Answers for the lesson Write Linear Equations in Slope-Intercept Form

Transcription

1 LESSON 4.1 Answers for the lesson Write Linear Equations in Slope-Intercept Form Skill Practice 1. slope. You can substitute the slope for m and the y-intercept for b to get the equation of the line.. y x y x 1. y x 6. y 7x y } x 9 8. y } 4 x 6 9. A 1. y x y 1 } x 1. y x y } x y x 1. y x 16. The given slope and y-intercept were interchanged in the slopeintercept form of the equation; y x The slope should be 4 }, y 4 } x y } 1 x y 4x 1 4. y 1 } 4 x 1 1. y 4 } x. y x 4. y x 4. y x 8. y x 6. y 4 7. y.6x y 8 } x 1 9. y 4x 4. y x 1 1. y x 1 7. y 1 } 4 x. y 4 } x 1 4. y x 4. y } x 1 6. y x y x y x The equations in slope-intercept form of the two lines are k: y x 1 and l: y } 1 x 1 1. The parameter m changed from to } 1 and the parameter b changed from 1 to Sample answer: A health club offers an aerobics membership that charges $9 plus $4 per class. 41. y x

2 4. y 1 } x 4. No; the slope of the line is undefined, the equation is x, which is not in slopeintercept form. 44. Find the slope by substituting the values: } b 1 m b m. The 1 y-intercept is when x, so the y-intercept is b. If you substitute (1, b m) into the equation y mx 1 b, you get b m m 1 b which is a true statement. Problem Solving 4. a. C 44m 1 48 b. $1 46. C.99e ; $ C h 1 ; $4 48. a. a.7e 1 b. dependent variable: a, independent variable: e c. Substitute for e to get approximately. 49. a. x (years since 197) y (km ) b The area of the glaciers changed 1.1 square kilometers between every 1 year interval. c. y.11x 1.;.11 km. a. 81 million gal b. y 1,,h c. h ; water is only released for hours after 1 A.M. 1. a. t.7d 1 b. 16 min. a. d b. d ft c. 44. ft 7 } 1 e } e ft 1 4 ft 1 9

3 LESSON 4. Answers for the lesson Use Linear Equations in Slope-Intercept Form Skill Practice 1. y-intercept. It is the point where x is.. y x 4. y x 9. y x 1 6. y x 1 7. y } 4 x 1 8. y } 1 x } was substituted for x instead of y and 6 was substituted for y instead of x, (6) 1 b, 1 1 b, 9 b should have been substituted for m, not b, 81 (18) 1 b, b, b $ y x y 7x y } x y x y } 4 x 1 } y } 1 x } y 4x y } x 1 } y } 1 x 1 } 1. y } 7 6 x 1 } y } 1 x } 4. y x 7. y x y x 4. y 1 } x y } 4 x 7. y x 8. y x 6 9. D. y 1 } 4 x 1 1. y } x 1 6. y 1 } x 1. y 6x 4 4. Yes; you can find the slope and then substitute m and the coordinates of the point in y mx 1 b, solve for b, and write the equation.. Yes; you can substitute m and the coordinates of the point in y mx 1 b, solve for b, and write the equation. 6. No; many lines have the same slope but different y-intercepts. 7. Yes; you can find the slope of the line, then substitute the y-intercept for b, and write the equation. 8. y } x } 1 97

4 9. y } 9 x } 1 4. y } x 1 } The lines y } x } 1 and y } 9 x } 1 and the lines y } 9 x } 1 and y } x 1 } 11 intersect because they have different slopes; the lines y } x } 1 and y } x 1 } 11 will not intersect, they have the same slope, so they are parallel. 4. The three points lie on the same line. If you find the equation of the line between two of the points and then check to see that the third point is a solution, you can see that all three points are on the line y } 4 x The three points do not lie on the same line. If you find the equation of the line between two of the points and then check to see that the third point is a solution, you can see they do not lie on the same line. 44. The three points do not lie on the same line. If you find the equation of the line between two of the points and then check to see that the third point is a solution, you can see they do not lie on the same line. 4. The three points do not lie on the same line. If you find the equation of the line between two of the points and then check to see that the third point is a solution, you can see they do not lie on the same line ; find the equation of the line through (, ) and (, ) to be y } 1 x 1 4, then substitute 6 for x to find k. Problem Solving 47. } ft/yr; 6 ft a. $.9 b. $ min or 1 h min; substitute for m, for x, and 8 for y into the equation y mx 1 b to find b. Then substitute for x into the equation y x 1 to solve for y. 98

5 . a. $8 b. y 7.8t 1 8 c. $ a. about 84 newspapers b. y 11.8x 1 84 c. about 98 newspapers. a. 17,81 airports b. y 17x 1 17,81 c.. a. d 81t 1 4 b d 18t The slope is the rate that the hurricane is traveling, the y-intercept represents the distance from the town at 1 P.M. c. 1 A.M.; find the t-intercept to find the value of t when the distance to the town is ; substitute for d and solve for t; t 1, so you need to add 1 hours to 1 P.M. to get 1 A.M. 4. a. d 1t 1 6 b. The rate of change, 1 meters per second, represents the skater s top racing speed; the initial value, 6, represents the distance the skater traveled from a stand-still to where he reached his top racing speed; d meters 11 } seconds meters (x seconds) 1 6 meters. c. 4 sec; the total distance is the length of the race track times laps, 6 meters. If you substitute 6 for d, t 4. 99

6 Mixed Review y x 6. y } 7 x 61. y x 4 6. y 9x y 4 } x y 1 } 1 x 1 1

7 LESSON 4. Answers for the lesson Write Linear Equations in Point-Slope Form Skill Practice ; (, ). Find the slope and substitute it for m in the equation y y 1 m(x x 1 ). Then pick one of the points and substitute the coordinates in for y 1 and x 1. y (x 1). y 1 (x ) 4. y (x ) 1.. y 1 1 6(x 7) 6. y 1 1 (x ) 7. y (x 1 8) y (x ) 8. y 6 } (x 1 6) 9. y 1 9(x 1 11) 1. y 1 9 } 7 (x 1 ) 11. y 1 1 } (x ) 1. C 1. The form y y 1, so the left side should be y () or y 1 ; y 1 (x 1). 16. y 1 (x 6) 1

8 17.. y 1 (x ) or y 1 (x 1) 1. y 4 (x 1) or y 1 (x 1 ) y 8 (x 4). y 4 } 1 (x 1 ) or y } 1 (x 1 1) 18.. y (x 7) or y 1 (x ) 4. y 1 } 1 (x 6) or y 1 } 1 (x 1). y 1 1 } (x 1 4) or y 1 (x 1 1) 4 y 1 7 } (x 6) 19. y 4 (x ) 6. y } (x 4) or 4 y 1 } (x 1 4) 4 7. y 1 8(x 1 ) or y 6 8(x 4) 8. y 1 19 } 16 (x 1 ) or y 1 } 16 (x ) 9. A point was not substituted into the equation, the y-coordinates of the two points were substituted;. B y } (x 1). 1

9 1. No; because the increase is not at a constant rate, the situation cannot be modeled by a linear equation.. Yes; because the rate is increasing at a constant rate, the situation can be modeled by a linear equation. Sample answer: y 1. } 1 (x 1). No; because the increase is not at a constant rate, the situation cannot be modeled by a linear equation. 4. Yes; because the rate is decreasing at a constant rate, the situation can be modeled by a linear equation. Sample answer: y 16 (x 1 ). ; y 8 (x ) or y 6 (x 4) 6. ; y (x 4) or y (x ) Problem Solving 7. a. y 1x 1 b. $17 8. Since the cost increases at a constant rate of $1714 per month, the situation can be modeled by a linear equation; $9; $ y 1,x 1 67,; $17, 4. a. y 1.4x 1 b. 4.4 gal 41. a. Since the cost increases at a constant rate of $.49 per print, the situation can be modeled by a linear equation. b. Sample answer: y (x 1) c. $1.49 d. $ a. y.4x b. about 6.64 million metric tons 4. a. y (x 6) b ft/sec 44. a. y.41791x b. 1.9 billion lb; find the number of cans recycled per pound of aluminum in by substituting for x to get about.9 cans per pound. Divide.8 billion aluminum cans by.9 cans per pound to find the number of pounds of aluminum. 14

10 LESSON 4.4 Answers for the lesson Write Linear Equations in Standard Form Skill Practice 1. standard form. slope-intercept form. point-slope form 4. Find the slope of the line then substitute the slope and one of the points into the point-slope form. Collect variables on one side and constants on the other side. 1. Sample answers are given.. x 1 y, x 1 y 6. x 1 y, 1x 1 y 7. x y 9, x 1 4y x 4y, 6x 8y 4 9. x y 4, 6x y 8 1. x 4y, 4x 1 8y x 1 y 1. x 1 y 1 1. x 1 y 14. 4x 1 y 1. } x 1 y } 6 x 1 y } x 1 y } x 1 y } x 1 y 1. 4x 1 y 1. 1 } x 1 y 1. y. y, x 4. y, x. y, x 1 6. y, x 7. y 4, x 1 8. y, x 6 9. (1, 4) was substituted incorrectly, 1 should be substituted for x and 4 substituted for y. A(1) (4), A 1 1, A 7.. } x 6 1 } y 1; Sample answer: 4 I solved the equations x 1 () 1 and () 1 y 1 to find a, the x-intercept, and b, the y- intercept, respectively. Then I substituted the values of a and b into the general intercept form. 1. 4; 4x 1 y. } 1 ; } 1 x 4y 1. 4; x 4y 1 16

11 4. 11; 8x 1 11y 4. ; x y 6. } 7 ; x 1 } 7 y 4 7. a } b x 1 y a Problem Solving 8..p 1 1.v ; Sample answer: 1 phlox plants and vinca plants; phlox plants and vinca plants; 6 phlox plants and 1 vinca plants 9. a. 1 oz b. 1c 1 1w 1 c. 1 corn, wheat; corn, 4 wheat; corn, 8 wheat 4. n 1 t 1; The n-intercept,, is the number of nights of boarding the dog at the kennel without any treats. The t-intercept,, is the number of treats that can be bought without boarding the dog for any nights. 41. a s 16 b c. Large rafts Small rafts a..7b 1 s 6 b. 18 subway rides; if you ride the bus 6 times, it costs (.7)($6) $4 without the pass. The pass costs $6, you need to spend $18 on subway rides, $18 4 $

12 4. 1 w 6; Sample answer: Length (ft) Width (ft) a..4x 1.6y 1 b. 1 ml c. 4 ml Mixed Review of Problem Solving 1. a. $1 per month; $1 b. C 1m 1 1 c. $6. a. d.t 1 b. 19 mi. a. Because the cost increases at a constant rate of $16 per person, the situation can be modeled by a linear equation. b. C 16p c. $19 4. V t; substitute 1 for V and solve for t; t minutes.. a. Because the same length of path is paved each day, the situation can be modeled by a linear equation. b. d 1 } t 1 1 c. days 6. Sample answer:.n 1.1d Nickels 7. $19; Dimes

13 8. $69.;

14 LESSON 4. Answers for the lesson Write Equations of Parallel and Perpendicular Lines Skill Practice 1. perpendicular. Identify the slopes of the lines. If the slopes are negative reciprocals, then the lines are perpendicular.. y x 1 4. y } x 1. y } x 1 6. y x y 6x y } 1 x 4 9. y x y x 11. y x 1 1. parallel: none; perpendicular: a and b 1. parallel: a and b; perpendicular: none 14. parallel: none; perpendicular: none 1. parallel: none; perpendicular: a and b 16. D 17. The line through points (6, 4) and (4, 1) is perpendicular to the line through points (1, ) and (4, 1); the slope of the line through the points (6, 4) and (4, 1) is }, the slope of the line through the points (1, ) and (4, 1) is }. The slopes are negative reciprocals, so the lines are perpendicular. 18. y x 19. y 1 } x 1. y 1 } x 1 1. y x 1 4. y 7 } x 1. y } 4 x 4 4. y } x 1. y } 1 x } 1 6. y x (, 1) was substituted incorrectly, should be substituted for x, and 1 should be substituted for y; 1 () 1 b, b, b. 8. B 111

15 9. Yes; the slope of the line through (4, ) and (, 1) is 4 and the slope of the line through (, ) and (1, ) is } 1. The slopes are 4 negative reciprocals, so the lines are perpendicular.. Sample answer: y x 1 1 and y x 1 ; y 1 } x 1 1. m x 1 x } y 1 y Problem Solving. a. y x 1 8 b. y x 1 8 c. No; the slopes and are not negative reciprocals.. a. w 1 d 1 6; w d 1 6 b. 1, lb; 1, lb c. The graphs of the lines are parallel because they have the same slope,. The w-intercept of the second line is more than the w-intercept of the first line. 4. Parallel: nd Street and Park Street; the slope of both streets is }. Since they have the same slope, the streets are parallel. Perpendicular: nd Street and Sea Street, Park Street and Sea Street; the slope of Sea Street is }, which is the negative reciprocal of }, the slope of nd Street and Park Street. Since the slopes are negative reciprocals, the streets are perpendicular.. Different registration fees; because the lines are parallel, the rate of change, the monthly fee, for each must be equal. Therefore, the students paid different registration fees. 6. a. C 8.7m 1 49 b. C 8.7m c. The graphs of the lines are parallel; they have the same slope, 8.7. The C-intercept of the second graph is 1 more than the C-intercept of the first graph. d. $1; $1 11

16 7. a. y.x 1 ; y.x 1 b. The graphs of the lines are parallel; they have the same slope,.. The y-intercept of the second line is less then the y-intercept of the first line. c. ; 1; the x-intercepts show the number of months of nonuse it would take for the value of the gift card to be $. 11

17 LESSON 4.6 Answers for the lesson Fit a Line to Data Skill Practice 1. increase. When data have a positive correlation, the dependent variable tends to increase as the independent variable increases. When data have a negative correlation, the dependent variable tends to decrease as the independent variable increases. When data have relatively no correlation where there is no apparent relationship between the independent variable and the dependent variable.. positive correlation 4. relatively no correlation. negative correlation 6. Sample answer: y 11.x Sample answer: y 4.x C 9. The line does not have approximately half the data above it and half below it. 1. The independent variable is x, not y; the dependent variable decreases as x increases. 11. Sample answer: The amount of time driving a car and the amount of gas left in the gas tank. 11

18 relatively no correlation; no; because there is relatively no correlation in the data you cannot write an equation positive correlation; Sample answer: y 1.49x 1 relatively no correlation 1. Line b; line a has too many points below the line, but line b has about half the points above the line and about half the points below the line. Problem Solving 16. a b. Positive correlation; the larger the home range size the larger the percent of pacing time. c. No; it is below the expected percent of time spent pacing. 116

19 17. a b. Sample answer: y.x c. Sample answer:. degrees per kilometer The growth rate of alligator is slightly greater than the growth rate of alligator Sample answer: y 1.6x 1. a. Sample answer: y 1.x 1 b. Sample answer: 1. min per day c. No; it will continue through June and then start decreasing. 1. a. Sample answer: h 17.8y b. Sample answer: m.78h. c. Sample answer: m 1.884y ; the function models the amount of money, m, spent on the Internet as a function of the number of years, y, since

20 1. d. Yes; if you substitute the number of years since 1999 for y, you get about the amount of money, m, given in the data. 118

21 LESSON 4.7 Answers for the lesson Predict with Linear Models Skill Practice 1. linear interpolation. Extrapolation is finding an approximate value ouside the range of known values. Interpolation is finding an approximate value within the range of known values.. y.6x 1.; y 8.x 1.1; y 1.7x 1 ; y.x 1.; } To find the zero of a function, substitute for y, not x; 14. B.x,.x, x }. 1. a and b were not substituted correctly; y 4.47x

22 16. Sample answer: Temperature on a mountain depends on the elevation. The zero indicates the elevation at which the temperature is 8F. 17. a. No; the data would first have a posititve slope and then a negative slope. b. You could fit a line to the data for the first 1 years and then fit another line to the data for the following 1 years. Problem Solving 18. a. 19. a b. y 1x 8 c. $ b. y.x 1 1. c. about 8.7 ft. a. y 1.x 1 14 b a. y 197.6x 1 4 b. about 17.9; 17.9 years from 198 the number of people living in high noise areas will be ; no.. a. Sample answer: b. Answers may vary; best fitting line: y.x ; answers may vary. 11

23 . c. The slope,., is the change in the cost (in thousands of dollars per thousand miles) of a local car of the same model, make, and year, and the y-intercept, 19.7, is the predicated selling price (in thousands of dollars) for a local car of the same model, make, and year with a mileage of.. a ,,, 1, 1 1 There is relatively no correlation in either scatter plot. b. No; because you cannot find a line of best fit for either correlation, you cannot use the mallard duck population to predict the total duck population. Mixed Review of Problem Solving 1. a.. ; Value (millions of dollars) Years since 199 b. y 1x 1 16 c. about $1 million per year d. 1

24 . Sample answer: y 8x positive correlation. a. y 1.1x b. about 1.1 percent per year c. 8.; 8. years after 1998, or 6, the percent of revenue from U.S. music sales made through music clubs will be. 6. C 4g 1., C 4g 1 1.7; the graphs have the same slopes, but different C-intercepts. 1