SECTION 6.6 Phase Shift; Sinusoidal Curve Fitting 453 The graph of π π ( ) ( ) = − = ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ y x x 3sin 2 3sin 2 2 may also be obtained using transformations. See Figure 79. Figure 79 21 2p 1 1 2 p–– 2 p 2 2 p 2 2 p 2 p 2 Multiply by 3; Vertical stretch by a factor of 3 Replace x by 2x; Horizontal compression by a factor of Replace x by x 2 Shift right units (a) y 5 sin x (b) y 5 3 sin x (c) y 5 3 sin (2x) (d) y 5 3 sin [2 (x 2 )] 5 3 sin (2x 2 p) y 23 3 y x 2p x 23 3 y p p p p x 23 3 y x ; ( , 21) 3p––– 2 ( , 23) 3p––– 2 ( , 1) p–– 2 ( , 3) p–– 2 ( , 3) ( , 3) p–– 4 p–– 4 ( , 23) ( , 23) 3p––– 4 3p––– 4 Check: Figure 80 shows the graph of π ( ) = − Y x 3sin 2 1 using a TI-84 Plus CE. To graph a sinusoidal function of the form ω φ ( ) = − + y A x B sin , first graph the function ω φ ( ) = − y A x sin and then apply a vertical shift. See Example 2. 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 2cos 4 3 1, and graph the function. Solution EXAMPLE 2 Step 1 Compare π π ( ) ( ) = + = ⎡ + ⎢ ⎣ ⎤ ⎦ ⎥ y x x 2cos 4 3 2cos 4 3 4 to ω φ ω φ ω ( ) ( ) = − = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ y A x A x cos cos Note that ω = = A 2, 4, and φ π = −3 . The graph is a cosine curve with amplitude = A 2, period π ω π π = = = T 2 2 4 2 , and phase φ ω π = =− shift 3 4 . Step 2 The graph of π ( ) = + y x 2 cos 4 3 lies between −2 and 2 on the y-axis. One cycle begins at φ ω π = =− x 3 4 and ends at φ ω π ω π π π = + =− + =− x 2 3 4 2 4 . To find the five key points, divide the interval π π ⎡ − − ⎢ ⎣ ⎤ ⎦ ⎥ 3 4 , 4 into four subintervals, each of length π π ÷ = 2 4 8 , by finding the following values of x: π π π π π π π π π π π π π − −+=− −+=− −+=− −+=− 3 4 3 4 8 5 8 5 8 8 2 2 8 3 8 3 8 8 4 1st x-coordinate 2nd x-coordinate 3rd x-coordinate 4th x-coordinate 5th x-coordinate Figure 80 3 23 2 3p 2 p 2 NOTE The interval defining one cycle can also be found by solving the inequality π π ≤ + ≤ x 0 4 3 2 Then π π π π − ≤ ≤− − ≤ ≤− x x 3 4 3 4 4 j (continued)
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