1. Objective: Sketch the graphs of tangent and cotangent functions

2.  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.

3.  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.

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

5.  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

6.  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:

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

8.  1.

9.  2.

10.  3.

11.  4.

12.  5.