Graphs of the Other Trigonometric Functions: Learn It 4

Graphing Transformations of y = sec x and y = csc x

For shifted, compressed, and/or stretched versions of the secant and cosecant functions, we can follow similar methods to those we used for tangent and cotangent. That is, we locate the vertical asymptotes and also evaluate the functions for a few points (specifically the local extrema). If we want to graph only a single period, we can choose the interval for the period in more than one way. The procedure for secant is very similar, because the cofunction identity means that the secant graph is the same as the cosecant graph shifted half a period to the left. Vertical and phase shifts may be applied to the cosecant function in the same way as for the secant and other functions. The equations become the following.

[latex]y=A\sec(Bx−C)+D[/latex]
[latex]y=A\csc(Bx−C)+D[/latex]

features of the graph of [latex]y=A\sec(Bx−C)+D[/latex]

  • The amplitude is |A|.
  • The period is [latex]\frac{2\pi}{|B|}[/latex].
  • The domain is [latex]x\ne \frac{C}{B}+\frac{\pi}{2|B|}k[/latex], where k is an odd integer.
  • The range is [latex]( -\infty, -|A|] \cup [|A|, \infty )[/latex]
  • The vertical asymptotes occur at [latex]x=\frac{C}{B}+\frac{\pi}{2|B|}k[/latex], where k is an odd integer.
  • There is no amplitude.
  • [latex]y=A\sec(Bx)[/latex] is an even function because cosine is an even function.

features of the graph of [latex]y=A\csc(Bx−C)+D[/latex]

  • The amplitude is |A|.
  • The period is [latex]\frac{2\pi}{|B|}[/latex].
  • The domain is [latex]x\ne\frac{C}{B}+\frac{\pi}{2|B|}k[/latex], where k is an integer.
  • The vertical asymptotes occur at [latex]x=\frac{C}{B}+\frac{\pi}{2|B|}k[/latex], where k is an odd integer.
  • There is no amplitude.
  • [latex]y=A\sec(Bx)[/latex] is an even function because cosine is an even function.
  • The vertical asymptotes occur at [latex]x=\frac{C}{B}+\frac{\pi}{|B|}k[/latex], where k is an integer.
  • There is no amplitude.
  • [latex]y=A\csc(Bx)[/latex] is an odd function because sine is an odd function.
How To: Given a function of the form [latex]f(x)=A\sec (Bx−C)+D[/latex], graph one period.

  1. Express the function given in the form [latex]y=A\sec(Bx−C)+D[/latex].
  2. Identify the stretching/compressing factor, |A|.
  3. Identify B and determine the period, [latex]\frac{2\pi}{|B|}[/latex].
  4. Identify C and determine the phase shift, [latex]\frac{C}{B}[/latex].
  5. Draw the graph of [latex]y=A\sec(Bx)[/latex]. but shift it to the right by [latex]\frac{C}{B}[/latex] and up by D.
  6. Sketch the vertical asymptotes, which occur at [latex]x=\frac{C}{B}+\frac{\pi}{2|B|}k[/latex], where k is an odd integer.
Graph one period of [latex]y=4\sec \left(\frac{\pi}{3}x−\frac{\pi}{2}\right)+1[/latex].

Graph one period of [latex]f(x)=−6\sec(4x+2)−8[/latex].

How To: Given a function of the form [latex]f(x)=A\csc(Bx−C)+D[/latex], graph one period.
  1. Express the function given in the form [latex]y=A\csc(Bx−C)+D[/latex].
  2. Identify the stretching/compressing factor, |A|.
  3. Identify B and determine the period, [latex]\frac{2\pi}{|B|}[/latex].
  4. Identify C and determine the phase shift, [latex]\frac{C}{B}[/latex].
  5. Draw the graph of [latex]y=A\csc(Bx)[/latex] but shift it to the right by and up by D.
  6. Sketch the vertical asymptotes, which occur at [latex]x=\frac{C}{B}+\frac{\pi}{|B|}k[/latex], where k is an integer.
Sketch a graph of [latex]y=2\csc\left(\frac{\pi}{2}x\right)+1[/latex]. What are the domain and range of this function?

Given the graph of [latex]f(x)=2\cos\left(\frac{\pi}{2}x\right)+1[/latex], sketch the graph of [latex]g(x)=2\sec\left(\frac{\pi}{2}x\right)+1[/latex] 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.