<< Chapter < Page Chapter >> Page >

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.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

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
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Precalculus' conversation and receive update notifications?

Ask