1-5 Parent Functions and Transformations

Transcription

1 Describe the following characteristics of the graph of each parent function: domain, range, intercepts, symmetry, continuity, end behavior, and intervals on which the graph is increasing/decreasing. 1. f (x) = x D = {x x R}, R = {y y Z}. The graph has a y-intercept at (0, 0) and x-intercepts for {x 0 x < 1, x R}. The graph has no symmetry. The graph has a jump discontinuity for {x x Z}. = and =. The graph is constant for {x x Z}. The graph increases for {x x Z}. 3. f (x) = x 3 D = {x x R}, R = {y y R}. The graph has an intercept at (0, 0). The graph is symmetric with respect to the origin. The graph is continuous. = and =. The graph is increasing on (, ). 5. f (x) = c D = {x x R}, R = {y y = c, c R}. If c = 0, all real numbers are x-intercepts. If c 0, there are no x- intercepts. The graph has a y-intercept at (0, c). If c 0, the graph is symmetric with respect to the y-axis. If c = 0, the graph is symmetric with respect to the x-axis, y-axis, and origin. The graph is continuous. and. The graph is constant on (, ). Use the graph of f (x) = to graph each function. 7. g(x) = g(x) = f (x 4). Therefore, g(x) is the graph of f (x) = translated 4 units to the right. esolutions Manual - Powered by Cognero Page 1

2 9. g(x) = 4 g(x) = f (x + 6) 4. Therefore, g(x) is the graph of f (x) = translated 6 units left and 4 units down. Use the graph of f (x) = to graph each function. 11. g(x) = + 4 g(x) = f (x) + 4. Therefore, g(x) is the graph of f (x) = translated 4 units up. esolutions Manual - Powered by Cognero Page 2

3 13. g(x) = + 8 g(x) = f (x 6) + 8. Therefore, g(x) is the graph of f (x) = translated 6 units right and 8 units up. Describe how the graphs of f (x) = x and g(x) are related. Then write an equation for g(x). 15. The graph of g(x) is the graph of f (x) translated 5 units to the right when g(x) = [[x 5]], or translated 5 units down when g(x) = [[x]] The graph of g(x) is the graph of f (x) reflected in the y-axis and translated 5 units right when g(x) = [[5 x]], or reflected in the y-axis and translated 5 units up when g(x) = [[ x]] + 5. esolutions Manual - Powered by Cognero Page 3

4 19. PROFIT An automobile company experienced an unexpected two-month delay on manufacturing of a new car. The projected profit of the car sales before the delay p (x) is shown below. Describe how the graph of p (x) and the graph of a projection including the delay d(x) are related. Then write an equation for d(x). Since there is a two-month delay, the graph of g(x) is the graph of p (x) translated 2 units (months) to the right. The equation for d(x) can be written by replacing x with x 2 in p (x). So, d(x) = 10(x 2) 3 70(x 2) (x 2) 2. Describe how the graphs of f (x) = x and g(x) are related. Then write an equation for g(x). 21. The graph of g(x) is the graph of f (x) translated 5 units down; g(x) = x The graph of g(x) is the graph of f (x) translated 1 unit to the right and 2 units down; g(x) = x 1 2. esolutions Manual - Powered by Cognero Page 4

5 Identify the parent function f (x) of g(x), and describe how the graphs of g(x) and f (x) are related. Then graph f (x) and g(x) on the same axes. 25. g(x) = 3 g(x) = 3f (x + 8), the graph of g(x) is the graph of f (x) translated 8 units to the left and expanded vertically. The translation left is represented by the addition of 8 on the inside of f (x). The expansion is represented by the coefficient of 3 on the outside of f (x). 27. g(x) = 2[[x 6]] g(x) = 2f (x 6), so the graph of g(x) is the graph of f (x) translated 6 units to the right and expanded vertically. The translation left is represented by the subtraction of 6 on the inside of f (x). The expansion is represented by the coefficient of 2 on the outside of f (x). 29. g(x) = 2 x + 5 g(x) = 2f (x + 5), so g(x) is the graph of f (x) translated 5 units to the left, expanded vertically, and reflected in the x- axis. The translation left is represented by the addition of 5 on the inside of f (x). The expansion is represented by th coefficient of 2 on the outside of f (x). The reflection is represented by the negative coefficient on the outside of f (x). esolutions Manual - Powered by Cognero Page 5

6 31. g(x) = g(x) = f (x + 3), so g(x) is the graph of f (x) translated 3 units to the left and compressed vertically. The translation left is represented by the addition of 3 on the inside of f (x). The compression is represented by the coefficient of on the outside of f (x). Graph each function. 33. Draw circles at ( 6, 2) and (4, 0.25). Draw dots at ( 6, 0.16) and (2, 6) because g( 6) = 0.16 and g(2) = 6. esolutions Manual - Powered by Cognero Page 6

7 35. Draw circles at ( 3, 8) and (3, 9). Draw dots at ( 1, 7) and (4, 2) because h( 1) = 7 and h(4) = Draw circles at ( 1, 4.5) and (3, 1). Draw dots at ( 1, 2) and (3, 6.5) because f ( 1) = 2 and f (3) = 6.5. esolutions Manual - Powered by Cognero Page 7

8 39. BUSINESS A no-contract cell phone company charges a flat rate for daily access and $0.10 for each minute. The cost of the plan can be modeled by c(x) = [[x]], where x is the number of minutes used. a. Describe the transformation(s) of the parent function f (x) = [[x]] used to graph c(x). b. The company offers another plan in which the daily access rate is $2.49, and the per-minute rate is $0.05. What function c(x) can be used to describe the second plan? c. Graph both functions on the same graphing calculator screen. d. Would the cost of the plans ever equal each other? If so, at how many minutes? a. The graph of c(x) is the graph of f (x) compressed vertically and translated 1.99 units up. b. The per-minute rate is the rate that is affected by the variable, so c(x) = [[x]]. c. d. Yes; the plans will equal each other at 10 minutes. Use the zoom function of the calculator to find the intersection of the graphs. Notice that the intersection is the segment from x = 10 to x = 11. esolutions Manual - Powered by Cognero Page 8

9 Use the graph of f (x) to graph g(x) = f (x) and h(x) = f ( x ). 41. f (x) = To graph g(x) = f (x), reflect the range with respect to the x-axis for all elements of the domain where f (x) is less than zero. To graph h(x) = f ( x ), replace the range for x < 0 with a reflection of the range for x > 0 with respect to the y-axis. esolutions Manual - Powered by Cognero Page 9

10 43. f (x) = x 4 x 3 4x 2 To graph g(x) = f (x), reflect the range with respect to the x-axis for all elements of the domain where f (x) is less than zero. To graph h(x) = f ( x ), replace the range for x < 0 with a reflection of the range for x > 0 with respect to the y-axis. esolutions Manual - Powered by Cognero Page 10

11 45. f (x) = + 5 To graph g(x) = f (x), reflect the range with respect to the x-axis for all elements of the domain where f (x) is less than zero. To graph h(x) = f ( x ), replace the range for x < 0 with a reflection of the range for x > 0 with respect to the y-axis. esolutions Manual - Powered by Cognero Page 11

12 47. TRANSPORTATION In New York City, the standard cost for taxi fare is shown. One unit is equal to a distance of 0.2 mile or a time of 60 seconds, when the car is not in motion. a. Write a greatest integer function f (x) that would represent the cost for x units of cab fare, where x > 0. Round to the nearest unit. b. Graph the function. c. How would the graph of f (x) change if the fare for the first unit increased to $3.70? Graph the new function. a. When there is only a fraction of a unit, we must round up. For example, if 3.4 units are used, the customer will be chard for 4 units. To accomplish this, use [[x + 1]] when x is not a whole number. b. c. The graph of f (x) is translated 0.5 unit up. esolutions Manual - Powered by Cognero Page 12

13 Write and graph the function with the given parent function and characteristics. 49. f (x) = ; expanded vertically by a factor of 2, translated 7 units to the left and 5 units up g(x) = + 5 The shift 5 units up is represented by an addition of 5 after f (x), or f (x) + 5. The shift 7 units left is represented by the addition of 7 inside f (x), or f (x + 7). The vertical expansion by a factor of 2 is represented by the coefficient 2 outside f (x), or 2f (x). Therefore, g(x) = 2f (x + 7) + 5. PHYSICS The distance an object travels as a function of time is given by f (t) = at 2 + v 0 t + x 0, where a is the acceleration, v 0 is the initial velocity, and x 0 is the initial position of the object. Describe the transformation(s) of the parent function f (t) = t 2 used to graph f (t) for each of the following. 51. a = 2, v 0 = 2, x 0 = 0 Substitute the values then complete the square to identify the transformations. translated one unit left; translated one unit down esolutions Manual - Powered by Cognero Page 13

14 53. a = 4, v 0 = 8, x 0 = 1 translated 2 units to the left; expanded vertically; translated 7 units down Write an equation for each g(x). 55. The parent function is f (x) = Therefore, we have g(x) =. The graph of g(x) appears to be f (x) shifted 4 units up and 3 units to the right Use (4, 6) to determine if there is a dilation. There appears to be no dilation, so g(x) = + 4. esolutions Manual - Powered by Cognero Page 14

15 57. The parent function is f (x) =. The graph of g(x) appears to be f (x) shifted 4 units left and 6 units down. There is also an obvious dilation. So far, we have g(x) = 6. Let a represent the dilation and use (0, 2) to solve for a. There is no reflection, so a = 4 and g(x) = SHOPPING The management of a new shopping mall originally predicted that attendance in thousands would follow f (x) = for the first 60 days of operation, where x is the number of days after opening and x = 1 corresponds with opening day. Write g(x) in terms of f (x) for each situation below. a. Attendance was consistently 12% higher than expected. b. The opening was delayed 30 days due to construction. c. Attendance was consistently 450 less than expected. a. A consistent percentage change is represented by a dilation, or a coefficient in front of f (x). Therefore, g(x) = 1.12f (x). b. There is no affect on f (x). While the opening is delayed, the number of days after the opening, which determines the domain of the function, is unaffected. c. To represent a consistently less value, subtract the difference from f (x). Therefore, g(x) = f (x) esolutions Manual - Powered by Cognero Page 15

16 Identify the parent function f (x) of g(x), and describe the transformation of f (x) used to graph g(x). 61. The parent function is f (x) = x 3. The graph is reflected in the x-axis because it resembles an upside-down version of the parent graph. Use like points to gauge the translation. In the parent graph, the point of inflection, or the point where the graph appears to curve in at the middle is located at x = 0. The point of inflection of g(x) is 2 units up and 4 units to the right. So far, we have g(x) = a(x 4) where a represents the unknown compression or expansion. Use the given point to identify the value of a. Substituting 3 for a, g(x) = 3(x 4) 3 + 2, which means that the parent graph was expanded vertically. Therefore, the graph of g(x) is the graph of f (x) translated 4 units to the right, expanded vertically, reflected in the x- axis, and translated 2 units up.. esolutions Manual - Powered by Cognero Page 16

17 63. The parent function is f (x) = of the parent graph.. The graph is reflected in the x-axis because it resembles an upside-down version Use like points to gauge the translation. In the parent graph, the point where the graph begins is located at x = 0. In g(x), this point located 3 units to the right and 5 units up. So far, we have g(x) = a + 5 where a represents the unknown compression or expansion. Use the given point to identify the value of a. Therefore, the graph of g(x) is the graph of f (x) translated 3 units to the right, reflected in the x-axis, and translated 5 units up. esolutions Manual - Powered by Cognero Page 17

18 Use f (x) to graph g(x). 65. g(x) = 3f (x) 6 f(x) is dilated by a factor of 3 and then translated down 6. Do this to each piece of the graph. Remember that in the graph of f (x), (x, y) = (x, f (x)). Therefore, in the graph of g(x), (x, y) = (x, g(x)) or (x, 3f (x) 6). 1 st Segment 2 nd (x, f (x)) ( 6, 6) to ( 2, 4) (x, g(x)) ( 6, 12) to ( 2, 18) ( 2, 1) to (6, ( 2, 3) to (6, Segment 1) 9) Ray (6, 4) (6, 6) esolutions Manual - Powered by Cognero Page 18

19 67. g(x) = 2f (x) + 1 f(x) is dilated by a factor of 2 and then translated up 1. Do this to each piece of the graph. Remember that in the graph of f (x), (x, y) = (x, f (x)). Therefore, in the graph of g(x), (x, y) = (x, g(x)) or (x, 2f (x) + 1). 1 st Segment 2 nd (x, f (x)) ( 6, 6) to ( 2, 4) (x, g(x)) ( 6, 11) to ( 2, 9) ( 2, 1) to (6, ( 2, 1) to (6, Segment 1) 3) Ray (6, 4) (6, 7) esolutions Manual - Powered by Cognero Page 19

20 Use f (x) = 4 to graph each function. 69. g(x) = 3f (x) g(x) = f (2x + 1) + 8 esolutions Manual - Powered by Cognero Page 20

21 73. ERROR ANALYSIS Danielle and Miranda are describing the transformation g(x) = [[x + 4]]. Danielle says that the graph is shifted 4 units to the left, while Miranda says that the graph is shifted 4 units up. Is either of them correct? Explain. Sample answer: Both; for the greatest integer function, a shift of a units left is identical to a shift of a units up. 75. Writing in Math Explain why order is important when transforming a function with reflections and translations. Sample answer: Order is important because different graphs can be obtained depending on the order the transformations are performed. For example, if (a, b) is on the original graph and there is a translation 6 units up and then a reflection in the x-axis, the resulting point will be (a, b 6). However, if (a, b) is reflected in the x-axis first and then translated 6 units up, the resulting point will be (a, b + 6). REASONING Determine whether the following statements are sometimes, always, or never true. Explain your reasoning. 77. If f (x) is an odd function, then f ( x) = f (x). Sometimes; sample answer: f (x) = x 3 is an odd function and f ( x) f (x) when x = 1. However, f (x) = 0 is an odd function and f ( x) = f (x) for all x. 79. CHALLENGE Describe the transformation of f (x) = if ( 2, 6) lies on the curve. Sample answer: The graph of g(x) = is the graph of f (x) = translated 6 units to the left and 8 units down. 81. Writing in Math Use words, graphs, tables, and equations to relate parent functions and transformations. Show this relationship through a specific example. See students work. Find the average rate of change of each function on the given interval. 83. g(x) = x 2 6x + 1; [4, 8] esolutions Manual - Powered by Cognero Page 21

22 Use the graph of each function to describe its end behavior. Support the conjecture numerically. 85. q(x) = 0; Sample answer: As x, the denominator of the fraction will increase and the value of the fraction will approach 0, so g(x) will approach p (x) = 1; Sample answer: As x, the fraction will get closer and closer to, so p (x) will approach 1. Use the graph of each function to estimate its y-intercept and zero(s). Then find these values algebraically. 89. From the graph, it appears that f (x) will intersect the y-axis at (0, 0). Find f (0). Because f (0) = 0, there is a y-intercept at (0, 0). From the graph, it appears that there is an x-intercept near x = 1, x = 0, and x = 2. Let f (x) = 0 and solve for x. Therefore, the zeros of f are 0, 2, and 1. esolutions Manual - Powered by Cognero Page 22

23 91. GOVERNMENT The number of times each of the first 42 presidents vetoed bills are listed below. What is the standard deviation of the data? 2, 0, 0, 7, 1, 0, 12, 1, 0, 10, 3, 0, 0, 9, 7, 6, 29, 93, 13, 0, 12, 414, 44, 170, 42, 82, 39, 44, 6, 50, 37, 635, 250, 181, 21, 30, 43, 66, 31, 78, 44, 25 Enter the data in your calculator and find the standard deviation of the population of data. The standard deviation is about SAT/ACT The figure shows the graph of y = g(x), which has a minimum located at (1, 2.5). What is the maximum value of the function h(x) = 3g(x) 1? A 0 B 1 C 2 D 3 E It can not be determined from the information given The graph of h(x) is the graph of g(x) expanded vertically by a factor of 3, reflected in the x-axis, and translated 1 unit down. The reflection maps the minimum located at (1, 2) to (1, 2), and the translation shifts the point down 1 unit to (1, 1). Due to the reflection, this point is now the location of the maximum of h(x). The vertical expansion does not affect the maximum point on the graph. So, the maximum value of h(x) is 1, and the correct answer is B. esolutions Manual - Powered by Cognero Page 23

24 95. What is the range of y =? F {y y ±2 } G {y y 4} H {y y 0} J {y y 0} The smallest possible value for x 2 is 0. Therefore, the smallest possible value of is or 4. As x approaches positive or negative infinity, x 2 approaches positive infinity. Therefore, the value of approaches infinity. The correct choice is G. also esolutions Manual - Powered by Cognero Page 24