A C E. Answers Investigation 4. Applications. x y y

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1 Answers Applications 1. a. No; 2 5 = 0.4, which is less than c. Answers will vary. Sample answer: 12. slope = 3; y-intercept can be found by counting back in the table: (0, 5); equation: y = 3x a. slope = 1 The slope is the change in the y-values compared to the change in the x-values between two points in the table. As the x-values go up by 1, the y-values go up by 1. So, the slope is 1 : 1 or 1. x y d. Answers will vary based on the graphs that students drew for part (c) and can include any equation of the form y = 2 x + c where c is any constant 5 number. The equation that corresponds to the graph above is y = 2 5 x a. horizontal distance: 3; vertical distance: 3 vertical change 3 Slope 1 horizontal change 3 3. slope = 3; y-intercept = slope = 0.5; y-intercept = 0 5. slope = 3; y-intercept = 0 6. slope = 5; y-intercept = 2 7. a. i, iii, iv ii, vi c. i, vi d. i, iv, vii e. iv, v, vi 8. slope = 2; y-intercept = 0; equation: y = 2x 9. slope = 1; y-intercept = 3.5; equation: y = x slope = 2; y-intercept can be found by counting back in the table: (0, 1); equation: y = 2x slope = 2; y-intercept = 5; equation: y = 5 2x 14. a. slope = 2 The slope is the change in the y-values compared to the change in the x-values between two points in the table. x y a. b = 8.5 b = 2 c. In part (a), the y-intercept is 8.5 and the rate of change is 3.5. In part (b), the y-intercept is 2 and the rate of change is 5. d. The x-intercept in part (a) is about 2.4, and 16. a. in part (b), it is 2.To find the x-intercept 5 symbolically, you need to substitute 0 for y and solve for x: 0 = x and 0 = 5x 2. Thus, (2.43, 0) and (0.4, 0) are the x-intercepts. slope = 1 c. y-intercept = 0; the y-intercept is where the graph crosses the y-axis, so you could look at the graph. d. y = x e. Possible answers: (6, 6), ( 1, 1) Moving Straight Ahead 1

2 Answers 17. a. 18. a. slope = 1 c. y-intercept = 0; the y-intercept is where the graph crosses the y-axis, so you could look at the graph. d. y = x e. Possible answers: (0, 0), ( 3, 3) c. y-intercept = 6; the y-intercept is where the graph crosses the y-axis, so you could look at the graph. d. y = 6 e. Possible answers: (1, 6), (2, 6) 20. B, E, and J 21. A, F, and K 22. C, D, G, and H 23. a. y = x 1 corresponds to graph B y = 2 corresponds to graph D c. y = 1 x corresponds to graph E a. x y slope = 1 c. y-intercept = 5; the y-intercept is where the graph crosses the y-axis, so you could look at the graph d. y = x 5 c. slope = 1 d. Answers will vary. Possible answer: The distance y a walker covers in x seconds walking at a constant rate of 1 meter per second. 25. a. x y a. e. Possible answers: (1, 6), ( 1, 4) slope = 0 c. slope = 2 d. Answers will vary. Possible answer: The amount of money y Mary has at the end of x weeks if she starts out owing $2, but she receives $2 each week in allowance. Moving Straight Ahead 2

3 Answers 26. a. x y i. Answers will vary based on the graph students drew. For the graph of the equation y = 1 x + 3, students 3 could pick (0, 3) and (3, 2). ii. Since the slope is given, the students just need to find the y-intercept of their graph. Again, they should have an equation of the form y = 1 x + When students have 3 c. slope = 0.5 d. Answers will vary. Possible answer: The amount of money, y, Mary has at the end of x days if she starts the week by receiving her $2 allowance, but she buys an apple for $0.50 each day. 27. Answers will vary. Possible answers: a. Students may graph any line of the form y = 3x + b, where b is any constant, not necessarily the same in both equations. Possible graph for parts (a) and (b): i. Answers will vary based on the graph students drew. For the graph of the equation y = 3x + 2, students could pick (0, 2) and (1, 5). ii. Since the slope is given, the students just need to find the y-intercept of their graph. Again, they should have an equation of the form y = 3x + When students have shared their equations, you may want to ask if it makes sense that their points from part (a) are on the graph of their equation and how they could use their equation to check this. shared their equations, you may want to ask if they notice anything about the angles formed by the two graphs. c. The lines are perpendicular. 28. a. The equation of the given line is y = 4x. The equation of a line parallel to this one is y = 4x + 5. (Students may choose any line with the same slope of 4 and a y-intercept other than 0.) The equation of a line perpendicular to 1 the given line is y x a. i. slope = 10, y = 10x ii. slope = 10, y = 10x iii. slope = 10, y = 10x The slopes of the three lines are the same. c. The three graphs have the same slope and the same equation. The scales on the axes are different, which makes the graphs look different. 30. a. i. Such a line exists, but the y-intercept of such a line must be 0 or negative, and the slope must be 0 or negative. For example, the line y = 2x 1 does not pass through the first quadrant. ii. Such a line does exist. The line y = 2 passes through only the first and second quadrants. iii. No such line exists because a line must pass through at least two quadrants. i. Such a line must have a y-intercept that is 0 or negative and a slope that is 0 or negative. Moving Straight Ahead 3

4 Answers ii. Such a line must have the equation y = a or y = ax, where a is any number not equal to zero. Note: Students have not been formally introduced to lines of the form x = b for b a constant, so you need not be concerned with these lines at this time. c. i. y = 2x 1 ii. y = Lines whose slopes are negative reciprocals of each other are perpendicular. Another example would be 1 3 x = y and 3x = y. When these equations are graphed and the angle formed by their intersection is a right angle, these lines are perpendicular. This conjecture is true even if the y-intercept is not zero. For example, if you graph 1 x + 5 = y and 3x + 5 = y, 3 the angle formed by their intersection is also a right angle. 32. Answers will vary. Possible answer: y = 7, y = 11, y = 2x + 1, y = 2x Opposite sides of a parallelogram are parallel, so the first two lines are parallel, and the second two lines are parallel. The coordinates are: top left vertex: ( 2, 11); top right vertex: (5, 11); bottom left vertex: ( 4, 7); bottom right vertex: (3, 7). 33. Answers will vary. Possible answer: y = 2x + 3, y = 1, and y = x; two of the lines must have slopes which are opposites and reciprocals of each other. The third line must cross the other two lines. Another type of possibility is y = 0, x = 0, and y = x 1. The lines y = 0 and x = 0 are perpendicular. 34. Two lines are parallel if they have the same slope. Two lines are perpendicular if their slopes are negative reciprocals of each other. In addition, two different lines are parallel if they are both of the form x = a and x = b, while two lines are perpendicular if they are of the form x = a and y = 35. a. Meifeng has = 5 payments left. $75 + 3($15) = $120 c. A = w, where A is the amount owed, and w is the number of weeks. d. If you graphed this equation, it would be a line with y-intercept $120 and 15 slope a. Robert is laying 60 stones in 90 minutes, which is 2 stones every 3 minutes or 40 stones an hour. He has 220 stones left, so it should take about = 5.5 hours at the same rate. So, he should finish at about 9:00 p.m. Because he has 120 stones laid at 2:00 P.M., he is 3 hours into the work, so he would have started at about 11:00 A.M. c. Answers will vary. Students may say that an equation will allow for a more precise answer but that graphs and tables allow for more visual inspection and estimation. 37. a. T = 30 3n y-intercept = 30; the y-intercept gives the starting temperature after 0 hours. c. Slope = 3; the slope tells you the rate of change (decrease) in temperature for each hour. c. Slope = -3; the slope tells you the rate of change (decrease) in temperature for each hour. Moving Straight Ahead 4

5 Answers 38. a. m = 0.50n Here, n is in dollars (If n is in cents, the equation becomes m = 50n.) slope = a. About 10,914; since 2.39P + P = 37,000, P = 10, So, the population was approximately 10,914 people in ,430; P = 2.39(37,000) + 37,000 = 125,430 people in a. ATerrance = m AKatrina = 34 Looking at the graphs makes it clear that Terrance s phone bill will be cheaper for less than 38 minutes. For example, we can see that for zero minutes (the y-intercept), the cost is $14.95 for Terrance and $34 for Katrina. Despite this early advantage, Terrance s bill will eventually be higher than Katrina s because the rate of change, or slope, is greater. c. The costs of the two plans will be the same when ATerrance = AKatrina. Note that the two bills are never exactly equal. They are closest when m = 38, at which point Terrance s bill is $33.95 and Katrina s bill is $34. Students can do this problem on the graphing calculator table, on a graph, or using a paper-and-pencil method. m = 38. d. The new plan, represented by A = m, will be cheaper than Terrance s bill if the time charged is more than 40.2 minutes. It will be cheaper than Katrina s bill if the time charged is less than 36 minutes. 41. a. Juan s method is correct; the perimeter increases by 1 each time. Natalie makes a common mistake. To convince students that she is incorrect, you should be able to apply her reasoning from Figure 1 to Figure 3 by tripling the perimeter of Figure 1. But this would mean that Figure 3 has a perimeter of 9, which we can see is not true. Steven s method does not work because there are segments that are no longer part of the perimeter when another triangle is added on to the existing figure. All three students are correct in their reasoning and their equations. All the expressions are equivalent. Students may justify each equation separately using the picture, or, if they recognize that the expressions are equivalent, they may justify one from the diagram and then say that the others are true by transforming one expression into a different, equivalent one. 42. a. Possible solutions include S = 2n 1 and S = n + (n 1) Answers will vary. Students might say something about how an arbitrary figure might look, such as having n squares vertically and n - 1 horizontally. Another way to describe each figure is to note that each row and column has n squares, but that the corner is double counted. c. P = 4n (there are n length units on each side) P = 2n + 2(n 1) + 2 2n represents the edge length on the left and bottom. 2(n 1) represents the inside corner edge lengths along the top and right sides. 2 represents the two 1-unit sides on the top and right sides. Moving Straight Ahead 5

6 Answers Connections a. х = 3; 2(3) + 3 = 9 х = 12; 1 2 (12) + 3 = 9 c. х = 1.5 or 3 2 ; = 9 2 d. х = 8.5; = 9 e. х = 15; = = a. x = 3; 3 + 6(3) = 4(3) + 9 = 21 х = 3; 6(3) + 3 = 4(3) + 9 = 21 c. х = 6; 6(6) 3 = 4(6) + 9 = 33 d. x = 6 3 or ; = = a. Yes; the ratios of their sides are all 2 : 3, since from smaller to larger the ratios are 2 : 3, 4 : 6, 6 : 9, 8 : 12, which all are equivalent to 2 : ; the slope of the diagonal line is 2 3 since the ratio of rise to run is 2 to 3. The slope is related to the similar rectangles because the ratio of the adjacent sides for each rectangle is the same as the ratio of rise to run, which is the slope. c. Neither rectangle is in the set, because neither has a slope of 2. However, the first 3 is similar to those shown in the graph; it is just oriented differently. 47. a. Before liftoff, the rocket is stationary (it is on the launch pad awaiting ignition). At time = 0, the engines are ignited. It appears that the rocket rises rapidly in the first 2 seconds, and then gradually tapers off. Before liftoff, the slope is 0. The rate of increase in altitude during this time (while the rocket is stationary) is a. х + 5 = 9, so х = 4. Check: 2(4 + 5) = B 2 х + 10 = х 8, so х = 18. Check: 2( ) = 18 8 = 26 c. 2 х + 10 = х, so х = 10. Check: 2( ) = 10 d. 2 х + 10 = 15; so х = 50. a. х = or Check: 2( ) = 2( 7.5) = 15 х = a. n = 15 n = 15 c. n = 2 d. n = 72 e. Answers will vary. Sample answer: y = 4 6 x a. n = 13.5 n = 400 c. n = 10 Moving Straight Ahead 6

7 Answers Extensions 53. a. Temperature should decrease as altitude increases, so t = a makes more sense. t = (620) = F c. Yes; A temperature of 44 F makes sense for Detroit in March. 54. a. Answers will vary. Possible answer: A table because the information would be easy to read. kilometers = 1.6 х miles; y = 1.6x Moving Straight Ahead 7