436 CHAPTER 6 Trigonometric Functions 5 Find an Equation for a Sinusoidal Graph Using a Graph to Find an Equation for a Sinusoidal Function Find an equation for the sinusoidal graph shown in Figure 64. Solution EXAMPLE 8 The graph has the characteristics of a cosine function. Do you see why? The maximum value, 3, occurs at = x 0. So the equation can be viewed as a cosine function ω( ) = y A x cos with = A 3 and period = T 1. Then π ω = 2 1, so ω π = 2 . The cosine function whose graph is shown in Figure 64 is ω π ( ) ( ) = = y A x x cos 3cos 2 Check: Using a graphing utility, graph π ( ) = Y x 3cos 2 1 and compare the result with Figure 64. Figure 64 x y 23 3 1 1 –– 2 5 –– 4 1 –– 2 1 –– 4 3 –– 4 2 1 –– 4 2 Period Using a Graph to Find an Equation for a Sinusoidal Function Find an equation for the sinusoidal graph shown in Figure 65. EXAMPLE 9 The graph is sinusoidal, with amplitude = A 3. The period is 8, so π ω = 2 8, or ω π = 4 . Since the graph passes through the origin but is decreasing near the origin, the graph is that of a sine function reflected about the x -axis. This requires that = − A 3. The sine function whose graph is given in Figure 65 is ω π( ) ( ) = = − y A x x sin 3sin 4 Check: Using a graphing utility, graph π( ) = − Y x 3sin 4 1 and compare the result with Figure 65. Solution NOTE The equation could also be viewed as a cosine function with a horizontal shift, but viewing it as a sine function is easier, since the graph passes through the origin. j Figure 65 4 24 24 10 Now Work PROBLEMS 59 AND 63 1. Now Work 1. Modeling 1.ExplainingConcepts CalculusPreview 1.InteractiveFigure ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 6.4 Assess Your Understanding 1. Use transformations to graph = y x3 .2 (pp. 112–120) 2. Use transformations to graph = y x2 . (pp. 112–120)
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