For Lesson 4.8: 61. How can you restrict the domain of each equation so that it is a function? 1 62. Describe what happens to the graph of y = ᎏᎏ near the value(s) of x for which x the graph is not a function. 4.8 Functions and Relations 261 • Practice Levels A, B, and C (Chapter 4 Resource Book, p. 112) • Reteaching with Practice (Chapter 4 Resource Book, p. 115) • See Lesson 4.8 of the Personal Student Tutor For more Mixed Review: • Search the Test and Practice Generator for key words or specific lessons. 261 4 ASSESS DAILY HOMEWORK QUIZ Transparency Available 1. Does the graph represent a function? Explain your answer. y 3 ⫺1 ⫺1 冋 冋 9 63. 12 64. –11 7 冋 冋 1 3 x ⫺3 Yes; no vertical line passes through two points of the graph. 3 66. º6 º5 MATRICES Find the sum or the difference of the matrices. (Review 2.4) 63–66. 4 º8 º5 3 See margin. º6.5 º4.2 2.4 º5.1 63. º 64. + 册 册 º4.1 4.3 冋 冋 º9.3 0.7 9.8 65. º12.3 12.4 1 ⫺3 MIXED REVIEW º17.9 º0.1 º13.2 4 º3 5 2. Evaluate ƒ(x) = –3x + 2 when x = –1, 0, and 3. 5, 2, –7 3. Find the slope of the graph of the linear function ƒ if ƒ(3) = –2 and ƒ(0) = 1. –1 册 册 1 º3 0 7 0 6.2 65. º2.5 3.4 册 冋 º5 º7 册冋 册 冋 º12 º3.6 5.9 º4.4 º 9.8 º4.3 º5.8 º9 7.4 0 册 冋 9 66. º4 º5 3.7 1 º7 0 册 冋 4.3 º3 册冋 6 º6 1 + º2 º1 0 册 3 4 5 º5 º4 1 册 SOLVING EQUATIONS Solve the equation if possible. (Review 3.4) 67. 4x + 8 = 24 4 68. 3n = 5n º 12 6 70. º5y + 6 = 4y + 3 1 }} 71. 3b + 8 = 9b º 7 3 69. 9 º 5z = º8z º3 1 2}} 2 72. º7q º 13 = 4 º 7q no solution GRAPHING LINES In Exercises 73–78, write the equation in slope-intercept form. Then graph the equation. (Review 4.6 for 5.1) 73–78. See margin. 73. 2x º y + 3 = 0 74. x + 2y º 6 = 0 75. y º 2x = º7 76. 5x º y = 4 77. x º 2y + 4 = 2 78. 4y + 12 = 0 79. 1 CHARITY WALK You start 5 kilometers from the finish line and walk 1ᎏᎏ 2 kilometers per hour for 2 hours, where d is your distance from the finish line. Graph the situation. (Review 4.6 for 5.1) See margin. EXTRA CHALLENGE NOTE Challenge problems for Lesson 4.8 are available in blackline format in the Chapter 4 Resource Book, p. 119 and at www.mcdougallittell.com. QUIZ 3 Self-Test for Lessons 4.7 and 4.8 Solve the equation graphically. Check your solution algebraically. (Lesson 4.7) 1 2 2. 6x º 12 = º9 }} 1. 4x + 3 = º5 º2 1 2 3 3 5. ᎏᎏx + 5 = ºᎏᎏx º 8 º13 6. ᎏᎏx + 2 = ºᎏᎏx º 6 3 3 4 4 1 º5}} 3 Evaluate the function when x = 3, x = 0, and x = º4. (Lesson 4.8) 1 2 4. º5x º 4 = 3x º}} ADDITIONAL TEST PREPARATION 1. WRITING Explain the relationship between a relation and a function. A function is a relation 7. h(x) = 5x º 9 8. g(x) = º4x + 3 9. ƒ(x) = 1.75x º 2 3.25; º2; º9 6; º9; º29 º9; 3; 19 2 1 4 3 10. h(x) = º1.4x 11. ƒ(x) = ᎏᎏx + 9 9}4}; 9; 8 12. g(x) = ᎏᎏx + ᎏᎏ 4 7 2 7 º4.2; 0; 5.6 2; }}; º2 7 In Exercises 13–18, graph the function. (Lesson 4.8) 13–18. See margin. 13. ƒ(x) = º5x 14. h(x) = 4x º 7 15. ƒ(x) = 3x º 2 in which each value in the domain is associated with exactly one value in the range. 2. OPEN ENDED Give an example of a linear function that models a real-world problem. Explain the significance of one point on the graph of the function. 2 16. h(x) = ᎏᎏx º 1 5 See sample answer at right. 19. 73–79. See Additional Answers beginning on page AA1. 262 1 1 17. ƒ(x) = ᎏᎏx + ᎏᎏ 4 2 18. g(x) = º6x + 5 LUNCH BOX BUSINESS You have a small business making and delivering box lunches. You calculate your average weekly cost y of producing x lunches using the function y = 2.1x + 75. Last week your cost was $600. How many lunches did you make last week? Solve algebraically and graphically. (Lesson 4.7) 250 lunches Chapter 4 Graphing Linear Equations and Functions Additional Test Preparation Sample answer: 2. I send my film away for processing. The company charges me $4.10 for shipping and $3.75 for processing each roll of film. The equation y = 3.75x + 4.10, where 262 3. 8x º 7 = x 1 x = the number of rolls of film that are processed and y = the total cost, models the situation; the point (4, 19.1) represents the cost ($19.10) of processing 4 rolls of film.