124 CHAPTER 2 Functions and Their Graphs In Problems 65–68, the graph of a function f is illustrated. Use the graph of f as the first step toward graphing each of the following functions: (a) F x f x 3 ( ) ( ) = + (b) G x f x 2 ( ) ( ) = + (c) P x f x ( ) ( ) = − (d) H x f x 1 2 ( ) ( ) = + − (e) Q x f x 1 2 ( ) ( ) = (f) g x f x ( ) ( ) = − (g) hx f x2 ( ) ( ) = 65. x y 24 2 (24, 22) (4, 0) (2, 2) (0, 2) 4 22 66. x y 24 22 2 (24, 22 ) (22, 22) (4, 22) 4 (2, 2) 4 2 22 67. 2p p 1 21 x y p– 2 ( , 1) p– 2 p– 22 (2 , 21) p– 2 68. 2p p 1 21 x y (2p, 21) (p, 21) p– 2 p– 2 2 Mixed Practice In Problems 69–76, complete the square of each quadratic expression.Then graph each function using graphing techniques. (If necessary, refer to Section A.3 to review completing the square.) 69. f x x x2 2 ( ) = + 70. f x x x6 2 ( ) = − 71. f x x x8 1 2 ( ) = − + 72. f x x x4 2 2 ( ) = + + 73. f x x x 2 12 19 2 ( ) = − + 74. f x x x 3 6 1 2 ( ) = + + 75. f x x x 3 12 17 2 ( ) =− − − 76. f x x x 2 12 13 2 ( ) =− − − Applications and Extensions 77. Suppose that the x-intercepts of the graph of y f x( ) = are 5− and 3. (a) What are the x-intercepts of the graph of y f x 2 ? ( ) = + (b) What are the x-intercepts of the graph of y f x 2 ? ( ) = − (c) What are the x-intercepts of the graph of y f x 4 ? ( ) = (d) What are the x-intercepts of the graph of y f x ? ( ) = − 78. Suppose that the x-intercepts of the graph of y f x( ) = are 8− and 1. (a) What are the x-intercepts of the graph of y f x 4 ? ( ) = + (b) What are the x-intercepts of the graph of y f x 3 ? ( ) = − (c) What are the x-intercepts of the graph of y f x 2 ? ( ) = (d) What are the x-intercepts of the graph of y f x ? ( ) = − 79. Suppose that the function y f x( ) = is increasing on the interval 1, 5 . [ ] − (a) Over what interval is the graph of y f x 2 ( ) = + increasing? (b) Over what interval is the graph of y f x 5 ( ) = − increasing? (c) Is the graph of y f x( ) = − increasing, decreasing, or neither on the interval 1, 5 ? [ ] − (d) Is the graph of y f x ( ) = − increasing, decreasing, or neither on the interval 5, 1 ? [ ] − 80. Suppose that the function y f x( ) = is decreasing on the interval 2, 7 . [ ] − (a) Over what interval is the graph of y f x 2 ( ) = + decreasing? (b) Over what interval is the graph of y f x 5 ( ) = − decreasing? (c) Is the graph of y f x( ) = − increasing, decreasing, or neither on the interval 2, 7 ? [ ] − (d) Is the graph of y f x ( ) = − increasing, decreasing, or neither on the interval 7, 2 ? [ ] − 81. The graph of a function f is illustrated in the figure. (a) Graph y f x . ( ) = (b) Graph y f x . ( ) = 23 3 (1, 1) (2, 0) (21, 21) (22, 21) 2 22 x y 82. The graph of a function f is illustrated in the figure. (a) Graph y f x . ( ) = (b) Graph y f x . ( ) = 23 3 (1, 1) (2, 0) (21, 21) (0, 21) (22, 0) 22 2 x y 83. Suppose 1,3 ( ) is a point on the graph of y f x . ( ) = (a) What point is on the graph of y f x 3 5? ( ) = + − (b) What point is on the graph of y f x 2 2 1? ( ) = − − + (c) What point is on the graph of y f x2 3 ? ( ) = + 84. Suppose 3, 5 ( ) − is a point on the graph of y g x . ( ) = (a) What point is on the graph of y g x 1 3? ( ) = + − (b) What point is on the graph of y g x 3 4 3? ( ) = − − + (c) What point is on the graph of y g x3 9 ? ( ) = + 85. Graph the following functions using transformations. (a) f x x int ( ) ( ) = − (b) g x x int ( ) ( ) = − 86. Graph the following functions using transformations. (a) f x x int 1 ( ) ( ) = − (b) g x x int 1 ( ) ( ) = − 87. (a) Graph ( ) = − − f x x 3 3 using transformations. (b) Find the area of the region that is bounded by f and the x-axis and lies below the x-axis. 88. (a) Graph f x x 2 4 4 ( ) = − − + using transformations. (b) Find the area of the region that is bounded by f and the x-axis and lies above the x-axis.
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