GRAPHS OF TANGENT AND COTANGENT A brief journey

GRAPHS OF TANGENT AND COTANGENT A brief journey into Section 4. 5 a

Analysis of the Tangent Function Domain: All reals except odd multiples of Range: by Continuous on its domain Increasing on each interval in its domain Symmetry: Origin (odd function) No Local Extrema V. A. : for all odd integers k Unbounded H. A. : None End Behavior: and do not exist (DNE)

Analysis of the Tangent Function by What is the period of the tangent function? ? ? Period: How do we know that these are the vertical asymptotes? They are where cos(x) = 0!!! How do we know that these are the zeros? They are where sin(x) = 0!!!

Analysis of the Tangent Function The constants a, b, h, and k influence the behavior of in much the same way that they do for the sinusoids… • The constant a yields a vertical stretch or shrink. • The constant b affects the period. • The constant h causes a horizontal translation • The constant k causes a vertical translation Note: Unlike with sinusoids, here we do not use the terms amplitude and phase shift…

Analysis of the Cotangent Function The graph of this function will have asymptotes at the zeros of the sine function and zeros at the zeros of the cosine function. by Vertical Asymptotes: Zeros:

Guided Practice Describe the graph of the given function in terms of a basic trigonometric function. Locate the vertical asymptotes and graph four periods of the function. Start with the basic tangent function, horizontally shrink by a factor of 1/2, and reflect across the x-axis. Since the basic tangent function has vertical asymptotes at all odd multiples of , the shrink factor causes these to move to all odd multiples of. Normally, the period is , but our new period is. Thus, we only need a window of horizontal length to see four periods of the graph…

Guided Practice Describe the graph of the given function in terms of a basic trigonometric function. Locate the vertical asymptotes and graph four periods of the function. by

Guided Practice Describe the graph of the given function in terms of a basic trigonometric function. Locate the vertical asymptotes and graph two periods of the function. Start with the basic cotangent function, horizontally stretch by a factor of 2, vertically stretch by a factor of 3, and vertically translate up 1 unit. The horizontal stretch makes the period of the function The vertical asymptotes are at even multiples of . .

Guided Practice Describe the graph of the given function in terms of a basic trigonometric function. Locate the vertical asymptotes and graph two periods of the function. How would you graph this with your calculator? OR by