Objective: Sketch the graphs of tangent and cotangent functions

 Since the values of are positive in quadrants I and III, and negative in quadrants II and IV, so the period is. A convenient interval for this purpose is because, although the endpoints are not in the domain, since those values are undefined, and exists for all other values in the interval.  So increases without bound as approaches from the left and decreases without bound as approaches from the right. So the graph has vertical asymptotes at.

 To sketch the graph of the basic tangent function by hand, it helps to note three key points in one period of the graph and the vertical asymptotes.  Tangent key points:  The amplitude of a tangent function is not defined.

 Domain:  Range:  Vertical Asymptote:  Symmetry: Odd  Period:

 A convenient interval for is because, although the endpoints are not in the domain, since those values are undefined, and exists for all other values in the interval.  In the interval, the values of are positive and increase without bound. In the interval, the values of are negative and decreases without bound. So the graph has vertical asymptotes at

 To sketch the graph of the basic cotangent function by hand, it helps to note three key points in one period of the graph and the vertical asymptotes.  Cotangent key points:

 Domain:  Range:  Vertical Asymptote:  Symmetry: Odd  Period:

 1.

 2.

 3.

 4.

 5.