1. Graphing Cotangent

2. Objective To graph the cotangent

3. y = cot x Recall that –cot  =. –cot  is undefined when y = 0. –y = cot x is undefined at x = 0, x =  and x = 2 .

4. Domain/Range of Cotangent Function Since the function is undefined at every multiple of , there are asymptotes at these points. Graphs must contain the dotted asymptote lines. These lines will move if the function contains a horizontal shift, stretch or shrink. There are asymptotes at every multiple of . The domain is (- ,  except k  ) The range of every cot graph is (- ,  ).

5. Period of the Function This means that one complete cycle occurs between zero and . The period is .

6. Max and Min Cotangent Function Range is unlimited; there is no maximum. Range is unlimited; there is no minimum.

7. Parent Function Key Points x = 0: asymptote. The graph approaches  as it approaches this asymptote. x =  : asymptote. The graph approaches -  as it approaches this asymptote.

8. Graph of Parent Function y = cot x

9. The Graph: y = a cot b(x-c) +d a = vertical stretch or shrink If |a| > 1, there is a vertical stretch. If 0 < |a| < 1, there is a vertical shrink. If a is negative, the graph reflects about the x-axis.

10. y = 4 cot x

11. The Graph: y = a cot b (x-c) +d b= horizontal stretch or shrink. Period =. If |b| > 1, there is a horizontal shrink. If 0 < |b| < 1, there is a horizontal stretch.

12. y = cot 2x

13. The Graph: y = a cot b(x- c ) +d c = horizontal shift. If c is negative, the graph shifts left c units. If c is positive, the graph shifts right c units.

14. y = cot (x - )

15. The Graph: y = a cot b(x-c) + d d= vertical shift. If d is positive, the graph shifts up d units. If d is negative, the graph shifts down d units.

16. y = cot x - 4

17. To find the asymptotes

18. y = cot ( 2 x + ) + 2

19. y = - 2cot ( ½ x - ) - 3