SECTION 6.5 Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions 445 The Graph of the Cotangent Function cot = y x The graph of = y x cot can be obtained in the same way as the graph of = y x tan . The period of = y x cot is π. Because the cotangent function is not defined for integer multiples of π, concentrate on the interval π ( ) 0, . Table 10 lists some points on the graph of π = < < y x x cot , 0 . To determine the behavior of = y x cot near 0 and π, use the identity = x x x cot cos sin . As x approaches 0 but remains greater than 0, the value of cos x is close to 1, and the value of sin x is positive and close to 0. The ratio x x cos sin is positive and large; so as →x 0 from the right, then →∞ x cot . Using limit notation, = ∞ → + x lim cot . x 0 Similarly, as x approaches π but remains less than π, the value of cos x is close to −1, and the value of sin x is positive and close to 0. The ratio = x x x cos sin cot is negative and large; so as π →x from the left, then →−∞ x cot . Using limit notation, = −∞ π→ − x lim cot . x Figure 69 shows the graph. Figure 70 shows the graph using Desmos. Figure 69 = −∞< <∞ y x x cot , , x not equal to integer multiples of π −∞< <∞ y , x p 2p –2p –p y –1 1 3p –– 2 – –p– 2 p– 2 3p –– 2 3p ––– 4 5p –– 2 x 5 x 5 x 5 x 5 0 ( , 21) p–– 4 ( , 1) x 5 Figure 70 2 Graph Functions of the Form tan ω( ) = + y A x B and cot ω( ) = + y A x B For tangent functions, there is no concept of amplitude since the range of the tangent function is ( ) −∞ ∞, . The role of A in ω( ) = + y A x B tan is to provide the magnitude of the vertical stretch. The period of π = y x tan is , so the period of ω π ω ( ) = + y A x B tan is , caused by the horizontal compression of the graph by a factor of ω 1 . Finally, the presence of B indicates a vertical shift. x = y x cot ( ) x y, π 6 3 π( ) 6 , 3 π 4 1 π( ) 4 , 1 π 3 3 3 π⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ 3 , 3 3 π 2 0 π( ) 2 , 0 π2 3 − 3 3 π − ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ 2 3 , 3 3 π3 4 −1 π ( ) − 3 4 , 1 π5 6 − 3 π ( ) − 5 6 , 3 Table 10 Graphing a Function of the Form tan ω( ) = + y A x B Graph = − y x 2tan 1. Use the graph to determine the domain and the range of the function = − y x 2tan 1. EXAMPLE 1 (continued)
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