652 CHAPTER 6 The Circular Functions and Their Graphs 33. y = -1 + 2 tanx 34. y = 3 + 1 2 tanx 35. y = -1 + 1 2 cot12x - 3p2 36. y = -2 + 3 tan14x + p2 37. y = 1 - 2 cot c 2 ax + p 2b d 38. y = -2 + 2 3 tan a 3 4 x - pb Connecting Graphs with Equations Determine the simplest form of an equation for each graph. Choose b 70, and include no phase shifts. (Midpoints and quarter points are identified by dots.) See Example 6. 39. x y p 2 3p 2 2 –p –2 0 2 40. y x 0 2p p –2 2 41. 0 –1 1 y p 3 2p 3 x 42. x y 1 6 –p p 6 p 2 0 43. x y p 2 p –p 2 –p –2 0 2 3 1 44. x y p 2 p –2 –4 0 2 Concept Check Decide whether each statement is true or false. If false, explain why. 45. The least positive number k for which x = k is an asymptote for the tangent function is p 2 . 46. The least positive number k for which x = k is an asymptote for the cotangent function is p 2 . 47. The graph of y = tanx in Figure 42 suggests that tan1-x2 = tanx for all x in the domain of tan x. 48. The graph of y = cot x in Figure 45 suggests that cot1-x2 = -cot x for all x in the domain of cot x. Work each problem. 49. Concept Check If c is any number, then how many solutions does the equation c = tanx have in the interval 1-2p, 2p4? 50. Concept Check Consider the function defined by ƒ1x2 = -4 tan12x + p2. What is the domain of f ? What is its range? 51. Show that tan1-x2 = -tanx by writing tan1-x2 as sin1-x2 cos1-x2 and then using the relationships for sin1-x2 and cos1-x2. 52. Show that cot1-x2 = -cot x by writing cot1-x2 as cos1-x2 sin1-x2 and then using the relationships for cos1-x2 and sin1-x2.
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