Chapter 7 Algebra: Graphs, Functions, and Linear Systems

Chapter 7 Algebra: Graphs, Functions, and Linear Systems Check Points 7.1 1.. x y x ( x, y ) y ( ) 7 (,7) y ( ) 6 (,6) 1 y ( 1) 1 ( 1,) 0 y (0) 0 (0,) 1 y (1) 1 (1, ) y () (,) y () 1 (,1). a. Without the discount pass With the discount pass x y x ( x, y ) x y 10 x ( x, y ) 0 y (0) 0 (0,0) 0 y 10 0 10 (0,10) y () (,) y 10 1 (,1) y () 8 (,8) y 10 1 (,1) 6 y (6) 1 (6,1) 6 y 10 6 16 (6,16) 8 y (8) 16 (8,16) 8 y 10 8 18 (8,18) 10 y (10) 0 (10,0) 10 y 10 10 0 (10,0) 1 y (1) (1, ) 1 y 10 1 (1, ) Copyright 01 Pearson Education, Inc. 79

Chapter 7 Graphs, Functions, and Linear Systems c. The graphs intersect at (10,0). This means that if the bridge is used ten times in a month, the total monthly cost is $0 with or without the discount pass.. a. f( x) x f (6) (6) 9 c. gx ( ) x 10 g( ) ( ) 10 6 hr () r 7r h( ) ( ) 7( ) 6. a. A car s required stopping distance at 0 miles an hour on dry pavement is about 190 feet. f( x) 0.087x 0.x66.6 f (0) 0.087(0) 0.(0) 66.6 191 6. x f( x) x ( x, y ) or x,f(x) x g( x) x ( x, y) or x,f(x) f ( ) ( ) (, ) g( ) ( ) 7 (, 7) 1 f ( 1) ( 1) ( 1, ) 1 g( 1) ( 1) ( 1, ) 0 f (0) (0) 0 (0,0) 0 g(0) (0) (0, ) 1 f (1) (1) (1, ) 1 g(1) (1) 1 (1, 1) f () () (,) g() () 1 (,1) The graph of g is the graph of f shifted vertically down units. 80 Copyright 01 Pearson Education, Inc.

Section 7.1 Graphing and Functions 7. a. y is a function of x.. y is a function of x. c. y is not a function of x. Two values of y correspond to an x-value. 8. a. The concentration is increasing from 0 to hours. The concentration is decreasing from to 1 hours.. c. The maximum concentration of 0.0 mg per 100 ml occurs after hours. d. None of the drug is left in the body. e. The graph defines y as a function of x because no vertical line intersects the graph in more than one point.. Concept and Vocabulary Check 7.1 1. x-axis.. y-axis. origin. quadrants; four. x-coordinate; y-coordinate 6. 6. solution; satisfies 7. y; x; function 8. x; 6 9. more than once; function 7. Exercise Set 7.1 1. 8. Copyright 01 Pearson Education, Inc. 81

Chapter 7 Graphs, Functions, and Linear Systems 9. 16. 10. 17. 11. 18. 1. 19. 1. 0. 1. 1. x 1 0 1 y x 7 1 1 7 1. 8 Copyright 01 Pearson Education, Inc.

Section 7.1 Graphing and Functions. y x x 1 0 1 11 6 6 11 7. x 1 0 1 1 1 1 x 1 0 1 y. x 1 0 1 y x 1 0 1 8. y x 1 0 1 1 7 1 x 1. x 1 0 1 y x 1 0 1 9. x 1 0 1 y x 7 8 1 0 1 8 7. x 1 0 1 y x1 1 1 7 0. x 1 0 1 y x 1 8 9 1 0 7 6 6. x 1 0 1 y x 10 8 6 0 1. x 1 0 1 y x 1 1 Copyright 01 Pearson Education, Inc. 8

Chapter 7 Graphs, Functions, and Linear Systems. x 1 0 1 y x 1 1 0 1 0 1 0. gx ( ) x 1 a. g() () 1 1 g( ) ( ) 1 161 1 h r r 1.. f(x) = x a. f(8) = 8 = f(1) = 1 =. f(x) = x 6 a. f(9) = 9 6 =. a. h 16 8 h 1 1 8 hr () r f() = 6 =. f(x) = x a. f(7) = (7) = 1 = 19 f(0) = (0) = 0 = 6. f(x) = x a. f(7) = (7) = 8 = f(0) = (0) = 0 = g x x 7. 8. 9. 1 a. g 11 g 11 gx ( ) x a. g() () 9 1 g( ) ( ) 9 1 gx ( ) x a. g() () 16 1 a... h() () () 0 6 h( 1) ( 1) f x x x 1 a. f 1 (9) 9 1 189 1 6 16 11 f 1 1 1 19 f( x) x x a. f () () () () 8 1 8 18 g( ) ( ) 9 7 f ( 1) ( 1) ( 1) 8 Copyright 01 Pearson Education, Inc.

Section 7.1 Graphing and Functions x. f( x) x a. 6 f (6) 1 6 6 6 f ( 6) 1 6 6 9. x f x x 1 1 0 1 1 0 1 6. f( x) x x a. f () 1 f ( ) 1 0. x f( x) x 1 7. x f x x 1 0 0 1 1 0 1 1 1 0 0 1 1 8. x f( x) x 1 1 0 1 1 1. x f x x 0 1 1 0 1 Copyright 01 Pearson Education, Inc. 8

Chapter 7 Graphs, Functions, and Linear Systems. x f( x) ( x 1) 1 1 0 0 1 1. y is a function of x. 6. y is a function of x. 7. y is a function of x. 8. y is not a function of x. All values of y correspond to an x-value. 9. y is not a function of x. Two values of y correspond to an x-value. 60. y is not a function of x. Two values of y correspond to an x-value. 61. y is a function of x. 6. y is not a function of x. Two values of y correspond to an x-value.. x f x x 1 6 7 1 0 0 1 1 6. g 1 1 f g f 1 10 6. g f g f 1 1 8 1 8 8 8 6 8 76 6. 1 6 6 6 166 6. x f( x) ( x 1) 61 6 8 8 1 1 0 0 1 1 8 66. 1 6 196 916 96 660 67. 86 Copyright 01 Pearson Education, Inc.

Section 7.1 Graphing and Functions 68. 78. a. G(10) 0.01(10) (10) 60 69 In 1990, the wage gap was 69%. This is represented as (10, 69) on the graph. G (10) underestimates the actual data shown by the bar graph by %. 69. 70. 71. The coordinates of point A are (,7). When the football is yards from the quarterback, its height is 7 feet. 7. The coordinates of point B are (8,7). When the football is 8 yards from the quarterback, its height is 7 feet. 7. The coordinates of point C are approximately (6, 9.). 7. The coordinates of point D are approximately (, 9.). 7. The football s maximum height is 1 feet. It reaches this height when it is 1 yards from the quarterback. 76. The football s height is feet when it is caught by the receiver. The receiver is 0 yards from the quarterback when he catches the ball. 77. a. G(0) 0.01(0) (0) 60 81 In 010, the wage gap was 81%. This is represented as (0,81) on the graph. 79. f (0) 0. 0 6 0 1000 0. 00 70 1000 160 70 1000 60 1000 0 Twenty-year-old drivers have 0 accidents per 0 million miles driven. This is represented on the graph by point (0,0). 80. f (0) 0. 0 6 0 1000 0. 00 1800 1000 1000 1800 1000 00 Fifty-year-old drivers have 00 accidents per 0 million miles driven. This is represented on the graph by point (0,00). 81. The graph reaches its lowest point at x. f () 0. 6 1000 0. 0 160 1000 810 160 1000 810 1000 190 Drivers at age have 190 accidents per 0 million miles driven. This is the least number of accidents for any driver between ages 16 and 7. 8. Answers will vary. One possible answer is age 16 and age 7. f (16) 0. 16 6 16 1000 0. 6 76 1000 10. 76 1000 6. f (7) 0. 7 6 7 1000 0. 76 66 1000 190. 66 1000 6. Both 16-year-olds and 7-year-olds have approximately 6. accidents per 0 million miles driven. 89. makes sense G (0) underestimates the actual data shown by the bar graph by %. Copyright 01 Pearson Education, Inc. 87

Chapter 7 Graphs, Functions, and Linear Systems 90. does not make sense; Explanations will vary. Sample explanation: The notation f( x ) does not mean multiplication of f by x. 91. makes sense 9. makes sense 9. f( 1) g( 1) 1 ( ) 9. f(1) g(1) ( ). Step 1. Plot the y-intercept of (0, 1) Step. Obtain a second point using the slope m. Rise m Run Starting from the y-intercept move up units and move units to the right. This puts the second point at (, 6). Step. Draw the line through the two points. f g( 1) f( ) 1 9. f g(1) f( ) 96. Check Points 7. 1. Find the x-intercept by setting y = 0 x (0) 6 x 6 x ; resulting point (, 0) Find the y-intercept by setting x = 0 (0) y 6 y 6 y ; resulting point (0, ) Find a checkpoint by substituting any value. (1) y 6 y 6 y y ; resulting point 1,. Solve for y. xy 0 y x0 y x 0 y x0 m and the y-intercept is (0, 0). Draw horizontal line that intersects the y-axis at.. a. 6 m 6 ( ) 1 ( ) 7 7 m 1 6. Draw vertical line that intersects the x-axis at. 88 Copyright 01 Pearson Education, Inc.

Section 7. Linear Functions and Their Graphs 7. The two points shown on the line segment for Medicare are (007, 6) and (016, 909). 11 m 0.8 010 1970 0 For the period from 1970 through 010, the percentage of married men ages 0 to decreased by 0.8 per year. The rate of change 0.8% is per year. 8. a. The y-intercept is 8 and the slope is Change in y 8 16 m 0. Change in x 0 0 0 The equation is Cx ( ) 0.x 8. Cx ( ) 0.x8 C(60) 0.(60) 8 7. The model projects that 7.% of the U.S. population will be college graduates in 00. Concept and Vocabulary Check 7. 1. x-intercept. y-intercept Exercise Set 7. 1. Find the x-intercept by setting y = 0 x y x 0 x ; resulting point (, 0) Find the y-intercept by setting x = 0 0 y y y ; resulting point (0, ). Find the x-intercept by setting y = 0 x y x 0 x ; resulting point (, 0) Find the y-intercept by setting x = 0 0 y y ; resulting point (0, ) y y 1. x x1. y mx b; slope; y-intercept. ; 6. 0, ; ; 7. horizontal 8. vertical. Find the x-intercept by setting y = 0 x (0) 1 x 1 x ; resulting point (, 0) Find the y-intercept by setting x = 0 (0) y 1 y 1 y ; resulting point (0, ) Copyright 01 Pearson Education, Inc. 89

Chapter 7 Graphs, Functions, and Linear Systems. Find the x-intercept by setting y = 0 x (0) 10 x 10 x ; resulting point (, 0) Find the y-intercept by setting x = 0 (0) y 10 y 10 y ; resulting point (0, ) 7. Find the x-intercept by setting y = 0 x (0) 1 x 1 x ; resulting point (, 0) Find the y-intercept by setting x = 0 (0) y 1 0y 1 y 1 y ; resulting point (0, ). Find the x-intercept by setting y = 0 x 06 x 6 x ; resulting point (, 0) Find the y-intercept by setting x = 0 (0) y 6 y 6; resulting point (0, 6) 8. Find the x-intercept by setting y = 0 x (0) 6 x 6 x ; resulting point (, 0) Find the y-intercept by setting x = 0 (0) y 6 0y 6 y 6 y ; resulting point (0, ) 6. Find the x-intercept by setting y = 0 x (0) 6 x 6; resulting point (6, 0) Find the y-intercept by setting x = 0 0y 6 y 6 y ; resulting point (0, ) 9. 10. 11. 1. 1. 6 1 m 1; line falls. 1 m ; line falls. 1 1 1 m ; line rises. 1 m ; line rises. ( 1) 1 m ; line falls. 1 1 90 Copyright 01 Pearson Education, Inc.

Section 7. Linear Functions and Their Graphs 1. 1. 16. 17. 18. 19. 0. ( ) m 1 ; line falls. 6 m ; 0 Slope undefined. Line is vertical. ( ) 9 m ; 0 Slope undefined. Line is vertical. 8 0 8 m ; line falls. 0 m 9 0 9 ; line rises. 0 11 0 m 0; line is horizontal. 7 0 m 0 ; line is horizontal. 1 ( ). y = x + Slope:, y-intercept: Plot point (0, ) and second point using rise m 1 run. y = x + Slope:, y-intercept: Plot point (0, ) and second point using rise m 1 run 1. y = x + Slope:, y-intercept: Plot point (0, ) and second point using rise m 1 run. 1 y x Slope: 1, y-intercept: Plot point (0, ) and second point using m 1 rise. run. y = x + 1 Slope:, y-intercept: 1 Plot point (0, 1) and second point using rise m 1 run Copyright 01 Pearson Education, Inc. 91

Chapter 7 Graphs, Functions, and Linear Systems 6. 1 y x Slope: 1, y-intercept: Plot point (0, ) and second point using m 1 rise. run 9. y x Slope:, y-intercept: Plot point (0, ) and second point using rise m. run 7. f( x) x Slope:, y-intercept: Plot point (0, ) and second point using rise m. run 0. y x Slope:, y-intercept: Plot point (0, ) and second point using rise m. run 8. f( x) x Slope:, y-intercept: Plot point (0, ) and second point using m rise. run 1. f( x) x or f( x) x 0 Slope:, y-intercept: 0 Plot point (0, 0) and second point using rise m. run 9 Copyright 01 Pearson Education, Inc.

Section 7. Linear Functions and Their Graphs. f( x) x or f( x) x 0 Slope:, y-intercept: 0 Plot point (0, 0) and second point using rise m. run c. 6. a. y = x y x or y x 0. a. x y 0 y = x or y = x + 0 Slope y-intercept = 0 c. Slope = y-intercept = 0 c. 7. a. x y y x. a. x + y = 0 y = x or y = x + 0 Slope = y-intercept = c. Slope = y-intercept = 0 c. 8. a. x + y = y = x +. a. y = x y xor y x 0 Slope = y-intercept = c. Slope = y-intercept = 0 Copyright 01 Pearson Education, Inc. 9

Chapter 7 Graphs, Functions, and Linear Systems 9. a. 7xy 1 y 7x1 7 y x7 7 Slope = y-intercept = 7 c.. y =. y = 0. a. xy 1 y x1 y x Slope y-intercept = c.. y =. x = 1. y = 6. x = 9 Copyright 01 Pearson Education, Inc.

Section 7. Linear Functions and Their Graphs 7. x + 1 = 0 or x = 1. Ax ByC Ax C By A C x y B B The slope is A C and the y intercept is. B B 8. x + = 0 or x =. y 1 y 6 y y y 9. 0. 1.. m 0 a a a b 0 b b Since a and b are both positive, Therefore, the line falls. b0 b b m 0 a a a Since a and b are both positive, Therefore, the line falls. bc b c m a a 0 The slope is undefined. The line is vertical. ac c a m a a b b a is negative. b b is negative. a Since a and b are both positive, a is positive. b Therefore, the line rises. 6. 1 y 1 y 1 y 6 6 y 61y 18 y 6 y 7. m1, m, m, m 8. b, b1, b, b 9. a. Change in y 9 9 0 m 0. Change in x 6 0 For each year of aging, the percentage of Americans reporting a lot of stress decreases by 0.%. The rate of change is 0.% per year of aging.. Ax By C By Ax C A C y x B B A The slope is B and the y intercept is C B. 60. a. Change in y 6 16 m 0.16 Change in x 180 80 100 For each minute of brisk walking, the percentage of patients with depression in remission increased by 0.16%. The rate of change is 0.16% per minute of brisk walking. Copyright 01 Pearson Education, Inc. 9

Chapter 7 Graphs, Functions, and Linear Systems 61. Px ( ) 0.7x 18 6. Px ( ) 0.x 6. a. Find slope by using the endpoints of the line segment. 90 60 80 m 16 010 1980 0 The value of b is the y-intercept, or 60. W( x) mxb W( x) 16x60 W( x) 16x60 W( x) 16(0) 60 1100 In the year 00, the number of bachelor s degrees that will be awarded to women will be about 1,100,000. 6. a. Find slope by using the endpoints of the line segment. 710 70 0 m 8 010 1980 0 The value of b is the y-intercept, or 70. M( x) mxb M( x) 8x70 M( x) 8x70 M( x) 8(0) 70 790 In the year 00, the number of bachelor s degrees that will be awarded to men will be about 790,000. 7. does not make sense; Explanations will vary. Sample explanation: Either point can be considered, x, y. x y or 1 1 7. does not make sense; Explanations will vary. Sample explanation: Since college cost are going up, this function has a positive slope. 76. does not make sense; Explanations will vary. Sample explanation: The slope of lines whose equations are in this form can be determined in several ways. One such way is to rewrite the equation in slope-intercept form. 77. makes sense 78. false; Changes to make the statement true will vary. A sample change is: It is possible for m 79. false; Changes to make the statement true will vary. A sample change is: Vertical lines can not be expressed in slope-intercept forn. 80. true 81. false; Changes to make the statement true will vary. A sample change is: The line y x 7 is 7 equivalent to y x which has a y-intercept of 7. 8. First, find the slope using the points 0, and 100,1. 1 180 9 m 100 0 100 The slope, m, is 9 and the y-intercept, b, is. Instead of y mx b we will use F mc F mcb 9 F C 96 Copyright 01 Pearson Education, Inc.