PMI Unit 2 Working With Functions

Transcription

1 Vertical Shifts Class Work 1. a) 2. a) 3. i) y = x 2 ii) Move down 2 6. i) y = x ii) Move down 1 4. i) y = 1 x ii) Move up 3 7. i) y = e x ii) Move down 4 5. i) y = x ii) Move up 1

2 Vertical Shifts Homework 8. a) 9. a) 10. i) y = x 13. i) y = x ii) Move up 3 ii) Move up i) y = 1 x ii) Move down i) y = log (x) ii) Move down i) y = cos (x) ii) Move up (2x + 3)(x 1) 16. y3 4x x3 3x 2 +5x x x + 9

3 Horizontal Shifts Class Work 19. a) 20. a) 21. i) y = x i) y = x ii) Move right 2 ii) Move right i) y = 1 x ii) Move left i) y = e x ii) Move right i) y = x ii) Move left 1

4 Horizontal Shifts Homework 26. a) 27. a) 28. i) y = x i) y = x ii) Move right 3 ii) Move right i) y = 1 x ii) Move right i) y = log (x) ii) Move right i) y = x ii) Move left x 3 3x 2 + 3x (2x 5)(3x 2) 35. ac6 b x 2

5 Reflections Class Work 37. a) 38. a) 39. i) y = x i) y = x ii) Reflect over x-axis ii) Reflect over x-axis 40. i) y = 1 x ii) Reflect over y-axis 43. i) y = e x ii) Reflect over y-axis 41. i) y = x ii) Reflect over y-axis

6 Reflections Homework 44. a) 45. a) 46. i) y = x i) y = 1 x ii) Reflect over y-axis ii) Reflect over x-axis 47. i) y = x 3 ii) Reflect over x-axis 50. i) y = log (x) ii) Reflect over x-axis 48. i) y = cos (x) ii) Reflect over x-axis x 3 +12x 2 + 6x x 3 +11x x7 z 5 y 5

7 Vertical Stretches and Shrinks Class Work 55. a) 56. a) 57. i) y = x i) y = x ii) Vertical stretch of 2 ii) Vertical shrink of i) y = 1 x ii) Vertical stretch of i) y = e x ii) Vertical stretch of i) y = x ii) Vertical shrink of 1 2

8 Vertical Stretches and Shrinks Homework 62. a) 63. a) 64. i) y = x i) y = x ii) Vertical stretch of 4 ii) Vertical stretch of i) y = 1 x ii) Vertical shrink of i) y = log (x) ii) Vertical shrink of i) y = cos (x) ii) Vertical shrink of x x x 2 +36x xz y 5

9 Horizontal Stretches and Shrinks Class Work 73. a) 74. a) 75. i) y = x i) y = x ii) Horizontal shrink of 2 ii) Horizontal stretch of i) y = 1 x ii) Horizontal shrink of i) y = e x ii) Horizontal shrink of i) y = x ii) Horizontal stretch of 1 2

10 Horizontal Stretches and Shrinks Homework 80. a) 81. a) 82. i) y = x i) y = x ii) Horizontal shrink of 4 ii) Horizontal shrink of i) y = 1 x ii) Horizontal stretch of i) y = log (x) ii) Horizontal stretch of i) y = x 2 ii) Horizontal stretch of x x (3x + 5)(3x 5) 89. Unfactorable x 7 15x 5

11 Combining Transformations Class Work 91. a) 92. a) 93. i) y = x i) y = x ii) Reflect over y-axis, move right 3, vertical ii) Horizontal shrink of 3, move right 2, vertical stretch of 4, move up 6 stretch of 2, move down i) y = 1 x ii) Horizontal shrink of 2, move left 1, vertical stretch of 3, move down i) y = e x ii) Horizontal shrink of 2, reflect over y-axis, vertical stretch of 3, move up i) y = x ii) Reflect over y-axis, move right 3,vertical stretch of 2, move up 4

12 Combining Transformations Homework 98. a) 99. a) 100. i) y = x 3 ii) Horizontal shrink of 2, move right 3, vertical stretch of 4, move up i) y = x ii) Reflect over y-axis, move down i) y = 1 x ii) Move right 2, vertical stretch of 3, move down i) y = log (x) ii) Reflect over y-axis, move right 2, move up i) y = x ii) Horizontal stretch of 1, move right 6, reflect 3 over x-axis x 2 24x m (5x + 1)(5x 1) 108. Unfactorable 5n

13 Operations with Functions Class Work 109. a) h(x) = 3x 2 + 3x 2 5 lim lim x b) 11 c) -3 d) a) h(x) = (3x 2 4)( 3x 2 1) lim lim x b) 24 c) -4 d) a) h(x) = 3x2 4 3x 2 1 b) c) -4 d) a) h(x) = 6x 2 3 3x 2 5 lim b) 7 c) -11 d) a) h(x) = 3x2 4 ( 3x 2 1) 2 b) 8 9 c) -4 x x lim lim x x x lim lim x 1 3 lim x 1 x 3 d) (4x + 9)(4x 9) x x 2 + 4x + 6

14 Operations with Functions Homework 118. a) h(x) = x + 5 (2x + 1) 2 lim lim x x b) c) 1 d) a) h(x) = x x + 5 lim 0 lim x b) 2.45 c) -2 d) a) h(x) = (2x+1)2 x+5 b) 3.67 c) 1 2 d) 27 x lim lim x x 121. a) h(x) = 5(2x + 1) 2 2 x + 5 lim lim x b) 40.1 c) 1 d) a) h(x) = x+5 (2x+1) 4 b) -.03 c) -2 d) x lim 0 lim 0 x x 123. y = 2(x 2) y = 3 x y = 2x + 1

15 Composite Functions Class Work 126. a) f(g(x)) = 6x + 10 b) a) f(g(x)) = 25x 2 10x + 2 b) a) f(g(x)) = 2 2x 2 11 b) a) f(g(x)) = x+2 x+1 b) a) f(g(x)) = x 1 3 b) x10 5y m 7 n 7 Composite Functions Homework 135. a) f(g(x)) = 2x + 2 b) a) f(g(x)) = 8x x + 13 b) a) f(g(x)) = 1 3x 2 8 b) a) f(g(x)) = 2 x x b) a) f(g(x)) = x b) y = x y = (x 2) a6 b 9

16 Inverse Functions Class Work 143. i) f 1 (x) = x i) f 1 (x) = 3+2x 4x ii) D: all reals R: all reals lim lim x x 144. i) f 1 (x) = ± x 1 2 ii) D: x 0 R: y lim lim 1 x 2 x i) f 1 (x) = 10 x 2 ii) D: x 1 R: all reals lim x undefined lim x ± 145. i) f 1 (x) = ± x ii) D: all reals R: y 2 lim x 2 lim x ii) D: x 1 R: all real numbers lim x 8 ± lim x undefined 148. f g(x) = x 4 + 2x (4x 5y)(4x + 5y) x 12 y H. shrink 3, reflect y, 2

17 Inverse Functions Homework 152. i) f 1 (x) = x i) f 1 (x) = 5x 2 3x ii) D: all reals R: all reals lim x lim x 153. i) f 1 (x) = ± 3 2 x + 9 ii) D: x 0 R: y lim lim 5 x 3 x i) f 1 (x) = e x + 2 ii) D: x 6 R: all reals lim x undefined lim x ± 154. i) f 1 (x) = ± x ii) D: all reals R: y > 2 lim 2 lim x x ii) D: all reals R: 2 y 2 lim x ± lim x ± 157. x 4 4x 3 + 6x 2 4x (8x 1)(2x + 3) x 6 y , v. stretch 2, reflect x, 5

18 Piecewise Functions Class Work 161. a. f(-2) = 465 b. f(1) = 8 c. f(4) = 74 d. D: all reals R: y a. f(-5) = -1 b. f(0) = -4 c. f(4) = 6 d. D: all reals R: y < 5 e a. f(-2) = -10 b. f(0) = 2 c. f(4) = 4 d. D: all reals R: 2 y < 3 e b = -3 e. Piecewise Functions Homework 165. a. f(-2) = -2 b. f(0) = 6 c. f(3) = -3 d. D: all reals R: y < a. f(-5) = -2 b. f(0) = -4 c. f(4) = 1 d. D: all reals R: y < 1 e a. f(-2) = 8 3 b. f(0) = 0 c. f(4) = 64 3 d. D: all reals R: all reals e a = -1.2 b = -3.6

19 e x (9x + 4y)(9x 4y) x 7 y , reflect x, 5 Unit Review Questions Multiple Choice 173. Describe the transformation of the parent function f(x) = x 2 to g(x) = x 2 1 a. shift left 1 b. shift right 1 c. shift down 1 d. shift up Describe the transformation of the parent function f(x) = x to g(x) = x+1 a. shift left 1 b. shift right 1 c. shift down 1 d. shift up Describe the transformation of the parent function f(x)= [x] to g(x)= [2x] a. horizontal stretch of scale factor 2 b. horizontal stretch of scale factor 1 / 2 c. vertical stretch of scale factor 2 d. vertical stretch of scale factor 1/ Describe the transformation of the parent function f(x)= 1 to g(x)= 2 x x a. horizontal stretch of scale factor 2 b. horizontal stretch of scale factor 1 / 2 c. vertical stretch of scale factor 2 d. vertical stretch of scale factor ½ 177. Describe the transformation of the parent function f(x)= log(x) to g(x)= log(-x) a. horizontal reflection b. vertical reflection c. does not affect f(x) since it is symmetrical d. not possible because log(x) is undefined for negatives 178. The order of the following transformation of h(x) = x to h(x) = 4 3 x + 5 is a. Slide 3 right, stretch 4 vertically, slide 5 up b. Slide 3 left, stretch 4 vertically, slide 5 up c. Reflect over the y-axis, slide 3 right, stretch 4 vertically, slide 5 up d. Reflect over the y-axis, slide 3 left, stretch 4 vertically, slide 5 up 179. f(x) = 3x 2 2, g(x) = 4 x, and h(x) = f(x) g(x). h(3) =

20 a. 78 b. 26 c. 24 d f(x) = 3x 2 2, g(x) = 4 2x, and h(x) = f(x)/g(x). h(3) = a. -50 b. -25 c. 25 d f(x) = (3x) 2 4, g(x) = 5 4x, and h(x) = f(x)g(x). h(3) = a b c. -7 d f(x) = 3x 2 2, g(x) = 4 x, and h(x) = f(g(x)). h(3) = a. -5 b. -3 c. -1 d f(x) = 3x 2 2, g(x) = 4 x, and h(x) = g(f(x)). h(2a+1) = a. 12a 2 12a + 3 b a + 12a 2 c. 6a d. 12a Given f(x) = 2x 3 2, find f 1 (8) a b c. 2 d Given f(x) = 2x 3 2 and f 1 (a) = 3, find a. a b. 3 c. -56 d. undefined 2x 1 if x a(x) = {, find a(3). 4x + 2 if x > 3 a. 5 b. -10 c. 5 or -10 d. Undefined 3x + 2a if x < b(x) = {, find a such that b(x) is continuous 4a x if x 2

21 a. -8 b. -4 c. 4 d. 8 Unit Review Questions Extended Response 188. Given the function of f(x) as shown at the right a. b. c h(x) = { 2x2 + xm if x 3 xm 2 if x > 3 a. m=3,-2 b. For which value of m is the rate of change about h(3) the closest? c. Find h(3) in terms of m Given h(x) = x 2 x+3, describe the type and locations of any discontinuities of f(x) x 2 4 x 2 +5x+6 a. Vertical Asymptote at x=-2.5, Horizontal Asymptote at y=0 b. Vertical Asymptote at x=0, Horizontal Asymptote at y=1 c. Vertical Asymptote at x=-2, Horizontal Asymptote at y=0 d. Write a piecewise function f(x) so that the hole in g(x) is removed People enter a park at a rate of E(t) = { 8t if t 6 where t is the number of hours after opening. 2t if t > 6 4t if t 6 People leave the park at a rate of L(t) = {. The park is open 12 hours a day. 3t + c if t > 6

22 a. Write an, P(t) equation for the rate of change in the number of people in the park in terms of E(t) and L(t). b. Create the piecewise function for P(t). c. Find c so that there is no one in the park at closing. d. Does the answer in part c make sense? Explain.