Graphing Cotangent

Objective To graph the cotangent

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

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 (- ,  ).

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

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

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.

Graph of Parent Function y = cot x

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.

y = 4 cot x

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.

y = cot 2x

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.

y = cot (x - )

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.

y = cot x - 4

To find the asymptotes

y = cot ( 2 x + ) + 2

y = - 2cot ( ½ x - ) - 3