SECTION 6.4 Graphs of the Sine and Cosine Functions 433 The next example illustrates how these five key points can be used to obtain the graph of a sinusoidal function. How to Graph a Sinusoidal Function Using Key Points Graph ( ) = y x 3sin 4 using key points. Step-by-Step Solution Step 1 Determine the amplitude and period of the sinusoidal function. EXAMPLE 5 Comparing ( ) = y x 3sin4 to ω( ) = y A x sin , note that = A 3 and ω = 4, so the amplitude is = A 3 and the period is π ω π π = = 2 2 4 2 . Because the amplitude is 3, the graph of ( ) = y x 3sin 4 lies between −3 and 3 on the y-axis. Because the period is π 2 , one cycle begins at = x 0 and ends at π = x 2 . Divide the interval π ⎡ ⎢ ⎣ ⎤ ⎦ ⎥ 0, 2 into four subintervals, each of length π π ÷ = 2 4 8 , as follows: Step 2 Divide the interval π ω ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 0, 2 into four subintervals of the same length. π πππ ππ πππ ππ π π π π π ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ + ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ + ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ + ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 0, 8 8 , 8 8 8 , 4 4 , 4 8 4 , 3 8 3 8 , 3 8 8 3 8 , 2 The endpoints of the subintervals are π π π π 0, 8 , 4 , 3 8 , 2 . These numbers represent the x-coordinates of the five key points on the graph. Figure 61 ( , 23) (a) 3p––– 8 x y 3 23 y ( , 3) (0, 0) p–– 8 ( , 0) p–– 4 (2 , 23) p–– 8 ( , 3) p–– 8 ( , 0) (0, 0) p–– 4 ( , 0) (– , 0) ( , 0) p–– 2 ( , 23) 3p––– 8 ( , 3) 5p––– 8 x 3 –3 (b) y 5 3 sin (4x) 3p ––– 8 5p ––– 8 –p –– 4 p –– 4 p –– 2 –p –– 8 p –– 8 p–– 8 3p ––– 8 p –– 2 p –– 4 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. Plot the five key points obtained in Step 3, and fill in the graph of the sine curve as shown in Figure 61(a). Extend the graph in each direction to obtain the complete graph shown in Figure 61(b). Notice that additional key points appear every π 8 radian. Check: Graph ( ) = y x 3sin 4 using transformations. Which graphing method do you prefer? Now Work PROBLEM 37 USING KEY POINTS Step 3 Use the endpoints of the subintervals from Step 2 to obtain five key points on the graph. To obtain the y-coordinates of the five key points of ( ) = y x 3sin 4 , evaluate ( ) = y x 3 sin 4 at each endpoint found in Step 2. The five key points are then π π π π ( ) ( ) ( ) ( ) ( ) − 0,0 8 ,3 4 ,0 3 8 , 3 2 ,0 NOTE The five key points ( ) x y, also can be found using the five endpoints (Step 2) as the x-coordinates. Then the y-coordinates are the product of =A 3 and the y-coordinates of the five key points of = y x sin . j
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