626 CHAPTER 6 The Circular Functions and Their Graphs NOTE Look at the middle row of the table in Example 3. Dividing C 0, 2p b D into four equal parts gives the values 0, p 2 , p, 3p 2 , and 2p for this row, resulting here in values of -1, 0, or 1. These values lead to key points on the graph, which can be plotted and joined with a smooth sinusoidal curve. Guidelines for Sketching Graphs of Sine and Cosine Functions To graph y =a sin bx or y =a cos bx, with b 70, follow these steps. Step 1 Find the period, 2p b . Start at 0 on the x-axis, and lay off a distance of 2p b . Step 2 Divide the interval into four equal parts. (See the Note preceding Example 2.) Step 3 Evaluate the function for each of the five x-values resulting from Step 2. The points will be maximum points, minimum points, and x-intercepts. Step 4 Plot the points found in Step 3, and join them with a sinusoidal curve having amplitude a . Step 5 Draw the graph over additional periods as needed. EXAMPLE 4 Graphing y =a sin bx Graph y = -2 sin 3x over one period using the preceding guidelines. SOLUTION Step 1 For this function, b = 3, so the period is 2p 3 . The function will be graphed over the interval C 0, 2p 3 D . Step 2 Divide the interval C 0, 2p 3 D into four equal parts to obtain the x-values 0, p 6 , p 3 , p 2 , and 2p 3 . Step 3 Make a table of values determined by the x-values from Step 2. Step 4 Plot the points 10, 02, Ap 6 , -2B , A p 3 , 0B , A p 2 , 2B , and A 2p 3 , 0B , and join them with a sinusoidal curve having amplitude 2. See Figure 28. Step 5 The graph can be extended by repeating the cycle. Notice that when a is negative, the graph of y =a sin bx is a reflection across the x-axis of the graph of y = ∣ a∣ sin bx. S Now Try Exercise 29. x 0 p 6 p 3 p 2 2p 3 3x 0 p 2 p 3p 2 2p sin 3x 0 1 0 -1 0 −2 sin 3x 0 -2 0 2 0 x y 0 –1 1 2 –2 y = –2 sin 3x p 6 p 3 p 2 2p 3 Figure 28
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