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The cotangent graph has vertical asymptotes at each value of x where tan x = 0 ; we show these in the graph below with dashed lines. Since the cotangent is the reciprocal of the tangent, cot x has vertical asymptotes at all values of x where tan x = 0 , and cot x = 0 at all values of x where tan x has its vertical asymptotes.

A graph of cotangent of x, with vertical asymptotes at multiples of pi.
The cotangent function

Features of the graph of y = A Cot( Bx )

  • The stretching factor is | A | .
  • The period is P = π | B | .
  • The domain is x π | B | k , where k is an integer.
  • The range is ( , ) .
  • The asymptotes occur at x = π | B | k , where k is an integer.
  • y = A cot ( B x ) is an odd function.

Graphing variations of y = cot x

We can transform the graph of the cotangent in much the same way as we did for the tangent. The equation becomes the following.

y = A cot ( B x C ) + D

Properties of the graph of y = A Cot( Bx −c)+ D

  • The stretching factor is | A | .
  • The period is π | B | .
  • The domain is x C B + π | B | k , where k is an integer.
  • The range is ( −∞ , | A | ] [ | A | , ) .
  • The vertical asymptotes occur at x = C B + π | B | k , where k is an integer.
  • There is no amplitude.
  • y = A cot ( B x ) is an odd function because it is the quotient of even and odd functions (cosine and sine, respectively)

Given a modified cotangent function of the form f ( x ) = A cot ( B x ) , graph one period.

  1. Express the function in the form f ( x ) = A cot ( B x ) .
  2. Identify the stretching factor, | A | .
  3. Identify the period, P = π | B | .
  4. Draw the graph of y = A tan ( B x ) .
  5. Plot any two reference points.
  6. Use the reciprocal relationship between tangent and cotangent to draw the graph of y = A cot ( B x ) .
  7. Sketch the asymptotes.

Graphing variations of the cotangent function

Determine the stretching factor, period, and phase shift of y = 3 cot ( 4 x ) , and then sketch a graph.

  • Step 1. Expressing the function in the form f ( x ) = A cot ( B x ) gives f ( x ) = 3 cot ( 4 x ) .
  • Step 2. The stretching factor is | A | = 3.
  • Step 3. The period is P = π 4 .
  • Step 4. Sketch the graph of y = 3 tan ( 4 x ) .
  • Step 5. Plot two reference points. Two such points are ( π 16 , 3 ) and ( 3 π 16 , −3 ) .
  • Step 6. Use the reciprocal relationship to draw y = 3 cot ( 4 x ) .
  • Step 7. Sketch the asymptotes, x = 0 , x = π 4 .

The orange graph in [link] shows y = 3 tan ( 4 x ) and the blue graph shows y = 3 cot ( 4 x ) .

A graph of two periods of a modified tangent function and a modified cotangent function. Vertical asymptotes at x=-pi/4 and pi/4.

Given a modified cotangent function of the form f ( x ) = A cot ( B x C ) + D , graph one period.

  1. Express the function in the form f ( x ) = A cot ( B x C ) + D .
  2. Identify the stretching factor, | A | .
  3. Identify the period, P = π | B | .
  4. Identify the phase shift, C B .
  5. Draw the graph of y = A tan ( B x ) shifted to the right by C B and up by D .
  6. Sketch the asymptotes x = C B + π | B | k , where k is an integer.
  7. Plot any three reference points and draw the graph through these points.

Graphing a modified cotangent

Sketch a graph of one period of the function f ( x ) = 4 cot ( π 8 x π 2 ) 2.

  • Step 1. The function is already written in the general form f ( x ) = A cot ( B x C ) + D .
  • Step 2. A = 4 , so the stretching factor is 4.
  • Step 3. B = π 8 , so the period is P = π | B | = π π 8 = 8.
  • Step 4. C = π 2 , so the phase shift is C B = π 2 π 8 = 4.
  • Step 5. We draw f ( x ) = 4 tan ( π 8 x π 2 ) 2.
  • Step 6-7. Three points we can use to guide the graph are ( 6 , 2 ) , ( 8 , 2 ) , and ( 10 , 6 ) . We use the reciprocal relationship of tangent and cotangent to draw f ( x ) = 4 cot ( π 8 x π 2 ) 2.
  • Step 8. The vertical asymptotes are x = 4 and x = 12.

The graph is shown in [link] .

A graph of one period of a modified cotangent function. Vertical asymptotes at x=4 and x=12.
One period of a modified cotangent function

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Source:  OpenStax, Essential precalculus, part 2. OpenStax CNX. Aug 20, 2015 Download for free at http://legacy.cnx.org/content/col11845/1.2
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