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y = A sec ( B x C ) + D
y = A csc ( B x C ) + D

Features of the graph of y = A Sec( Bx C )+ D

  • The stretching factor is | A | .
  • The period is 2 π | B | .
  • The domain is x C B + π 2 | B | k , where k is an odd integer.
  • The range is ( , | A | ] [ | A | , ) .
  • The vertical asymptotes occur at x = C B + π 2 | B | k , where k is an odd integer.
  • There is no amplitude.
  • y = A sec ( B x ) is an even function because cosine is an even function.

Features of the graph of y = A Csc( Bx C )+ D

  • The stretching factor is | A | .
  • The period is 2 π | B | .
  • The domain is x C B + π 2 | 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 csc ( B x ) is an odd function because sine is an odd function.

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

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

Graphing a variation of the secant function

Graph one period of f ( x ) = 2.5 sec ( 0.4 x ) .

  • Step 1. The given function is already written in the general form, y = A sec ( B x ) .
  • Step 2. A = 2.5 so the stretching factor is 2 .5 .
  • Step 3. B = 0.4 so P = 2 π 0.4 = 5 π . The period is 5 π units.
  • Step 4. Sketch the graph of the function g ( x ) = 2.5 cos ( 0.4 x ) .
  • Step 5. Use the reciprocal relationship of the cosine and secant functions to draw the cosecant function.
  • Steps 6–7. Sketch two asymptotes at x = 1.25 π and x = 3.75 π . We can use two reference points, the local minimum at ( 0 , 2.5 ) and the local maximum at ( 2.5 π , −2.5 ) . [link] shows the graph.
    A graph of one period of a modified secant function, which looks like an upward facing prarbola and a downward facing parabola.
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Graph one period of f ( x ) = 2.5 sec ( 0.4 x ) .

This is a vertical reflection of the preceding graph because A is negative.

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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Do the vertical shift and stretch/compression affect the secant’s range?

Yes. The range of f ( x ) = A sec ( B x C ) + D is ( , | A | + D ] [ | A | + D , ) .

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

  1. Express the function given in the form y = A sec ( 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 sec ( B x ) . but shift it to the right by C B and up by D .
  6. Sketch the vertical asymptotes, which occur at x = C B + π 2 | B | k , where k is an odd integer.

Graphing a variation of the secant function

Graph one period of y = 4 sec ( π 3 x π 2 ) + 1.

  • Step 1. Express the function given in the form y = 4 sec ( π 3 x π 2 ) + 1.
  • Step 2. The stretching/compressing factor is | A | = 4.
  • Step 3. The period is
    2 π | B | = 2 π π 3        = 2 π 1 3 π        = 6
  • Step 4. The phase shift is
    C B = π 2 π 3     = π 2 3 π     = 1.5
  • Step 5. Draw the graph of y = A sec ( B x ) , but shift it to the right by C B = 1.5 and up by D = 6.
  • Step 6. Sketch the vertical asymptotes, which occur at x = 0 , x = 3 , and x = 6. There is a local minimum at ( 1.5 , 5 ) and a local maximum at ( 4.5 , 3 ) . [link] shows the graph.
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Graph one period of f ( x ) = 6 sec ( 4 x + 2 ) 8.

A graph of one period of a modified secant function. There are two vertical asymptotes, one at approximately x=-pi/20 and one approximately at 3pi/16.
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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