Answers for the lesson Plot Points in a Coordinate Plane

Transcription

1 LESSON 3.1 Answers for the lesson Plot Points in a Coordinate Plane Skill Practice 1. 5; No; the point could lie in either Quadrant II or Quadrant IV. 3. (3, 22) 4. (, 21) 5. (4, 4) 6. (24, 3) 7. (4, 21) 8. (3, ) 9. (25, 4) 1. (23, 22) 11. (24, 21) 12. (21, 2) 13. B U(, 6) Q(21, 5) R(23, ) T(23, 24) 2 S(, ) P(5, 5) V(1.5, 4) W(3, 22.5) 14. Quadrant I 15. Quadrant II 16. x-axis 17. origin 18. Quadrant III 19. y-axis 2. Quadrant I 21. Quadrant IV 22. The description of the location is backwards, the point is 6 units to the right of the origin and 6 units down. 23. B ,, 1, 2, , 27, 25, 23, , 23, 21, 1, 3 53

2 If the x-coordinate is, then the point is on the y-axis. If the y-coordinate is, then the point is on the x-axis. 34. Sample answer: 22, 21,, 1, X Y J 2 2 W rectangle; perimeter: 28 units, area: 48 square units. 29. Quadrant IV; the x-coordinate is positive and the y-coordinate is negative so the point is in Quadrant IV. 3. Quadrant IV; the x-coordinate is positive and the y-coordinate is negative so the point is in Quadrant IV. 31. Quadrant II; the x-coordinate is negative and the y-coordinate is positive so the point is in Quadrant II. 32. Quadrant III; the x-coordinate is negative and the y-coordinate is negative so the point is in Quadrant III. Z Decide on a side length of a square that is greater than 4, like 5, so the other points will be in different quadrants. Add 5 to the x-coordinate of J, 24, to find the point (1, 3) in Quadrant I. Then subtract 5 from the y-coordinate of J to find the point (24, 22) in Quadrant 3. Then add 5 to the x-coordinate of (24, 22) and add 5 to the y-coordinate of (1, 3) to find the point (1, 22) in Quadrant IV. 54

3 35. For (b, a): Quadrant II; since (a, b) is in Quadrant IV, a must be positive and b must be negative, so the coordinates of (b, a) must be negative and positive. For (2a, 22b): Quadrant I; since (a, b) is in Quadrant IV, a must be positive and b must be negative, so the coordinates of (2a, 22b) must both be positive. For (2b, 2a): Quadrant IV; since (a, b) is in Quadrant IV, a must be positive and b must be negative, so the coordinates of (2b, 2a) must be positive and negative. Problem Solving 36. a. Asia b. North America c. Asia d. South America e. North America f. Europe 37. There is exactly one low temperature for each day in February a. There is exactly one change in value for each day. 2 b. Sample answer: The change in value increases until day 4, and then decreases. 55

4 39. a a. 2.5, 1, 21, 21, 22, 23, 3, 23, b. (2, 8), (2.5, 22), (1, 23), (21, 3), (21, 23), (, ) b. Sample answer: From 1992 to 1999 the federal deficit was decreasing. 4. a. 3, 4, 5 29, 26, 23 b. 2 c. Sample answer: The diet is lowering the patient s LDL number. 2 c. Sample answer: A person who reported the same information that was measured. d. Quadrant IV; the person reported a greater height than was measured and a lesser weight than was measured. 56

5 LESSON 3.2 Answers for the lesson Graph Linear Equations Skill Practice 1. linear function No, to be in standard form it should be in the form Ax 1 By 5 C, so it should be 26x 1 y solution 4. solution 2 y 2 3x 5 5. solution 6. not a solution 7. not a solution 8. not a solution 14. y 1 4x The 8 should be substituted for x and 11 for y, Þ 23, so (8, 11) is not a solution B y 1 x 5 2 y 2x y 6x 1 3y 1 4x

6 x 2 2y y x 1 2y y x x C 24. A 25. B 59

7 y 5 3x y 5 26 y 22 y y 5 25x y 5 2x y 3 y y y y y 5 2x 2 1 6

8 32. 2 x 2 y 5 3 2x 2 2y 5 6 The equations are the same. Each term in the first equation was multiplied by 2 to get the second equation. Sample answer: 3x 2 3y D Problem Solving w 5 f domain: f 4, range: w 2; 2 lb a. d t domain: t 4, range: d 2; 125 mi b domain: d 4, range: 2 T 12; 128C domain: d 3, range: 2 T 95; 3 km 61

9 38. a. $19; substitute 3 for f and solve. b. 14, 39. a. 4. a. b. 13 } 1 yd; substitute 5 for C 3 and solve for f r 5 12t domain: t, range: r b. domain: t 4, range: r 48; the graph was a ray, but is now a segment. t (seconds) h (feet) a. 1, 6, 2, h 5 14t domain: t 72, range: 5 h y 5 2.5x y b. (12, 3); at 12 days the cost is the same for both payment plans c. Monthly; if he pays daily it will cost $37.5, if he pays monthly it will cost only $

10 LESSON 3.3 Answers for the lesson Graph Using Intercepts Skill Practice x-intercept 2. 24, 3; the x-intercept is when y is, so the point (24, ) gives the x-intercept. The y-intercept is when x is, so the point (, 3) gives the y-intercept. 2 (2, ) (, 22) 3. The intercepts are switched around; the x-intercept is 22, and the y-intercept is , , (2, ) 6. 6, , , , , , , } 2, , , (23, ) 2 (, 3) 19. ( 2 1 2, ) 2 (, 28) (, 5) 64

11 2. 1 ( 4, ) (18, ) 2 (, 22) (, 24.5) 21. (, 3) 3 ( 4, ) (1, ) (, 15) 2 (5, ) 25. (, 216) ( , ) (, 3) 2 65

12 26. (, 3) (18, ) (, 1 4) 2 ( 2 1 2, ) , , ,

13 D 38. C 39. B 4. A 41. Yes; yes; a horizontal line does not have an x-intercept if y Þ, a vertical line does not have a y-intercept if x Þ. 42. Sample answer: 15 and 3; k can be any multiple of both 3 and x 5 2 b } a, y 5 b Problem Solving 44. a. 2x 1 2y 5 72 b. 36, 36; Length (feet) Width (feet) 45. a. x 5 14, y 5 7; Large bottles (, 7) 4 2 (14, ) Small bottles b. Sample answer: 2 and 6, 4 and 5, 6 and a. x 5 64, y 5 128; Free throws Field goals b. The x-intercept means 64 field goals were scored and no free throws were scored. The y-intercept means that no field goals were scored and 128 free throws were scored. c. Sample answer: 4 field goals and 48 free throws, 5 field goals and 28 free throws, 6 field goals and 8 free throws. 67

14 46. d. 52 field goals 47. a. v-intercept: 12; f-intercept: 18; the v-intercept means there are no flowers planted, the f-intercept means there are no vegetables planted. b. Area (square feet) for flower plants Vegetable plants domain: v 12, range: f 18 c. 6 ft Sample answer: 1 h and 76 mi, 2 h and 64 mi, 3 h and 52 mi h. Sample answer: Since the tank will be empty when it needs to be refilled, replace w in the function with and then solve the resulting equation for t. 5. a. The B-intercept is the balance of the loan after weeks, the n-intercept is the amount of time it takes to pay off the loan. b. Balance (dollars) (, 18) B 18 2n Number of weeks c. domain: n 9, range: B 18; 9 wk d. The graph is two line segments; 11 payments. 68

15 Mixed Review of Problem Solving 1. a. 5; the number of all-day passes sold when no passes after 5 P.M. were sold. b. 1; the number of passes sold after 5 P.M. when no all-day passes were sold. c. Passes sold after 5 P.M All-day passes sold 2. a. There is exactly one cost for each time. b. Total spent (dollars) Time (months) yes; there is exactly one number of minutes for each day. 4. a. Yes; the domain is specified because you plan on hiking for 1 hours. Hiking distance (miles) b Time (hours) domain: t 1, range: d 3; 2 h 5. a. There is exactly one temperature departure for each month. 69

16 5. b. 2 domain: 1, 2, 3, 4, 5, and 6, range: 23, 21, 2, and 4 c. The temperature is below normal T-shirts; 6 7

17 LESSON 3.4 Answers for the lesson Find Slope and Rate of Change Skill Practice 1. slope 2. The line between the two points rises from left to right. 3. The denominator should be 2 2 5, not 5 2 2; m 5 } } positive; 2 } 3 6. negative; 2 1 } 2 5. undefined 7. The slope was calculated using run rise }, not } rise run ; m 5 } } } undefined } } undefined C 18. A 19. $2.25 per day, it costs $2.25 per day to rent a movie. 2., it does not matter how long a person stays at the park, the admission is the same Yes; the slope of the line containing both points is No; a line with an undefined slope is a vertical line, which is not the graph of a function. 35. Multiply y 2 2 y 1 } by } 21 x 2 2 x 1 21 to get 2y 2 1 y 1 } 2x 2 1 x 1, which, by the commutative property of addition, is equal to y 1 2 y 2 } x 1 2 x 2 ; it does not matter which point you choose to be (x 2, y 2 ) and which point is (x 1, y 1 ). Problem Solving 36. Sample answer: The water level decreases until 8 A.M. and then it increases until 12 P.M. 37. a. h to 1.5 h b h to 8.95 h 38. Sample answer: The altitude of the plane increases during the first 2 hours of the flight, then stays the same for about 45 minutes, and then decreases during the last hour and 45 minutes. 72

18 39. Sample answer: The elevation of the hiker increases for about 6 minutes, then stays the same for about 3 minutes, then decreases for the last 6 minutes. c. 4. a to 1998; about 215 students per year b to 2; about 15 students per year c. It increased. Sample answer: Although the number of engineering majors decreased, the decrease was offset by the greater increase in the number of biological science majors and liberal arts majors. 41. a. b. 73

19 LESSON 3.5 Answers for the lesson Graph Using Slope-Intercept Form Skill Practice parallel 2. y 5 mx 1 b; because m is the slope and b is the y-intercept , , 5. 23, , } 3, } 4, 8 9. A 1. C y 5 24x 1 1; 24, y 5 x 2 6; 1, y 5 2x 1 3; 2, y 5 23x 2 } 1 2 ; 23, 2 } y } 5 x 2 2; 2 2 } 5, y } 1 x 2 2; 2 1 } 1, B 18. A 19. C 2. The y-intercept is 21, not 1; (1, 3) (, 21)

20 blue and green 31. red, blue, and green 32. Not parallel; the slopes are 5 and Parallel; the slopes are both Parallel; the slopes are both Not parallel; the slopes are 24 and 2 1 } Sample answer: y 5 26x 1 5; the equation has the same slope as 6x 1 y 5 24, but a different y-intercept. 76

21 y 1 5mx 1 1 b and y 2 5 mx 2 1 b, so the slope of the line is y 2 2 y 1 } x 2 2 x 5 (mx 2 1 b) 2 (mx 1 1 b) }} 1 x 2 2 x 5 1 mx 2 2 mx 1 } x 2 2 x 5 m(x 2 2 x 1 ) } 1 x 2 2 x 5 m a. Total cost (dollars) 5 C 5 4t C 5 4t Problem Solving 4. a b. 41. a. c. $6 Distance (miles) Total cost (dollars) b. 3 mi C t C t Time (hours) d 5 65t 2 15 d 5 55t Time (hours) 43. a Time (hours) b. $5; $5; the difference is $5 no matter how many hours it takes to repair the car because the slopes are the same. Amount (dollars) a 5 14t a 5 12t Time (hours) The slopes are the amount of money earned per hour. The a-intercepts show the amount of money made at hours. b. $8 77

22 44. a b. Profit (dollars) P 5 5p 2 35 P 5 5p 2 5 b. A nonstudent pays more than a student after the first visit; a student pays more than a nonstudent when getting certified. Number of pictures c. Larger booth; if the artist rents the larger booth and sells all the paintings, the artist will make $55, if the artist rents the smaller booth and sells all the paintings, the artist will make only $ a. Total cost (dollars) C 5 3v 1 5 C Number of visits (1, 8); the point represents when the costs are equal. 78

23 LESSON 3.6 Answers for the lesson Model Direct Variation Skill Practice direct variation 2. No; a direct variation equation has a y-intercept of. 3. direct variation; not direct variation 5. not direct variation 6. direct variation; 1 } 3 7. direct variation; not direct variation 9. C 2 1. The coefficient of y should be 11. 1, not 3; y 5 } 5 x, the constant of 3 variation is }

24

25 y 5 x 34. y 5 2 } 9 x 35. y 5 4x 36. y } 2 x y } 26 x 38. Yes; the constants of variation are reciprocals of each other; if y 5 a 1 x, then a 1 5 } y and if x x 5 a 2 y, then a 2 5 x } y, which shows the constants of variation a 1 and a 2 are receiprocals of each other. 39. direct variation 23. y 5 2x; y 5 } 5 x; y 5 2 } 3 x; direct variation; y 5 5x 27. not a direct variation 28. Sample answer: Each of the ratios } y x should be equal, } 4 8 Þ } 6 16, so y does not vary directly with x. 29. y 5 3x 3. y 5 13x 31. y 5 1 } 2 x 32. y } 3 x Problem Solving 4. a. d 5 2r b. 3 m 41. a. v 5 } 3 t b. 12 h s 5 5d; 15 bags 43. a. Compare the ratios, } f, for all w data pairs (w, f ). Since the ratios all equal.25, f varies directly with w. b. f 5.25w; $.25 per pound; $7 44. a. Compare the ratios, } p, for all data pairs (, p). Since the ratios all equal 1.25, p varies directly with. b. p ; $1.25 per inch; 24 in. 82

26 45. a. Sample answer: d C (dollars) b. Cost (dollars) Number of days c. C 5 1.5d; yes; it s in the form y 5 ax; $ a. All of the ratios, m } t, are approximately equal to.4; m 5.4t;.4. b. about 26 field goals c. No; the ratios, } m t, for each season would not be equal to each other. 47. Because d 5 2r and r varies directly with p, you can write the equation r 5 ap. When d meters, r Substitute.65 for r when p 5 5 to get a. Solve to find a If you substitute ap for r into the equation d 5 2r, you get d 5 2(.13)p, giving the direct variation equation d 5.26p. 83

27 LESSON 3.7 Answers for the lesson Graph Linear Functions Skill Practice 1. function notation 2. Yes; they are all linear functions , 1, , 5, , 22, ,, ,, , 21, Because the graphs of g and f have the same slope, m 5 1, the lines are parallel. The y-intercept of the graph of g is 5 more than the y-intercept of the graph of f } 1 5, 3, 4 } } 1 2, 26, 23 } , 5, 1 } g(23) does not mean multiply 23 and g, it means to find the value of the function when g 5 23, g(23) D } C Because the graphs of h and f have the same slope, m 5 1, the lines are parallel. The y-intercept of the graph of h is 6 more than the y-intercept of the graph of f. 2 Because the graphs of q and f have the same slope, m 5 1, the lines are parallel. The y-intercept of the graph of q is 1 less than the y-intercept of the graph of f. 85

28 Because the graphs of m and f have the same slope, m 5 1, the lines are parallel. The y-intercept of the graph of m is 6 less than the y-intercept of the graph of f. Because the graphs of t and f have the same slope, m 5 1, the lines are parallel. The y-intercept of the graph of t is 3 less than the y-intercept of the graph of f Because the graphs of d and f have the same slope, m 5 1, the lines are parallel. The y-intercept of the graph of d is 7 more than the y-intercept of the graph of f. 2 Because the slope of the graph of r is greater than the slope of the graph of f, the graph of r rises faster from left to right. The y-intercept for both graphs is, so both lines pass through the origin. 86

29 Because the slope of the graph of w is greater than the slope of the graph of f, the graph of w rises faster from left to right. The y-intercept for both graphs is, so both lines pass through the origin. 2 Because the slope of the graph of h is negative, the graph of h falls from left to right. The y-intercept for both graphs is, so both lines pass through the origin. 33. Because the slope of the graph of k is negative, the graph of k falls from left to right. The y-intercept for both graphs is, so both lines pass through the origin. 2 Because the slope of the graph of g is less than the slope of the graph of f, the graph of g rises slower from left to right. The y-intercept for both graphs is, so both lines pass through the orgin. 87

30 34. b B Because the slope of the graph of m is negative, the graph of m falls from left to right. The y-intercept for both graphs is, so both lines pass through the origin. 36. a. Sample answer: f(x) 5 2x 1 3; Because the slope of the graph of g is greater than the slope of the graph of f, the graph of g rises faster from left to right. The y-intercept for both graphs is the same, so both lines pass through the point (, 3). 2 c. 2 Because the graphs of h and f have the same slope, m 5 2, the lines are parallel. The y-intercept of the graph of h is 3 less than the y-intercept of the graph of f. 88

31 37. Since the graphs of g and h have the same slope, m 5, the lines are parallel. The y-intercept of the graph of h is 2 less than the y-intercept of the graph of g x 1 14; 8x 1 7 Problem Solving 39. a. 4. a. Price (dollars) Years since 198 domain: x 2, range: 2.75 f(x) 4.75 b. 18; in 1998, 18 years after 198, the price of a movie ticket was $4.55. DVDs sold (in thousands) 2, 16, 12, 8, 4, Years since domain: x 5, range: 33 f(x) 21,58 b. 3; in 21, 3 years after 1998, the number of DVD players sold was 13,8,. Distance (miles) Time (hours) domain: x, range d(x) ; 1.5 h; substitute 15 for d(x) to get the equation x, solve for x. Fee (dollars) f(x) 5 4x g(x) 5 4x 1 75 Number of months Because the graphs of g and f have the same slope, m 5 4, the lines are parallel. The y-intercept of the graph of g is 8 less than the y-intercept of the graph of f. 89

32 43. Revenue (dollars) 44. a s(x) 5 16x r (x) 5 2x Number of tickets Because the slope of the graph of r is greater than the slope of the graph of s, the graph of r rises faster from left to right. The y-intercept for both graphs is, so both lines pass through the origin. b. Daily cost (dollars) Revenue (dollars) Number of pretzels Number of pretzels c. Sample answer: Find the values of y on each graph for any value of x. Subtract C(x) from R(x) to find the vendor s profit. 45. a. See graph in part (b); domain: 1 x 31, range: (x) Time (hours) b d(x) 24 l(x) l(x).3x Day of month domain: (x) 12.43, range: d(x) c. The graph of d is a reflection of the line. d. The number of hours of daylight equals the number of hours of darkness. 9

33 Mixed Review of Problem Solving Number of pages left 1. a. Drink mix (tablespoons) b. 8 tbsp 2. a. s 5 25t b. $ c. 133 adult tickets p m 1 Water (fluid ounces) 2 p m Time (minutes) Find the m-intercepts of each line and subract the m-intercept of your friend s line from the m-intercept of your line to find your friend takes 5 more minutes to read the essay. 4. Sample: 5. a. Average number of sunspots Distance (miles) Time (hours) Years since 1995 b to 1999; 36.1 sunspots per year c to 1997; 1.75 sunspots per year d. Subtract 17.5 from 11.9 and divide by 6 to find 15.6 sunspots per year. 91

34 6. 5 units down; 5 92