452 CHAPTER 6 Trigonometric Functions For the graphs of ω φ ( ) = − y A x sin or ω φ ω ( ) = − > y A x cos , 0, π ω φ ω = = = = A T Amplitude Period 2 Phase shift The phase shift is to the left if φ < 0 and to the right if φ > 0. Figure 78 ( , 23) 9p––– 4 ( , 0) 3p––– 2 ( , 3) 7p––– 4 ( , 23) (a) 5p––– 4 5p––– 4 ( , 0) 3p––– 2 ( , 3) 3p––– 4 3p––– 4 x 3 2 21 1 22 23 (p, 0) ( , 0) p–– 2 p–– 4 p–– 2 p–– 4 ( , 23) (b) 5p––– 4 5p––– 4 ( , 0) 5p––– 2 ( , 3) 3p––– 4 3p––– 4 x y y 3 2 21 2 1 22 23 9p––– 4 7p––– 4 p–– 4 (p, 0) (2p, 0) ( , 0) p–– 2 ( , 23) p–– 4 (2 , 0) p–– 2 (2 , 3) p–– 4 NOTE The interval defining one cycle can also be found by solving the inequality π π ≤ − ≤ x 0 2 2 Then π π π π ≤ ≤ ≤ ≤ x x 2 3 2 3 2 j Finding the Amplitude, Period, and Phase Shift of a Sinusoidal Function and Graphing It Find the amplitude, period, and phase shift of π ( ) = − y x 3sin 2 , and graph the function. EXAMPLE 1 Solution Use the same four steps used to graph sinusoidal functions of the form ω( ) = y A x sin or ω( ) = y A x cos given on page 434. S tep 1 Comparing π π ( ) ( ) = − = ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ y x x 3sin 2 3sin 2 2 to ω φ ω φ ω ( ) ( ) = − = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ y A x A x sin sin note that ω = = A 3, 2, and φ π = . The graph is a sine curve with amplitude = A 3, period π ω π π = = = T 2 2 2 , and phase φ ω π = = shift 2 . S tep 2 The graph of π ( ) = − y x 3sin 2 lies between −3 and 3 on the y -axis. One cycle begins at φ ω π = = x 2 and ends at φ ω π ω π π π = + = + = x 2 2 3 2 . To find the five key points, divide the interval π π ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 2 , 3 2 into four subintervals, each of length π π ÷ = 4 4 , by finding the following values of x : π π π π π π π π π π π π π + = + = + = + = 2 2 4 3 4 3 4 4 4 5 4 5 4 4 3 2 1st x -coordinate 2nd x -coordinate 3rd x -coordinate 4th x -coordinate 5th x -coordinate S tep 3 Use these values of x to determine the five key points on the graph: π π π π π ( ) ( ) ( ) ( ) ( ) − 2 , 0 3 4 , 3 , 0 5 4 , 3 3 2 , 0 S tep 4 Plot these five points and fill in the graph of the sine function as shown in Figure 78(a). Extend the graph in each direction to obtain Figure 78(b).
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