637 6.4 Translations of the Graphs of the Sine and Cosine Functions Combinations of Translations Further Guidelines for Sketching Graphs of Sine and Cosine Functions To graph y=c+a sin 3 b1x−d2 4 or y=c+a cos 3 b1x−d2 4 , with b 70, follow these steps. Method 1 Step 1 Find an interval whose length is one period 2p b by solving the threepart inequality 0 … b1x - d2 … 2p. Step 2 Divide the interval into four equal parts to obtain five key x-values. 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 points that intersect the line y = c (“middle” points of the wave). 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. Method 2 Step 1 Graph y = a sin bx or y = a cos bx. The amplitude of the function is a , and the period is 2p b . Step 2 Use translations to graph the desired function. The vertical translation is up c units if c 70 and down c units if c 60. The horizontal translation (phase shift) is to the right d units if d 70 and to the left d units if d 60. EXAMPLE 5 Graphing y =c +a sin 3 b 1x −d2 4 Graph y = -1 + 2 sin14x + p2 over two periods. SOLUTION We use Method 1. We first write the expression on the right side of the equation in the form c + a sin 3b1x - d24. y = -1 + 2 sin14x + p2, or y = -1 + 2 sin c 4 ax + p 4b d Rewrite by factoring out 4. Step 1 Find an interval whose length is one period. 0 … 4 ax + p 4b … 2p Three-part inequality 0 … x + p 4 … p 2 Divide each part by 4. - p 4 … x … p 4 Subtract p 4 from each part. Step 2 Divide the interval C - p 4 , p 4 D into four equal parts to obtain these x-values. - p 4 , - p 8 , 0, p 8 , p 4 Key x-values
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