649 6.5 Graphs of the Tangent and Cotangent Functions EXAMPLE 4 Graphing y =c +tan x Graph y = 2 + tan x. ANALYTIC SOLUTION Every value of y for this function will be 2 units more than the corresponding value of y in y = tan x, causing the graph of y = 2 + tan x to be translated up 2 units compared to the graph of y = tan x. See Figure 50. x y 0 1 –1 y = 2 + tan x –2 2 – 3p 2 3p 2 p 2 –p –p p 2 Figure 50 Three periods of the function are shown in Figure 50. Because the period of y = 2 + tan x is p, additional asymptotes and periods of the function can be drawn by repeating the basic graph every p units on the x-axis to the left or to the right of the graph shown. GRAPHING CALCULATOR SOLUTION Observe Figures 51 and 52. In these figures y2 = tan x is the red graph and y1 = 2 + tan x is the blue graph. Notice that for the arbitrarily chosen value of p 4 1approximately 0.785398162, the difference in the y-values is y1 - y2 = 3 - 1 = 2. This illustrates the vertical translation up 2 units. −4 4 p 2 p 2 − Figure 51 −4 4 p 2 p 2 − Figure 52 S Now Try Exercise 29. EXAMPLE 5 Graphing y =c +a cot1x −d2 Graph y = -2 - cot Ax - p 4 B . SOLUTION Here b = 1, so the period is p. The negative sign in front of the cotangent will cause the graph to be reflected across the x-axis, and the argument Ax - p 4 B indicates a phase shift (horizontal shift) to the right p 4 unit. Because c = -2, the graph will then be translated down 2 units. To locate adjacent asymptotes, because this function involves the cotangent, we solve the following. x - p 4 = 0 and x - p 4 = p x = p 4 and x = 5p 4 Add p 4 . Dividing the interval Ap 4 , 5p 4 B into four equal parts and evaluating the function at the three key x-values within the interval give these points. a p 2 , -3b , a 3p 4 , -2b , 1p, -12 Key points We plot these points and join them with a smooth curve approaching the asymptotes. This period of the graph, along with the one in the domain interval A - 3p 4 , p 4 B , is shown in Figure 53. S Now Try Exercise 37. y 0 –3 1 y = –2 x y = –2 – cot(x – ) 4 P p 5p 4 3p 4 p 2 p 4 –p 2 – 3p 4 – p 4 Figure 53
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