434 CHAPTER 6 Trigonometric Functions SUMMARY Steps for Graphing a Sinusoidal Function of the Form ω( ) = y A x sin or ω( ) = y A x cos Using Key Points Step 1 Determine the amplitude and period of the sinusoidal function. Step 2 Divide the interval π ω ⎡ ⎢ ⎣ ⎤ ⎦ ⎥ 0, 2 into four subintervals of the same length. Step 3 Use the endpoints of these subintervals to obtain five key points on the graph. Step 4 Plot the five key points, and draw a sinusoidal graph to obtain the graph of one cycle. Extend the graph in each direction to make it complete. CALCULATOR TIP To graph a sinusoidal function of the form ω( ) = y A x sin or ω( ) = y A x cos using a graphing utility, use the amplitude to set Ymin and Y , max and use the period to set Xmin and X . max ■ Graphing a Sinusoidal Function Using Key Points Graph π ( ) = − y x 2sin 2 using key points. Solution EXAMPLE 6 Figure 62 x y 2 –2 (b) y 5 2 sin (2 x) 3 5 1 21 x (a) 3 1 (2, 0) (4, 0) (0, 0) (22, 0) (1, 22) (5, 22) 2 –2 y (0, 0) (2, 0) (4, 0) (3, 2) (21, 2) (3, 2) (1, 22) p –– 2 (c) 2 22 0 4 Since the sine function is odd, use the equivalent form: π( ) = − y x 2sin 2 Step 1 Compare π( ) = − y x 2sin 2 to ω( ) = y A x sin . Then = − A 2 and ω π = 2 , so the amplitude is = − = A 2 2 and the period is π ω π π = = = T 2 2 2 4. The graph of π( ) = − y x 2sin 2 lies between −2 and 2 on the y-axis. One cycle begins at = x 0 and ends at = x 4. Step 2 Divide the interval [ ] 0,4 into four subintervals, each of length ÷ = 4 4 1. The x-coordinates of the five key points are + = + = + = + = 0 0 1 1 1 1 2 2 1 3 3 1 4 1st x-coordinate 2nd x-coordinate 3rd x-coordinate 4th x-coordinate 5th x-coordinate Step 3 Evaluate π( ) = − y x 2sin 2 at each of the five x-coordinates. • at = = − = x y 0, 2sin0 0 • at π = = − = − x y 1, 2sin 2 2 • at = = x y 2, 0 • at = = x y 3, 2 • at = = x y 4, 0 The five key points on the graph are ( ) ( ) ( ) ( ) ( ) − 0,0 1, 2 2,0 3,2 4,0 Step 4 Plot these five points, and fill in the graph of the sine function as shown in Figure 62(a). Extend the graph in each direction to obtain Figure 62(b). Figure 62(c) shows the graph of one period using a TI-84 Plus CE.
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