647 6.5 Graphs of the Tangent and Cotangent Functions Techniques for Graphing Guidelines for Sketching Graphs ofTangent and Cotangent Functions To graph y =a tan bx or y =a cot bx, with b 70, follow these steps. Step 1 Determine the period, p b . To locate two adjacent vertical asymptotes, solve the following equations for x: For y = a tan bx: bx = - p 2 and bx = p 2 . For y = a cot bx: bx = 0 and bx = p. Step 2 Sketch the two vertical asymptotes found in Step 1. Step 3 Divide the interval formed by the vertical asymptotes into four equal parts. Step 4 Evaluate the function for the first-quarter point, midpoint, and thirdquarter point, using the x-values found in Step 3. Step 5 Join the points with a smooth curve, approaching the vertical asymptotes. Indicate additional asymptotes and periods of the graph as necessary. Step 5 Join these points with a smooth curve, approaching the vertical asymptotes. See Figure 47. S Now Try Exercise 13. EXAMPLE 1 Graphing y =tan bx Graph y = tan 2x. SOLUTION Step 1 The period of this function is p 2 . To locate two adjacent vertical asymptotes, solve 2x = - p 2 and 2x = p 2 (because this is a tangent function). The two asymptotes have equations x = - p 4 and x = p 4 . Step 2 Sketch the two vertical asymptotes x = { p 4 , as shown in Figure 47. Step 3 Divide the interval A - p 4 , p 4 B into four equal parts to find key x-values. first-quarter value: - p 8 , middle value: 0, third-quarter value: p 8 Key x-values Step 4 Evaluate the function for the x-values found in Step 3. x - p 8 0 p 8 2 x - p 4 0 p 4 tan 2 x -1 0 1 Another period has been graphed, one half period to the left and one half period to the right. 0 1 y = tan 2x x y –1 Period: p 2 –p 2 –p 4 –p 8 p 8 p 4 p 2 Figure 47
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