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1 CHAPTER Algebra Toolbox CHAPTER Functions Graphs, and Models; Linear Functions Toolbox Exercises. {,,,,5,6,7,8} and { xx< 9, x N} Remember that x N means that x is a natural number.. (,7]. (,7 ] 5. (,) 6.. Yes.. No. A is not a subset of B. A contains elements, 7, and 0 which are not in B. 7. Note that 5 > x implies x< 5, therefore:. No. N {,,,,...}. Therefore, N. 5. Yes. Every integer can be written as a fraction with the denominator equal to. 6. Yes. Irrational numbers are by definition numbers that are not rational Integers. However, note that this set of integers could also be considered as a set of rational numbers. See question Rational numbers Irrational numbers 0. x >. x. x Copyright 0 Pearson Education. Inc.

2 CHAPTER Functions, Graphs, and Models; Linear Functions. 9. The x term has a coefficient of 5. The x term has a coefficient of 7. The constant term is.. 0. ( z 5z + 0z 6) + ( z + z z 5) ( z + z ) + ( z ) + ( 5z z ) + ( 0z) + ( 6 5) 7 0 z + z z + z x+ y x y x y + y + ( ) y 5y x y y + + ( x 5x) + ( 9 + 0) 7 9 y x y y x.. ( ) p+ d p+ d. ( ) x 7y 6x+ y. ab ( + 8c) ab 8ac. Yes, it is a polynomial with degree. 5. No, it is not a polynomial since there is a variable in a denominator. 6. No, it is not a polynomial since there is a variable inside a radical. 7. Yes, it is a polynomial with degree ( x y) ( x+ y) x y x y x 6y x 6y 6. ( x y) + xy 5( y xy) ( x y) 8x y+ xy 5y+ 5xy x+ y ( 8x x) ( xy 5xy) ( y 5y y) x+ 9xy 5y 8. The x term has a coefficient of. The x term has a coefficient of. The constant term is x( yz ) ( 5xyz x) 8xyz 8x 5xyz + x ( 8xyz 5xyz) ( 8x x) + + xyz 5x Copyright 0 Pearson Education. Inc.

3 CHAPTER Section. Section. Skills Check. Using Table A a. 5 is an x-value and therefore is an input into the function f( x ). f ( 5) represents an output from the function. 5. No. In the given table, x is not a function of y. If y is considered the input variable, one input will correspond with more than one output. Specifically, if y 9, then x or x Yes. Each input y produces exactly one output x. c. The domain is the set of all inputs. D:{ 9, 7, 5, 6,,7, 0}. The range is the set of all outputs. R:{,5,6,7,9,0 } d. Each input x into the function f yields exactly one output y f ( x).. Using Table B. a. 0 is an x-value and therefore is an input into the function gx ( ). g ( 7) represents an output from the function. c. The domain is the set of all inputs. D:{,, 0,,, 7,}. The range is the set of all outputs. R:{,5,7,8,9,0,5 } d. Each input x into the function g yields y g x. exactly one output ( ) f ( 9) 5 f 7 9 ( ) 7. a. f (), since x in the table corresponds with f( x ). f () 0 () 0 () 0 c. f (), since (, ) is a point on the graph. 8. a. ( ) 5 9. f, since (,5) is a point on the graph. f ( ) 8, since x in the table corresponds with f(x) 8. c. ( ) x f ( ) + ( ) y 0. g( ) 5 g() 8 0. x y Copyright 0 Pearson Education. Inc.

4 CHAPTER Functions, Graphs, and Models; Linear Functions. Recall that R( x) 5x+ 8. a. R( ) 5( ) R( ) 5( ) c. R () 5() Recall that Cs () 6 s. a. C () 6 () 6 (9) Yes. The graph passes the vertical line test. Each input matches with exactly one output. 8. No. The graph fails the vertical line test. Each input does not match with exactly one output. 9. No. If x, then y 5or y 7. One input yields two outputs. The relation is not a function. 0. Yes. Each input x yields exactly one output y. C( ) 6 ( ) 6 () a. Not a function. If x, then y or y 8. Yes. Each input yields exactly one output. c. C( ) 6 ( ) 6 () 6. Yes. Each input corresponds with exactly,0,,,. one output. The domain is { } The range is { 8,,,5,7}.. No. Each input x does not match with exactly one output y. Specifically, if x then y or y. 5. No. The graph fails the vertical line test. Each input does not match with exactly one output.. a. Yes. Each input yields exactly one output. Not a function. If x, then y or y 6.. a. Not a function. If x, then y or y. Function. Each input yields exactly one output.. a. Function. Each input yields exactly one output. Not a function. If x, then y or y Yes. The graph passes the vertical line test. Each input matches with exactly one output. Copyright 0 Pearson Education. Inc.

5 CHAPTER Section The domain is the set of all inputs. D:{,,,,, }. The range is the set of all outputs. R:{ 8,,,, 6} 6. The domain is the set of all inputs. D:{ 6,,, 0,, }. The range is the set of all outputs. R:{ 5,, 0,,, 6} 7. Considering y as a function of x, the domain is the set of all inputs, x. Therefore the domain is D:[ 0, 8]. The range is the set of all outputs, y. Therefore, the range is R:[, ]. 8. Considering y as a function of x, the domain is the set of all inputs, x. Therefore the domain is D:[, ]. The range is the set of all outputs, y. Therefore, the range is R:[, ].. The denominator of the fractional part of the function will be zero if x + 0 or x. This implies x. The domain is all real numbers except or in interval notation D:,,. ( ) ( ). The denominator of the fractional part of the function will be zero if x 6 0. Solving for x yields: x x 6 x This implies x. The domain is all real numbers except or in interval notation D:,,. ( ) ( ). No. If x 0, then (0) + y y y±. So, one input of 0 corresponds with outputs of and. Therefore the equation is not a function. 9. Since y x 6 will not be a real number if x 6< 0, the only values of x that yield real outputs to the function are values that satisfy x 6 0. Isolating x yields: x x 6 x,. Therefore the domain is D: [ ). Yes. Each input for x corresponds with exactly one output for y. 5. C π r, where C is the circumference and r is the radius. 6. D is found by squaring E, multiplying the result by, and subtracting Since y x 8 will not be a real number if x 8< 0, the only values of x that yield real outputs to the function are values that satisfy x 8 0. Isolating x yields: x x 8 x,. Therefore the domain is D: [ ) Copyright 0 Pearson Education. Inc.

6 6 CHAPTER Functions, Graphs, and Models; Linear Functions Section. Exercises 7. a. No. Each input (x, given day) corresponds with multiple outputs (p, stock prices). Stock prices fluctuate throughout the trading day. Yes. Each input (x, given day) corresponds with exactly one output (p, the stock price at the end of the trading day). The independent variable is x, the given day, and the dependent variable is p, the stock price at the end of the trading day. 8. a. Yes. Each input (m, miles traveled) corresponds with exactly one output (s, odometer reading). The independent variable is m, miles traveled, and the dependent variable is s, the odometer reading. Yes. Each input (s, odometer reading) corresponds with exactly one output (m, miles traveled). The independent variable is s, odometer reading, and the dependent variable is m, miles traveled. 9. a. Yes. Each input (a, age in years) corresponds with exactly one output (p, life insurance premium). The independent variable is a, age in years, and the dependent variable is p, life insurance premium. No. One input of $.8 corresponds with six outputs (a, age in years). 0. Yes. Each input (d, degree) corresponds with exactly one output (M, mean earnings). The independent variable is d, highest degree for females in 007, and the dependent variable is M, mean earnings in dollars.. Yes. Each input (m, month) corresponds with exactly one output (r, unemployment rate). The independent variable is m, the month during June to November 00, and the dependent variable is r, the U.S. unemployment rate for those months.. T, temperature, is a function of m, number of minutes after the power outage, since each value for m corresponds with exactly one value for T. The graph of the equation passes the vertical line test, which implies there is one temperature for each value of m, number of minutes after the power outage.. a. Yes. Each input (the barcode) corresponds with exactly one output (an item s price). No. Every input (an item s price) could correspond with more than one output (the barcode). Numerous items can have the same price but different barcodes.. a. Yes. Each input (a child s piano key) corresponds with exactly one output (a musical note). Since the domain is the set of all inputs into the function and there are keys on the child s piano keyboard, there are elements in the domain of the function. Yes. Each input (a note from the child s piano) corresponds with exactly one output (a piano key). Since the range is the set of all outputs from the function and there are keys on the child s piano keyboard, there are elements in the range of the function. 5. Yes. Each input (x, years) corresponds with exactly one output (V, value of the property). The graph of the equation passes the vertical line test. Copyright 0 Pearson Education. Inc.

7 CHAPTER Section Yes. Each input (d, depth) corresponds with exactly one output (p, pressure). The graph of the equation passes the vertical line test. 7. a. Yes. Each input (day of the month) corresponds with exactly one output (weight). The domain is the first days of May or D:{,,,,5,6,7,8,9,0,,,, }. c. The range is { 7,7,7,7,75,76,77,78 }. d. The highest weights were on May and May. e. The lowest weight was on May. f. Three days from May 8 until May. 8. a. No. One input of 75 matches with two outputs of 70 and 8. Yes. Each input (average score on the final exam) matches with exactly one output (average score on the math placement test). 9. a. P () $ If the car is financed over three years, the payment is $ C (5) $, c. t. If the total cost is $,0.08, then the car has been financed over four years, d. Since C(5) $,580.80, and C() $9,8.08, the savings would be C(5) C() $096.7, 50. a. The couple must make payments for 0 years. f 0,000 0 ( ) f ( 0,000) 0. It will take the couple 0 years to payoff a $0,000 mortgage at 7.5%. c. f ( i ) f ( ) 0,000 0,000 0 d. If A 0,000 then ( ) ( ) f A f 0, e. f ( i ) f ( ) if ( 0,000) i5 5 0,000 0,000 0 The expressions are not equal. 5. a. Approximately million f ( 90). Approximately million women were in the work force in 90. c. D: { 90,90,950,960,970, 980,990,000,005,00,05 d. Increasing. As the year increases, the number of women in the work force also increases. 5. a. When t 00, the ratio is approximately to. f (005). For year 005, the projected ratio of the working-age population to the elderly is to. c. The domain is the set of all possible inputs. In this example, the domain consists of all the years, t, represented in the figure. Specifically, the domain is {995,000,005,00,05, 00,05,00}. } Copyright 0 Pearson Education. Inc.

8 8 CHAPTER Functions, Graphs, and Models; Linear Functions d. As the years, t, increase, the projected ratio of the working-age population to the elderly decreases. Notice that the bars in the figure grow smaller as the time increases. 5. a. f ( 890) 6. f ( 00) 7. g ( 90).5 g ( 000) 5. c. If x 980, then f ( x ).7. The median age at first marriage for men in 980 was.7 years. d. f ( 00) 7..8 f ( 960) >. Therefore, the median age at first marriage for men increased between 960 and a. f (005) 9, 60 In year 005 there were 9,60 nonfatal firearm incidents. c. The maximum number of nonfatal firearm incidents is,060,800, occurring in year 99. Note that f (99),060, a. In 000,.5 million homes used the Internet. f ( 008) In 008, 78.0 million U.S. homes used the Internet. c f ( 998) 6.0. d. The function is increasing. Since 008, home use of the Internet in the U.S. has increased rapidly. 56. a. Yes. Each year, t, corresponds with exactly one number of U.S. farms, N. f (90) 6.(millions). f (90) represents the number of U.S. farms in millions in the year 90. c. If f() t., then t 990. d. f (009). implies that in 009, there were. million farms in the U.S. 57. a. f ( 995) In 995, the birth rate for U.S. girls ages 5 to 9 was 56.0 per 000 girls f ( 005) 0.5. c. The birth rate appears to be at its maximum (59.9) in the year 990. d. In the years 005 to 009, the birth rate appears to have increased until 007, then decreased. 58. a. f ( 990).. In 990 there were. workers for each retiree. 00. f ( 00). c. As the years increase, the number of workers available to support retirees decreases. Therefore, funding for social security into the future is problematic. Workers will need to pay a larger portion of their salaries to fund payments to retirees. 59. a. R ( 00) ( 00) 600. The revenue generated from selling 00 golf hats is $600. R ( 500) ( 500 ) $80,000 Copyright 0 Pearson Education. Inc.

9 CHAPTER Section a. C ( 00) ( 00) 600. The cost of producing 00 golf hats is $600. C ( 500) ( 500) $, a. ( ) ( ) f The monthly charge for using 000 kilowatt hours is $ ( ) ( ) f The monthly charge for using 500 kilowatt hours is $ a. ( ) P (500) 0.(500) 000 5,000 5, ,000 The profit generated from the production and sale of 500 ipod players is $98,000. P( 000),800,000,600, ,000 P 000 $98,000 50(000) 0.(000) 000 ( ) P The daily profit from the production and sale of 00 Blue Chief bicycles is $ a. ( ) ( ) ( ) ( ) ( ) ( ) P The daily profit from the production and sale of 60 Blue Chief bicycles is $ a. h () 6+ 96() 6() The height of the ball after one second is 86 feet. h ( ) 6+ 96( ) 6( ) c After three seconds the ball is 50 feet high. Test t. h ( ) 6+ 96( ) 6( ) Test t. h ( ) 6+ 96( ) 6( ) Test t 5. h ( 5) 6+ 96( 5) 6( 5) Since h() 86, and h(), and h() 50, and h(), and h(5) 86, it appears that the ball stops climbing after seconds and begins to fall. One might conclude that the ball reaches its maximum height at seconds since at and 5 seconds, and again at and seconds, the respective heights are the same. Copyright 0 Pearson Education. Inc.

10 0 CHAPTER Functions, Graphs, and Models; Linear Functions 65. a n 0 0.7n n n 7 Therefore the domain of R( n ) is all real numbers except or 7,, 7 7. In the context of the problem, n represents the factor for increasing the number of questions on a test. Therefore it makes sense that n is positive ( n > 0 ). function is,, the actual domain 0,. in the given physical context is [ ) 67. a. Since p is a percentage, 0 p 00. However in the given function, the denominator, 00 p, cannot equal zero. Therefore, p 00. The domain is 0 p < 00or, in interval notation, [ 0, 00 ). C ( ) ( ) 7, , ,000( 90) C ( 90),, a. Yes, since each value of s produces exactly one value of K c. Any input into the function must not create a negative number under the radical. Therefore, the radicand, s +, must be greater than or equal to zero. Isolating s yields: s + 0 s + 0 s s Therefore, the domain defined by the equation is all real numbers greater than or equal to or, in interval notation,,. c. Since s represents wind speed in the given function, and wind speed cannot be less than zero, the domain of the function is restricted based on the physical context of the problem. Even though the domain implied by the 68. a. Any input into the function must not create a negative number under the square root. Therefore, p + 0. Isolating p yields p + 0 p p Since the denominator cannot equal zero, p. Therefore the domain of q is,. In the context of the problem, p represents the price of a product. Since the price can not be negative, p 0. The domain is [ 0, ). Also, since q represents the quantity of the product demanded by consumers, q 0. The range is ( 0,00 ]. Copyright 0 Pearson Education. Inc.

11 CHAPTER Section. 69. a. V V (08 8) (60) 860 (08 7) (6),66 () () (08 ()) (8) (8) (08 (8)) First, since x represents a side length in the diagram, x must be greater than zero. Second, to satisfy postal restrictions, the length (longest side) plus the girth must be less than or equal to 08 inches. Therefore, Length + Girth 08 Length + x 08 x 08 Length 08 Length x Length x 7 Since x is greatest if the longest side is smallest, let the length equal zero to find the largest value for x. 0 x 7 x 7 Therefore the conditions on x are 0< x 7. If x 7, the length would be zero and the package would not exist. Therefore, in the context of the question, 0< x < 7and the corresponding domain for the function ( ) 0,7. V x is ( ) The maximum volume occurs when x 8. Therefore the dimensions that maximize the volume of the box are 8 inches by 8 inches by 6 inches, a total of 08 inches. 70. a. ( ) ( ) ( ) S The initial height of the bullet is meters. t Height ( ) ( ) ( ) S S 0 9 S 87. c. The bullet seems to reach a maximum height at 0 seconds and then begins to fall. See the table in part b) for further verification, using 9.5 seconds and 0.5 seconds. c. x Volume Copyright 0 Pearson Education. Inc.

12 CHAPTER Functions, Graphs, and Models; Linear Functions Section. Skills Check. a. x y x xy, 7 (, 7) 8 (, 8) (, ) 0 0 (0, 0) (, ) 8 (, 8) 7 (, 7) ( ) c. Your hand-drawn graph in part a) by plotting points from the table should match the calculator-drawn graph in part b).. x y c. Your hand-drawn graph in part a) by plotting points from the table should match the calculator-drawn graph in part b).. a. x y x + ( xy, ) 9 (, 9) 9 (, 9) (, ) 0 (0, ) (, ) 9 (, 9) 9 (, 9). x y Copyright 0 Pearson Education. Inc.

13 CHAPTER Section. 5. x y x y 5 5 undefined 6. x y y this window x 5, yes there is a turning point in 0. y x, yes there is a turning point in this window 7. x 0 y undefined Copyright 0 Pearson Education. Inc.

14 CHAPTER Functions, Graphs, and Models; Linear Functions. y x x in this window, yes there are turning points 5. a. y x+ 0. y x x points in this window +, yes there are turning y x+ 0. y 9/ ( x ) +, yes there is a turning point in this window View b) is better. 6. a. y x x +. y 0 / ( x ) +, yes there is a turning point in this window y x x + Copyright 0 Pearson Education. Inc.

15 CHAPTER Section. 5 y x x View b) is better. 7. a. ( x ) y x + 00 View b) is better. 9. When x 8 or x 8, y. When x 0, y 50. Letting y vary from 5 to 00 gives one view. y x + 50 ( x ) y x + 00 The turning point is at (0,50) and is a minimum point. 8. a. View b) is better. y x x When x 60, y 0. When x 0, y 0. When x 0, y 870. Letting y vary from 000 to 0 gives one view. y x x The turning point is at ( 0, 870) and is a minimum point. Copyright 0 Pearson Education. Inc.

16 6 CHAPTER Functions, Graphs, and Models; Linear Functions.. When x 0, y 50. When x 0, y 850. When x 0, y 0. Letting y vary from 00 to 00 gives one view.. Note that answers for the window may vary. y x x + 0 y x + x 5x The turning points are at ( 5,75), a maximum, and at (, 8), a minimum. Note that answers for the window may vary.. When x 8, y 0. When x 5, y 7. When x, y 7. Letting y vary from 0 to 0 gives one view. There is no turning point t St ( ) 5.t q f( q) q 5q Copyright 0 Pearson Education. Inc.

17 CHAPTER Section a. 8. c. Yes. Yes. 9. a.. a. f ( 0) ( 0) 5( 0) x 0 implies 0 years after 000. Therefore the answer to part a) yields the millions of dollars earned in 00.. a. f ( 0) 00( 0) 5( 0) 0, In 00, 0 x. Therefore, 9950 thousands of units or 9,950,000 units are produced in 00. c. Yes. Yes. Copyright 0 Pearson Education. Inc.

18 8 CHAPTER Functions, Graphs, and Models; Linear Functions Section. Exercises. a. ( ) ( ) y In 9, there were 6. million (6,,000) women in the workforce. In 00, x y 0.006( 0) 0.08( 0) Based on the model in 00, there were 76.7 million (76,7,000) women in the workforce.. a. x Year 970 For 990, x For 005, x For 995, x 5. Therefore, y 0.009(5) + 0.(5) The unemployment rate in Canada for 995 was 8.076%. c. For 008, x 8. Therefore, y 0.009(8) + 0.(8) The unemployment rate in Canada for 008 was.878%. c. xmin x max a. t Year 980 For 988, t For 000, t For 0, t a. P f() represents the value of P in 99 ( ). ( ) ( ) f () ,07 8,7 8,7 represents the cost of prizes and expenses in millions of dollars for state lotteries in 99. c. xmin x max a. t Year 995 For 996, t For 0, t P f(0) represents the value of P in 005 ( ). f (0) 5.8( 0) represents the percent of households with Internet access in 005. Considering the table, S 8 feet when x is or when x is. The height is the same for two different times because the height of the ball increases, reaches a maximum height, and then decreases. Copyright 0 Pearson Education. Inc.

19 CHAPTER Section. 9 c. From the table in part b), it appears the maximum height is 6 feet, occurring seconds into the flight of the ball. 0. a. 8. a. Use the Trace feature, and when t 5, S 5.55 When x 0, V 600,000 0, ,000 00,000 $00, a. ( ). a. c. If the year is 005, then t When t 5, S 0.07(5).85(5) The estimated number of osteopathic students in 005 is,555. ( ) F When x 6, y Therefore, when the median male salary is $6,000, the median female salary is $8,090. A f() represents the value of A in 00 ( ) + and A f(8) represents the value of A in ( ) f ().75( ) ( ) ,90.786,9 million dollars ( ) ( ) f (8) ,76.98,76 million dollars Copyright 0 Pearson Education. Inc.

20 0 CHAPTER Functions, Graphs, and Models; Linear Functions c. Yes, based on this model, the amount spent on advertising in the Yellow Pages will continue to increase until the year 06, when it will begin to decrease. d.. a. In 99, the number of arrests is 6. per 00,000 people. In 009, the number of arrests is 96. per 00,000 people. The number is decreasing over this period. e. Yes. Both indicate a decrease.. a. Bt t t ( ) L ( 6),5 c. The cost in 006 is approximately $,5 million ($,5,000,000).. a. ( ) f x 0.7x +.x The tax burden increased. Reading the graph from left to right, as t increases B(t) also increases. 5. Since the base year is 980, the graph in x 0 part a) shows years 980 ( ) through 00 ( x 0). c. The graph in part a) decreases for x 0. Therefore, the number of arrests per 00,000 people decreased after 990. Copyright 0 Pearson Education. Inc.

21 CHAPTER Section. 6. c. 7. An appropriate viewing would be [0, 9] by [0, 60,000]. 50. y x x a. Since the base year is 975, correspond to values of x between 0 and Since percentages are between 0 and 00, y must correspond to values between 0 and 00. c. 9. f () t,6.879t+,8.869 d. a. Since the base year is 990, correspond to values of t between 0 and 9 inclusive. For 990: f ( ) ( ) 0, ,8.869,8.869 For 009: f ( ) ( ) 9, , ,69.57 e. 0 corresponds to x When x 8, y Thus in 0, approximately 7.8% of high school seniors will have used cocaine. Copyright 0 Pearson Education. Inc.

22 CHAPTER Functions, Graphs, and Models; Linear Functions 5. a million, or 99,900, c. Years after 000 Population (millions) a. 5. a. c. Yes, the function is a good visual fit to the data. 55. a. In 00 the unemployment rate was 6.0%. Copyright 0 Pearson Education. Inc.

23 CHAPTER Section. c. 56. a. The dropout rate in 00 is 0.%. c. Copyright 0 Pearson Education. Inc.

24 CHAPTER Functions, Graphs, and Models; Linear Functions Section. Skills Check. Recall that linear functions must be in the form f( x) ax+ a. Not linear. The equation has a nd degree (squared) term. Linear. c. Not linear. The x-term is in the denominator of a fraction. 5x (0) 5 5x 5 x y-intercept: Let x 0 and solve for y. 5(0) y 5 y 5 y 5 x-intercept: (, 0), y-intercept: (0, 5). No. A vertical line is not a function. y y 6 6 m x x 8. y y. m x x ( 0) undefined Zero in the denominator creates an undefined expression. 5. The given line passes through (,0) and ( 0, ). Therefore the slope is y y m x x 0 0 ( ) 8. a. x-intercept: Let y 0 and solve for x. x + 5(0) 7 x 7 y-intercept: Let x 0 and solve for y. 0+ 5y 7 5y 7 7 y 5 y. x-intercept: (7, 0), y-intercept: (0,.) 6. Since the line is horizontal, the slope of the line is zero. m a. x-intercept: Let y 0 and solve for x. Copyright 0 Pearson Education. Inc.

25 CHAPTER Section. 5 y-intercept: Let x 0 and solve for y. 9. a. x-intercept: Let y 0 and solve for x. (0) 9 6x 0 9 6x 0+ 6x 9 6x+ 6x 6x 9 9 x 6 x.5 y 9(0) y 0 x-intercept: (0, 0), y-intercept: (0, 0). Note that the origin, (0, 0), is both an x- and y-intercept. To graph, use the slope, m 9 9, or find another point from the equation, like (,9) or (, 9). y-intercept: Let x 0 and solve for y. y 9 6(0) y 9 y x-intercept: (.5, 0), y-intercept: (0, ). Horizontal lines have a slope of zero. Vertical lines have an undefined slope.. Since the slope is undefined, the line is vertical.. a. Positive. The graph is rising. Undefined. The line is vertical.. a. Negative. The graph is falling. Zero. The line is horizontal. 5. a. m, b 8 0. a. x-intercept: Let y 0 and solve for x. 0 9x x 0 Copyright 0 Pearson Education. Inc.

26 6 CHAPTER Functions, Graphs, and Models; Linear Functions 8. a. x 6, vertical line undefined slope, no y-intercept 6. a. x+ y 7 y x+ 7 x + 7 y 7 y x+ 7 m, b 9. a. m, b 5 Rising. The slope is positive c. 0. a. m 0.00, b a. 5y y,horizontal line 5 m 0, b 5 Rising. The slope is positive. c.. a. m 00, b 50,000 Falling. The slope is negative. Copyright 0 Pearson Education. Inc.

27 CHAPTER Section. 7 c. 8. For a linear function, the rate of change is equal to the slope. y y m. x x 6 9. a. The identity function is y x. Graph ii represents the identity function.. Steepness refers to the amount of vertical change compared to the amount of horizontal change between two points on a line, regardless of the direction of the line. The steeper line would have the greater absolute value of its slope. Since the slope in exercise 9 is, and in exercise 0 it is 0.00, and in exercise it is 00, exercise 0 (m.00) is the least steep, followed by exercise 9 (m ). Exercise displays the greatest steepness since m 00 which gives For a linear function, the rate of change is equal to the slope. m.. For a linear function, the rate of change is equal to the slope. m. 5. For a linear function, the rate of change is equal to the slope. m For a linear function, the rate of change is equal to the slope. m For a linear function, the rate of change is equal to the slope. y y 7 0 m. x x 5 ( ) The constant function is y k, where k is a real number. In this case, k. Graph i represents a constant function. 0. The slope of the identity function is one ( m ).. a. The slope of a constant function is zero ( m 0 ). The rate of change of a constant function equals the slope, which is zero.. The rate of change of the identity function equals the slope, which is one. Section. Exercises. Yes, the function is linear since it is written in the form y ax+ The independent variable is x, the number of years after No it is not linear since the function does not fit the form y ax+ 5. a. The function is linear since it is written in the form y ax+ The slope is 0.6. The marriage rate has decreased at a rate of 0.6 per thousand each year since 980. Copyright 0 Pearson Education. Inc.

28 8 CHAPTER Functions, Graphs, and Models; Linear Functions 6. a. The function is linear since it can be written in the form y ax+ The slope is.. c. The rate at which sales grew during this period is. billion dollars per year. 8. a. y-intercept: Let x 0 and solve for y. y 88,000 00(0) 88,000 Initially the value of the building is $88,000. x-intercept: Let y 0 and solve for x. 7. a. x-intercept: Let p 0 and solve for x. 5p+ x 5 5(0) + x 5 x 5 05 x 7 The x-intercept is 05,0 7. p-intercept: Let x 0 and solve for p. 5 p+ x 5 5 p + (0) 5 5 p 5 p 8.6 c. 0 88, x 00x 88,000 88,000 x The value of the building is zero (the building is completely depreciated) after 60 months or 0 years. The y-intercept is ( 0,8.6 ). In 000, the percentage of high school students who had ever used marijuana was 8.6%. 9. a. Cultivation Area (hectares) The data can be modeled by a constant function. c. Integer values of x 0 on the graph represent years 000 and after. c. y 00 Copyright 0 Pearson Education. Inc.

29 CHAPTER Section. 9 d. Cultivation Area (hectares). a. m 6, 665 From 007 to 00, the number of tweets increased by 6,665 thousand per year. c. No, 00% is incorrect since 00% of T(9) is not equal to T(0). The correct percentage is an increase of 50%. 0. a. The data can be modeled by a constant function. Every input x yields the same output y. y.8 c. A constant function has a slope equal to zero. d. For a linear function the rate of change is equal to the slope. m 0.. a. For a linear function, the rate of change is equal to the slope. m The slope is negative. The percentage of adults who have tried cigarettes decreased at a rate of 0.50 per year between 99 and a. For a linear function, the rate of change is equal to the slope. m. The 7 slope is positive. For each one degree increase in temperature, there is a increase in the 7 number of cricket chirps per minute. More generally, as the temperature increases, the number of chirps increases.. a. The average rate of growth over this period of time is 0.65 percentage points per year a. Yes, it is linear. m c. For each one dollar increase in white median annual salaries, there is a dollar increase in minority median annual salaries. 6. a. To determine the slope, rewrite the equation in the form f( x) ax+ b or y mx+ 0 p 9x 0 0 p 9x+ 0 0 p 9x p x+ 0 9 m.6 0 Each year, the percentage of high school seniors using marijuana daily increases by approximately 0.6% (9/0) percentage points per year. Copyright 0 Pearson Education. Inc.

30 0 CHAPTER Functions, Graphs, and Models; Linear Functions 7. a. m From 990 to 050, the percent of the U.S. population that is black increased by percentage points per year. 8. a. p 8d 96 Solving for p : p 8d d + 96 p 8 96 p d p d + 6 Therefore, m For every one unit increase in depth, there is a corresponding 6 pound per square inch increase in pressure. 9. x-intercept: Let R 0 and solve for x. R x x 70x x ,0. The x-intercept is ( ) R-intercept: Let x 0 and solve for R. R x R (0) R 500 The R -intercept is ( 0,500 ). 50. a. m 5.0 y-intercept: b 6.5 After 995, the percent of the U.S. population with Internet access increased by 5.0 percentage points per year. 5. a. The rate of change of revenue for call centers in the Philippines from 006 to 00, was billion dollars per year. 00 corresponds to x When x 0, R 0.975(0).5 R Thus in 00, the revenue for call centers in the Philippines was 6. billion dollars. c. No, it would not be valid since the result would be a negative number of dollars. 5. a. m. y-intercept: b 6.05 The y-intercept represents the total amount spent for wireless communications in 995. Therefore in 995, the amount spent on wireless communication in the U.S. was 6.05 billion dollars. c. The slope represents the annual change in the amount spent on wireless communications. Therefore, the amount Copyright 0 Pearson Education. Inc.

31 CHAPTER Section. spent on wireless communications in the U.S. increased by. billion each year. y y 5. a. m x x 700, 000,0, , ,000 Based on the calculation in part a), the property value decreases by $6,000 each year. The annual rate of change is $6,000 per year. y y 5. a. m x x Based on the calculation in part a), the number of men in the workforce increased by 0. million (or,000) each year from 890 to Marginal profit is the rate of change of the profit function. y y m x x The marginal profit is $58 per unit. 56. Marginal cost is the rate of change of the cost function. m y y x x The marginal cost is $.80 per unit. 57. a. m 0.56 The marginal cost is $0.56 per unit. c. Manufacturing one additional golf ball each month increases the cost by $0.56 or 56 cents. 58. a. m 98 The marginal cost is $98 per unit. c. Manufacturing one additional television each month increases the cost by $ a. m.60 The marginal revenue is $.60 per unit. c. Selling one additional golf ball each month increases total revenue by $ a. m 98 The marginal revenue is $98 per unit c. Selling one additional television each month increases total revenue by $ The marginal profit is $9 per unit. Note that m The marginal profit is $99 per unit. Note that m 99. Copyright 0 Pearson Education. Inc.

32 CHAPTER Functions, Graphs, and Models; Linear Functions Section. Skills Check.. m, b. The equation is m 5, b. The equation is y x+. y 5x+. y y Slope: m x x ( ) 6 Equation: y y m( x x) y 7 ( x ) y 7 x y x+. m, b. The equation is y x+. y y 6 0. Slope: m x x ( ). m, b 8. The equation is y x 8. Equation: y y m( x x) y 6 ( x ) y 6 x y x+ 5. y y m( x x ) y ( 6 ) ( x ) y+ 6 x+ y x 6. y y m( x x) y ( x ( ) ) 7. x 9 y + y x y x+ 8. y 0 ( x ). Slope: y y m x x 0 5 ( ) 8 The line is horizontal. The equation of the line is y. y y 5. Slope: m undefined x x The line is vertical. The equation of the line is x 9.. With the given intercepts, the line passes through the points ( 5, 0) and (0, ). The slope of the line is y y 0 m. x x 0 ( 5) 5 Equation: y y m( x x) y y ( x+ 5) 5 y x+ 5 ( x ( )) 0 Copyright 0 Pearson Education. Inc.

33 CHAPTER Section.. With the given intercepts, the line passes through the points (, 0) and (0, 5). The slope of the line is y y m. x x 0 () Equation: 5. x+ y y x+ m y y m( x x) 5 y 0 ( x ) 5 y ( x ) 5 y x 5 Since the new line is parallel with the given line, the slopes of both lines are the same. Equation: 6. x+ y y x m y y m( x x) y y+ 6 x+ y x+ 6 ( 6) ( x ) Since the new line is parallel with the given line, the slopes of both lines are the same. Equation: y y m( x x) y y+ x+ 0 y x+ 7 ( ) ( x 5) 7. x+ y 7 y x+ 7 y x+ 7 7 y x+ m Since the new line is perpendicular with the given line, the slope of the new line is m, where m is the slope of the m given line. m. m Equation: 8. x+ y 8 y x 8 y x 8 y x m y y m( x x) y 7 9 y 7 x+ 9 y 7+ 7 x+ + 7 y x+ ( x ( )) Since the new line is perpendicular with the given line, the slope of the new line is m, where m is the slope of the m given line. m. m Copyright 0 Pearson Education. Inc.

34 CHAPTER Functions, Graphs, and Models; Linear Functions Equation: 9. Slope: y y m( x x ) y 5 8 y 5 x+ 8 y 5+ 5 x+ + 5 y x+ ( x ( )) ( ) ( ) y y m x x Equation: y y m( x x) y ( x ) y x y x+. y y m( x x) y ( 7) 8( x 0) y+ 7 8x y 8x 7. f( b) f( a) f() f( ) b a ( ) () ( ) The average rate of change between the two points is. 0. Slope: Equation: y y 7 8 m x x 6 ( ) y y m( x x ) ( ) y 7 ( x ) y 7 x y x 5. f( b) f( a) f() f( ) b a ( ) () ( ) For a linear function, the rate of change is equal to the slope. Therefore, m 5. The equation is y y m( x x ) y 5( x 0) y 5x y 5x+.. For a linear function, the rate of change is equal to the slope. Therefore, m 8. The equation is 5. The average rate of change between the two points is. f( b) f( a) f() f( ) b a ( ) 7 9 Copyright 0 Pearson Education. Inc.

35 CHAPTER Section f( b) f( a) f() f( ) b a ( ) 6 7. f( x+ h) 5 5( x+ h) 5 5x 5h f( x+ h) f( x) [ ] 5 5x 5h 5 5x 5 5x 5h 5 + 5x 5h f( x+ h) f( x) 5h 5 h h 8. f( x+ h) ( x+ h) + x+ h+ 0. f( x+ h) f( x) xh+ h h h h x h h x+ h ( ) ( ) f x+ h x+ h + ( + ) ( x xh h ) x + xh+ h + f( x+ h) f( x) x + 6xh+ h + x x 6xh h x 6xh + h f( x+ h) f( x) 6xh+ h h h h x h h 6x+ h ( 6 + ) f( x+ h) f( x) [ ] x+ h+ x+ x+ h+ x h. a. The difference in the y-coordinates is consistently 0, while the difference in the x-coordinates is consistently 0. Considering the scatter plot below, a line fits the data exactly. f( x+ h) f( x) h h h 9. f x h x h ( + ) ( + ) + f( x+ h) f( x) ( x xh h ) x xh h x xh h x x xh h x xh + h [0, 60] by [500, 800] Copyright 0 Pearson Education. Inc.

36 6 CHAPTER Functions, Graphs, and Models; Linear Functions y y Slope: m x x Equation: y y m( x x) y 585 ( x 0) y 585 x 0 y x a. The difference in the y-coordinates is consistently 9, while the difference in the x-coordinates is consistently 6. Considering the scatter plot below, a line fits the data exactly. Equation: y y m( x x ) y 6.5 ( x 9) 57 y 6.5 x y x y x Section. Exercises. Let x kwh used and let y monthly charge in dollars. Then the equation is y 0.x Let x minutes used and let y monthly charge in dollars. Then the equation is y 0.0x Let t number of years, and let y value of the machinery after t years. Then the equation is y 6,000,600t. [0, 0] by [ 0, 0] y y Slope: m x x ( 8) 0 hours 6. a. Sleep needed 8 + ( 8 0) 8+ 9 hours Sleep needed 8 + ( 8 ) ( ) c. Let x age in years, and let y hours of sleep. Using the results of parts a) and b), the two ordered pairs are (0, 0) and (, 9). Copyright 0 Pearson Education. Inc.

37 CHAPTER Section. 7 The resulting slope is Then using the point-slope form of a linear equation and one of the ordered pairs, (, 9): y 9 0.5( x ) y 9 0.5x+.5 y 0.5x+.5 d. Let x 8. y ( ) 7. a. Let x the number of years after 009, and let y the amount spent on Internet advertising. The linear equation modeling the spending is y.5x+.7 billion dollars. To estimate the spending in 05, let x The estimated spending is y.5(6) +.7 $58.7 billion. 8. a. Let x the number of years after 009, and let y the amount spent on magazine advertising. The linear equation modeling the spending is y 0.65x+ 5.5 billion dollars. To estimate the spending in 05, let x The estimated spending is y 0.65(6) $.6 billion. 9. a. From year 0 to year 5, the automobile depreciates from a value of $6,000 to a value of $,000. Therefore, the total depreciation is 6, or $5,000. Since the automobile depreciates for 5 years in a straight-line (linear) fashion, each year the value declines by 5,000 $5, c. Let t the number of years, and let s the value of the automobile at the end of t years. Then, based on parts a) and b) the linear equation modeling the value is s 5000t+ 6, P.5% ( 75,000) y 875y where y the number of years of service, and P the annual pension in dollars.. Notice that the x and y values always match. The number of deputies always equals the number of patrol cars. Therefore the equation is y x, where x represents the number of deputies, and y represents the number of patrol cars.. a. Notice that the y values are always the same, regardless of the x value. The market share (%) is constant. Therefore the equation is y 66, where x represents the number of months after April 00, and y represents Google s market share (%). For this time period, the rate of change of Google s market share (%) is zero. y y. m x x Equation: y y m( x x) y ( x 00) y x 7,00 y 58x,750 Copyright 0 Pearson Education. Inc.

38 8 CHAPTER Functions, Graphs, and Models; Linear Functions y y. m x x Equation: y y m( x x) y 680 ( x 00) y 680 x 600 y x y x+ 080 y y 5. m x x 700,000,0, , ,000 Equation: y y m( x x) y 700,000 6,000( x 0) y 700,000 6,000x+,0,000 y 6,000x+,90,000 V 6,000x+,90, a. At t 0, y 860,000. ( 0,860,000 ),( 5,0) y y m x x 0 860, , 000 5,00 Equation: y y m( x x) y 0,00( x 5) y,00x+ 860,000 y 860,000,00t where t the number of years, and y the property value. y y 7. m x x Equation: y y m( x x) y ( x 5) y x+.6 y x y 0.50x+. p. 0.50t where t the number of years after 960, and p the percent of adults who smoke cigarettes. 8. Let x median weekly income for whites, and y median weekly income for blacks. The goal is to write y f(x). y y changein y changein 00 m x x x Equation: y y m( x x) y ( x 676) y x 8. y 0.69x Copyright 0 Pearson Education. Inc.

39 CHAPTER Section a. Notice that the change in the x-values is consistently while the change in the y- values is consistently 0.05 percentage points per drink. Therefore the table represents a linear function. The rate of change is the slope of the linear function. vertical change 0.05 m 0.05 horizontal change Let x the number of drinks, and let y the blood alcohol percent. Using the slope 0.05 and one of points (5, 0.5), the equation is: y y m( x x) y ( x 5) y x 0.5 y 0.05x 50. a. Notice that the change in the x-values is consistently while the change in the y- values is consistently 0.07 percentage points per drink. Therefore the table represents a linear function. The rate of change is the slope of the linear function. vertical change 0.07 m 0.07 horizontal change Let x the number of drinks, and let y the blood alcohol percent. Using the slope 0.07 and one of the points (0, 0), the equation is y y m( x x) y ( x 0) y 0.07x y y m x x Equation: y y m( x x) y ( x 890) y x y x y 0.50x 9.6 Yes. Consider the following table of values based on the equation in comparison to the actual data points. x y Equation Values Actual Values c. They are the same since the points ( 890,8. ) and ( 990,68.5 ) were used to calculate the slope of the linear model. 5. a. Let x the year at the beginning of the decade, and let y average number of men in the workforce during the decade. Using points (890, 8.) and (990, 68.5) to calculate the slope yields: Copyright 0 Pearson Education. Inc.

40 0 CHAPTER Functions, Graphs, and Models; Linear Functions 5. a. Let t the year, and let p the percentage of workers in farm occupations. Using points (80, 7.8) and (005,.5) to calculate the slope yields: y y m x x Equation: y y m( x x) y.5 0.8( x 005) y.5 0.8x y 0.8x+ 76. y 0.8x+ 76. p 0.8t+ 76. Yes, the line appears to be a reasonable fit to the data. c. Each year, between 80 and 005, the percent of workers in farm-related jobs decreased by 0.8 percentage points. d. No. The percent of farm workers would become negative. y y 5. a. m x x The annual rate of change of expenditures between 990 and 008 is $5.57 billion dollars per year. c. No. Note in the chart that the change in expenditures from one year to the next is not constant. It varies. y y 5. a. m x x The average rate of change is 08 students per year. Note that it is the same as the answer in part a). c. Since enrollment is projected to increase, additional buildings may be necessary. y y 55. a. m x x The average rate of change of the birth rate between 957 and 009 was. births per 000 girls. c. The birth rate has decreased over this period by.% per year. d. Let x the number of years after 950, and let y the birth rate for U.S. girls. y y m( x x) y 9..( x 59) y 9..x+ 6.9 y.x y.x+ 0 Copyright 0 Pearson Education. Inc.

41 CHAPTER Section. y y 56. a. m x x a. It is the same as part a). The average rate of change in the number of women in the workforce between 950 and 00 is increasing at a rate of 0.79 million per year. y y m( x x ) y ( x 0) y x y 0.79x+ 8. c. f( b) f( a) f(009) f(960) b a ,67,78, ,0,55 9 8, ,66 The slope of the line connecting the two points is the same as the average rate of change between the two points. Based on part a), m 8,66. c. The equation of the secant line, using the unrounded slope from part a), is given by: y y m( x x) y,95 8, ( x 960) y,95 8, x 56,8,000 +,95 +,95 y 8, x 55,968, a. e. Points corresponding to 990 and 005. The points on the scatter plot between those two years do approximate a linear pattern. Answers may vary. f( b) f( a) f(5) f() b a On average from year to year 5, the worth of the investment increases by $0.5 per year. c. The slope is the same as the average rate of change, 0.5. y y m( x x ) y ( x ) y x 0.5 y 0.5x d. 59. a. No. The points in the scatter plot do not lie approximately on a line. f( b) f( a) f(950) f(90) b a The average rate of change is 0.96 million (96,00) women per year. d. No. The points on the scatter plot do not approximate a linear pattern. Copyright 0 Pearson Education. Inc.

42 CHAPTER Functions, Graphs, and Models; Linear Functions c. f( b) f( a) b a f(00) f(950) The average rate of change is approximately 0.98 million (98,000) women per year. d. Yes. Since the graph curves, the average rate of change is not constant. The points do not lie exactly along a line. 60. a. No. Yes. The points seem to follow a straight line pattern for years between 00 and 00. c. f( b) f( a) f(00) f(00) b a The average annual rate of change of the data over this period of time is 0.085percentage points per year. d. y y m x x ( ) y ( x 00) y x y 0.085x a. Let x the number of years since 950, and let y the U.S. population in thousands. Then, the average rate of change in U.S. population, in thousands, between 950 and 009 is given by: f( b) f( a) f(59) f(0) b a ,007 5, , The annual average increase in population is 6 thousand people, or approximately,.6 million people, per year. y y m x x ( ) y 5,7 6( x 0) y 5,7 6x + 5,7 + 5,7 y 6x+ 5,7 c. 975 corresponds to x 5. y 6x+ 5,7 y 6( 5) + 5,7 y 7,86 thousand The estimated U. S. population in 975 is 7,86,000 people. No, the values are different, but they are relatively close. d. The data in the table cannot be modeled exactly by a linear function. 6. a. Let x represent the number of clients in Group, and y represent the number of clients in Group. Then, Group Expense + Group Expense Total Expense 00x+ 00y 00,000 Copyright 0 Pearson Education. Inc.

43 CHAPTER Section. 00x+ 00y 00,000 00y 00x+ 00,000 00x + 00,000 y ,000 y x y.5x+ 500 The y-intercept is 500. If no clients from the first group are served, then 500 clients from the second group can be served. The slope is.5. For each one person increase in the number of clients served from the first group there is a corresponding decrease of.5 clients served from the second group. c. 0(.5) 5 Fifteen fewer clients can be served from the second group. Copyright 0 Pearson Education. Inc.

44 CHAPTER Functions, Graphs, and Models; Linear Functions Chapter Skills Check f ( ) 0. The table represents a function because each x matches with exactly one y. 7.. Domain {,,,,5, 7,9,, } Range { 9, 6,, 0,, 6, 9,, 5}. f () 0. Yes. The rate of change between any two pairs of values is constant. The slope is y y 6 9 m. x x ( ) Calculating the equation: y y m x x ( ) y 9 9 y 9 x 9 y x y x + 9 y x+ ( x ( )) y x + x a. c. C () 6 () 6 (9) 6 8 C( ) 6 ( ) 6 () C( ) 6 ( ) 6 () 6 The second view is better. 6. a. f ( ) Copyright 0 Pearson Education. Inc.

45 CHAPTER Skills Check 5 0. Since the slopes of perpendicular lines are negative reciprocals of one another, 6 6 m m y y 6 6. m x x 8 ( ) 6. a. Since y x 8 will not be a real number if x 8< 0, the only values of x that yield real outputs to the function are values that satisfy x 8 0. Isolating x yields: x x 8 x,. Therefore the domain is D: [ ) The denominator of the function will be zero if x 6 0 or x 6. This implies x 6. The domain is all real numbers except 6 or in interval notation D:,6 6,. ( ) ( ) 5. a. x-intercept: Let y 0 and solve for x. x (0) x x 6 y-intercept: Let x 0 and solve for y. (0) y y y x-intercept: (6, 0), y-intercept: (0, ). The slope of the line through the two given y y 8 6 points is: m x x ( ) The slope of the line from the given equation is /. Since the slopes are negative reciprocals of each other, the lines are perpendicular.. The slope of the given line is y y 7 7 m. x x 5 ( ) 6 6 Since two parallel lines have the same slope, the slope of any line parallel to this one is 7 also m Solving for y : x y y x+ y x+ y x Since the model is linear, the rate of change is equal to the slope of the equation. The slope, m, is. Copyright 0 Pearson Education. Inc.

46 6 CHAPTER Functions, Graphs, and Models; Linear Functions 7. The slope is m 6 The y-intercept is 0,. ( ) 8. Since the function is linear, the rate of change is the slope. 6 m. 9. y mx+ b y x+ 0. y y m( x x ) y ( 6) ( x ) y+ 6 x+ y x. The slope is y y 6 m. x x ( ). a. f ( x+ h) 5 ( x+ h) 5 x h f ( x+ h) f ( x) 5 ( x+ h) [ 5 x] c. 5 x h 5+ x h ( + ) ( ) f x h f x h h h. a. f ( x+ h) ( x h) x+ 0h 50 f ( x+ h) f ( x) [ 0x+ 0h 50] [ 0x 50] 0x+ 0h 50 0x+ 50 0h Solving for the equation: y y m( x x ) y 6 ( x ) y 6 x y x+ c. ( + ) ( ) f x h f x h 0h h 0. y x ( ) ( ) ( ) ( ) x 0: y 0 0 0,0 x : y 9,9 The average rate of change is y y m x x Copyright 0 Pearson Education. Inc.

47 CHAPTER Review 7 Chapter Review Exercises 5. a. Yes. If only the Democratic percentages are considered, each year matches with exactly one black voter percentage. f (99) 8. The table indicates that in 99, 8% of black voters supported a Democratic candidate for president. c. When f( y) 9, y 96. The table indicates that in 96, 9% of black voters supported a Democratic candidate for president. 6. a. The domain is { 960, 96, 968, 97, 976, 980, 98, 99, 996, 000, 00, 008. } ( ) f ( a) f b b a No, the rates of change are not equal. 9. a. Each amount borrowed matches with exactly one monthly payment. The change in y is 89.6 for each change in x of f (5,000) 8.. Therefore, borrowing $5,000 to buy a car from the dealership results in a monthly payment of $8.. c. If f( A) $58.9, then A $0, No. 98 was not a presidential election year. 0. a. Domain { 0,000, 5,000, 0,000, 5,000, 0,000 Range { 79.5, 68.87, 58.9, 8., 57.7 No. $,000 is not in the domain of the function. } } 8. a. For 968 to 008: y y m x x The average annual rate of change is the same as the slope in part a), therefore 0.5 percentage points per year. c. For 968 to 980:. a. ( ) ( ) f 8, , The predicted monthly payment on a car loan of $8,000 is $ Any positive input could be used for A. Borrowing a negative amount of money does not make sense in the context of the problem. Copyright 0 Pearson Education. Inc.