SECTION 2.5 Graphing Techniques: Transformations 123 15. x y 3 3 23 23 16. x y 5 3 23 21 17. x y 3 3 23 23 18. x y 8 6 26 24 19. x y 4 4 24 24 20. x y 3 3 23 23 21. x y 4 4 24 24 22. x y 3 3 23 23 In Problems 23–32, write the function whose graph is the graph of y x3 = but is: 23. Shifted to the right 4 units 24. Shifted to the left 4 units 25. Shifted up 4 units 26. Shifted down 4 units 27. Reflected about the y-axis 28. Reflected about the x-axis 29. Vertically stretched by a factor of 5 30. Horizontally stretched by a factor of 4 31. Horizontally compressed by a factor of 1 2 32. Vertically compressed by a factor of 1 4 In Problems 33–36, find the function that is finally graphed after each of the following transformations is applied to the graph of y x = in the order stated. 33. (1) Shift up 2 units (2) Reflect about the x-axis (3) Reflect about the y-axis 34. (1) Reflect about the x-axis (2) Shift right 3 units (3) Shift down 2 units 35. (1) Vertical stretch by a factor of 3 (2) Shift up 4 units (3) Shift left 5 units 36. (1) Shift up 2 units (2) Reflect about the y-axis (3) Shift left 3 units 37. If 3, 6 ( ) is a point on the graph of y f x , ( ) = which of the following points must be on the graph of y f x ? ( ) = − (a) 6, 3 ( ) (b) 6, 3 ( ) − (c) 3, 6 ( ) − (d) 3, 6 ( ) − 38. If 3, 6 ( ) is a point on the graph of y f x , ( ) = which of the following points must be on the graph of y f x ? ( ) = − (a) 6, 3 ( ) (b) 6, 3 ( ) − (c) 3, 6 ( ) − (d) 3, 6 ( ) − 39. If 1, 3 ( ) is a point on the graph of y f x , ( ) = which of the following points must be on the graph of y f x 2 ? ( ) = (a) 1, 3 2 ( ) (b) 2, 3 ( ) (c) 1, 6 ( ) (d) 1 2 , 3 ( ) 40. If 4, 2 ( ) is a point on the graph of y f x , ( ) = which of the following points must be on the graph of y f x2 ? ( ) = (a) 4, 1 ( ) (b) 8, 2 ( ) (c) 2, 2 ( ) (d) 4, 4 ( ) In Problems 41–64, graph each function using the techniques of shifting, compressing, stretching, and/or reflecting. Start with the graph of the basic function (for example, y x2 = ) and show all the steps. Be sure to show at least three key points. Find the domain and the range of each function. 41. f x x 1 2 ( ) = − 42. f x x 4 2 ( ) = + 43. g x x3 ( ) = 44. g x x 1 2 3 ( ) = 45. h x x 2 ( ) = + 46. h x x 1 ( ) = + 47. f x x 1 2 3 ( ) ( ) = − + 48. f x x 2 3 3 ( ) ( ) = + − 49. g x x 4 ( ) = 50. g x x 1 2 ( ) = 51. f x x 3 ( ) = − 52. f x x ( ) = − 53. f x x 2 1 3 2 ( ) ( ) = + − 54. f x x 3 2 1 2 ( ) ( ) = − + 55. g x x 2 2 1 ( ) = − + 56. g x x 3 1 3 ( ) = + − 57. h x x 2 ( ) = − − 58. h x x 4 2 ( ) = + 59. f x x 1 1 3 ( ) ( ) = − + − 60. f x x 4 1 ( ) = − − 61. g x x 2 1 ( ) = − 62. g x x 4 2 ( ) = − 63. h x x 1 2 ( ) = 64. h x x 1 3 3 ( ) = − +
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