SECTION 6.6 Phase Shift; Sinusoidal Curve Fitting 451 ‘Are You Prepared?’ Answers 1. = x 4 2. True Retain Your Knowledge Problems 56–65 are based on previously learned material. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for subsequent sections, a final exam, or later courses such as calculus. 56. Factor: − p q 125 8 3 6 57. Painting a Room Hazel can paint a room in 2 hours less time than her friend Gwyneth.Working together, they can paint the room in 2.4 hours. How long does it take each woman to paint the room by herself? 58. Solve: = − − 9 3 x x 1 5 2 59. Use the slope and the y -intercept to graph the linear function ( ) = − f x x 1 4 3 60. Find the domain of ( ) = − y x x log 4 . 4 61. If ( ) = + − f x x x 1 2 and ( ) = − g x x3 7, find ( )( ) g f 3 . 62. If ( ) = − f x x x3 , 2 find ( ) ( ) − − f x f c x c . 63. Find the intercepts of the graph of the function ( ) = + − + f x x x x 2 6 3 2 64. Complete the square in x to write + + x x2 26 2 in the form + u a . 2 2 65. Find the domain of ( ) = − − f x x5 2 3. 4 6.6 Phase Shift; Sinusoidal Curve Fitting OBJECTIVES 1 Graph Sinusoidal Functions of the Form ω φ ( ) = − + y A x B sin (p. 451) 2 Build Sinusoidal Models from Data (p. 455) 1 Graph Sinusoidal Functions of the Form ω φ ( ) = − + y A x B sin We have seen that the graph of ω ω ( ) = > y A x sin , 0, has amplitude A and period π ω = T 2 . One cycle can be drawn as x varies from 0 to π ω 2 or, equivalently, as ωx varies from 0 to π2 . See Figure 76. Now consider the graph of ω φ ( ) = − y A x sin which may also be written as ω φ ω ( ) = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ y A x sin where ω > 0 and φ (the Greek letter phi) are real numbers.The graph is a sine curve with amplitude A . As ω φ−x varies from 0 to π2 , one period is traced out. This period begins when ω φ φ ω − = = x x 0 or and ends when ω φ π φ ω π ω − = = + x x 2 or 2 See Figure 77. Notice that the graph of ω φ ω φ ω ( ) ( ) = − = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ y A x A x sin sin is the same as the graph of ω( ) = y A x sin , except that it has been shifted φ ω units (to the right if φ > 0 and to the left if φ < 0). This number φ ω is called the phase shift of the graph of ω φ ( ) = − y A x sin . Figure 77 One cycle of ω φ ( ) = − y A x sin , >A 0, ω φ > > 0, 0 x y A 2A Period 5 Phase shift 2p ––– v f–––v 2p ––– v f1––v Figure 76 One cycle of ω ω ( ) = > > y A x A sin , 0, 0 x y A 2A Period 5 2p –––v 2p –––v NOTE The beginning and end of the period can also be found by solving the inequality: ω φ π φ ω π φ φ ω π ω φ ω ≤ − ≤ ≤ ≤ + ≤ ≤ + x x x 0 2 2 2 j
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