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The domain of csc x was given to be all x such that x k π for any integer k . Would the domain of y = A csc ( B x C ) + D be x C + k π B ?

Yes. The excluded points of the domain follow the vertical asymptotes. Their locations show the horizontal shift and compression or expansion implied by the transformation to the original function’s input.

Given a function of the form y = A csc ( B x ) , graph one period.

  1. Express the function given in the form y = A csc ( B x ) .
  2. | A | .
  3. Identify B and determine the period, P = 2 π | B | .
  4. Draw the graph of y = A sin ( B x ) .
  5. Use the reciprocal relationship between y = sin x and y = csc x to draw the graph of y = A csc ( B x ) .
  6. Sketch the asymptotes.
  7. Plot any two reference points and draw the graph through these points.

Graphing a variation of the cosecant function

Graph one period of f ( x ) = −3 csc ( 4 x ) .

  • Step 1. The given function is already written in the general form, y = A csc ( B x ) .
  • Step 2. | A | = | 3 | = 3 , so the stretching factor is 3.
  • Step 3. B = 4 , so P = 2 π 4 = π 2 . The period is π 2 units.
  • Step 4. Sketch the graph of the function g ( x ) = −3 sin ( 4 x ) .
  • Step 5. Use the reciprocal relationship of the sine and cosecant functions to draw the cosecant function .
  • Steps 6–7. Sketch three asymptotes at x = 0 , x = π 4 , and x = π 2 . We can use two reference points, the local maximum at ( π 8 , −3 ) and the local minimum at ( 3 π 8 , 3 ) . [link] shows the graph.
    A graph of one period of a cosecant function. There are vertical asymptotes at x=0, x=pi/4, and x=pi/2.
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Graph one period of f ( x ) = 0.5 csc ( 2 x ) .

A graph of one period of a modified secant function, which looks like an downward facing prarbola and a upward facing parabola.
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Given a function of the form f ( x ) = A csc ( B x C ) + D , graph one period.

  1. Express the function given in the form y = A csc ( B x C ) + D .
  2. Identify the stretching/compressing factor, | A | .
  3. Identify B and determine the period, 2 π | B | .
  4. Identify C and determine the phase shift, C B .
  5. Draw the graph of y = A csc ( B x ) but shift it to the right by and up by D .
  6. Sketch the vertical asymptotes, which occur at x = C B + π | B | k , where k is an integer.

Graphing a vertically stretched, horizontally compressed, and vertically shifted cosecant

Sketch a graph of y = 2 csc ( π 2 x ) + 1. What are the domain and range of this function?

  • Step 1. Express the function given in the form y = 2 csc ( π 2 x ) + 1.
  • Step 2. Identify the stretching/compressing factor, | A | = 2.
  • Step 3. The period is 2 π | B | = 2 π π 2 = 2 π 1 2 π = 4.
  • Step 4. The phase shift is 0 π 2 = 0.
  • Step 5. Draw the graph of y = A csc ( B x ) but shift it up D = 1.
  • Step 6. Sketch the vertical asymptotes, which occur at x = 0 , x = 2 , x = 4.

The graph for this function is shown in [link] .

A graph of 3 periods of a modified cosecant function, with 3 vertical asymptotes, and a dotted sinusoidal function that has local maximums where the cosecant function has local minimums and local minimums where the cosecant function has local maximums.
A transformed cosecant function
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Given the graph of f ( x ) = 2 cos ( π 2 x ) + 1 shown in [link] , sketch the graph of g ( x ) = 2 sec ( π 2 x ) + 1 on the same axes.

A graph of two periods of a modified cosine function. Range is [-1,3], graphed from x=-4 to x=4.
A graph of two periods of both a secant and consine function. Grpah shows that cosine function has local maximums where secant function has local minimums and vice versa.
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Analyzing the graph of y = cot x

The last trigonometric function we need to explore is cotangent    . The cotangent is defined by the reciprocal identity cot x = 1 tan x . Notice that the function is undefined when the tangent function is 0, leading to a vertical asymptote in the graph at 0 , π , etc. Since the output of the tangent function is all real numbers, the output of the cotangent function is also all real numbers.

We can graph y = cot x by observing the graph of the tangent function because these two functions are reciprocals of one another. See [link] . Where the graph of the tangent function decreases, the graph of the cotangent function increases. Where the graph of the tangent function increases, the graph of the cotangent function decreases.

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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