# 2.2 Graphs of the other trigonometric functions  (Page 5/9)

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$y=A\mathrm{sec}\left(Bx-C\right)+D$
$y=A\mathrm{csc}\left(Bx-C\right)+D$

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

• The stretching factor is $\text{\hspace{0.17em}}|A|.$
• The period is $\text{\hspace{0.17em}}\frac{2\pi }{|B|}.$
• The domain is $\text{\hspace{0.17em}}x\ne \frac{C}{B}+\frac{\pi }{2|B|}k,$ where $\text{\hspace{0.17em}}k\text{\hspace{0.17em}}$ is an odd integer.
• The range is $\text{\hspace{0.17em}}\left(-\infty ,-|A|\right]\cup \left[|A|,\infty \right).$
• The vertical asymptotes occur at $\text{\hspace{0.17em}}x=\frac{C}{B}+\frac{\pi }{2|B|}k,$ where $\text{\hspace{0.17em}}k\text{\hspace{0.17em}}$ is an odd integer.
• There is no amplitude.
• $y=A\mathrm{sec}\left(Bx\right)\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}|A|.$
• The period is $\text{\hspace{0.17em}}\frac{2\pi }{|B|}.$
• The domain is $\text{\hspace{0.17em}}x\ne \frac{C}{B}+\frac{\pi }{2|B|}k,$ where $\text{\hspace{0.17em}}k\text{\hspace{0.17em}}$ is an integer.
• The range is $\text{\hspace{0.17em}}\left(-\infty ,-|A|\right]\cup \left[|A|,\infty \right).$
• The vertical asymptotes occur at $\text{\hspace{0.17em}}x=\frac{C}{B}+\frac{\pi }{|B|}k,$ where $\text{\hspace{0.17em}}k\text{\hspace{0.17em}}$ is an integer.
• There is no amplitude.
• $y=A\mathrm{csc}\left(Bx\right)\text{\hspace{0.17em}}$ is an odd function because sine is an odd function.

Given a function of the form $\text{\hspace{0.17em}}y=A\mathrm{sec}\left(Bx\right),\text{\hspace{0.17em}}$ graph one period.

1. Express the function given in the form $\text{\hspace{0.17em}}y=A\mathrm{sec}\left(Bx\right).$
2. Identify the stretching/compressing factor, $\text{\hspace{0.17em}}|A|.$
3. Identify $\text{\hspace{0.17em}}B\text{\hspace{0.17em}}$ and determine the period, $\text{\hspace{0.17em}}P=\frac{2\pi }{|B|}.$
4. Sketch the graph of $\text{\hspace{0.17em}}y=A\mathrm{cos}\left(Bx\right).$
5. Use the reciprocal relationship between $\text{\hspace{0.17em}}y=\mathrm{cos}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y=\mathrm{sec}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ to draw the graph of $\text{\hspace{0.17em}}y=A\mathrm{sec}\left(Bx\right).$
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 $\text{\hspace{0.17em}}f\left(x\right)=2.5\mathrm{sec}\left(0.4x\right).$

• Step 1. The given function is already written in the general form, $\text{\hspace{0.17em}}y=A\mathrm{sec}\left(Bx\right).$
• Step 2. $\text{\hspace{0.17em}}A=2.5\text{\hspace{0.17em}}$ so the stretching factor is $\text{\hspace{0.17em}}\text{2}\text{.5}\text{.}$
• Step 3. $\text{\hspace{0.17em}}B=0.4\text{\hspace{0.17em}}$ so $\text{\hspace{0.17em}}P=\frac{2\pi }{0.4}=5\pi .\text{\hspace{0.17em}}$ The period is $\text{\hspace{0.17em}}5\pi \text{\hspace{0.17em}}$ units.
• Step 4. Sketch the graph of the function $\text{\hspace{0.17em}}g\left(x\right)=2.5\mathrm{cos}\left(0.4x\right).$
• Step 5. Use the reciprocal relationship of the cosine and secant functions to draw the cosecant function.
• Steps 6–7. Sketch two asymptotes at $\text{\hspace{0.17em}}x=1.25\pi \text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x=3.75\pi .\text{\hspace{0.17em}}$ We can use two reference points, the local minimum at $\text{\hspace{0.17em}}\left(0,2.5\right)\text{\hspace{0.17em}}$ and the local maximum at $\text{\hspace{0.17em}}\left(2.5\pi ,-2.5\right).\text{\hspace{0.17em}}$ [link] shows the graph.

Graph one period of $\text{\hspace{0.17em}}f\left(x\right)=-2.5\mathrm{sec}\left(0.4x\right).$

This is a vertical reflection of the preceding graph because $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ is negative.

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

Yes. The range of $\text{\hspace{0.17em}}f\left(x\right)=A\mathrm{sec}\left(Bx-C\right)+D\text{\hspace{0.17em}}$ is $\left(-\infty ,-|A|+D\right]\cup \left[|A|+D,\infty \right).$

Given a function of the form $\text{\hspace{0.17em}}f\left(x\right)=A\mathrm{sec}\left(Bx-C\right)+D,\text{\hspace{0.17em}}$ graph one period.

1. Express the function given in the form $\text{\hspace{0.17em}}y=A\text{\hspace{0.17em}}\mathrm{sec}\left(Bx-C\right)+D.$
2. Identify the stretching/compressing factor, $\text{\hspace{0.17em}}|A|.$
3. Identify $\text{\hspace{0.17em}}B\text{\hspace{0.17em}}$ and determine the period, $\text{\hspace{0.17em}}\frac{2\pi }{|B|}.$
4. Identify $\text{\hspace{0.17em}}C\text{\hspace{0.17em}}$ and determine the phase shift, $\text{\hspace{0.17em}}\frac{C}{B}.$
5. Draw the graph of $\text{\hspace{0.17em}}y=A\text{\hspace{0.17em}}\mathrm{sec}\left(Bx\right)\text{\hspace{0.17em}}.$ but shift it to the right by $\text{\hspace{0.17em}}\frac{C}{B}\text{\hspace{0.17em}}$ and up by $\text{\hspace{0.17em}}D.$
6. Sketch the vertical asymptotes, which occur at $\text{\hspace{0.17em}}x=\frac{C}{B}+\frac{\pi }{2|B|}k,$ where $\text{\hspace{0.17em}}k\text{\hspace{0.17em}}$ is an odd integer.

## Graphing a variation of the secant function

Graph one period of $\text{\hspace{0.17em}}y=4\mathrm{sec}\left(\frac{\pi }{3}x-\frac{\pi }{2}\right)+1.$

• Step 1. Express the function given in the form $\text{\hspace{0.17em}}y=4\mathrm{sec}\left(\frac{\pi }{3}x-\frac{\pi }{2}\right)+1.$
• Step 2. The stretching/compressing factor is $|A|=4.$
• Step 3. The period is
• Step 4. The phase shift is
• Step 5. Draw the graph of $\text{\hspace{0.17em}}y=A\mathrm{sec}\left(Bx\right),$ but shift it to the right by $\text{\hspace{0.17em}}\frac{C}{B}=1.5\text{\hspace{0.17em}}$ and up by $\text{\hspace{0.17em}}D=6.$
• Step 6. Sketch the vertical asymptotes, which occur at $\text{\hspace{0.17em}}x=0,x=3,$ and $\text{\hspace{0.17em}}x=6.\text{\hspace{0.17em}}$ There is a local minimum at $\text{\hspace{0.17em}}\left(1.5,5\right)\text{\hspace{0.17em}}$ and a local maximum at $\text{\hspace{0.17em}}\left(4.5,-3\right).\text{\hspace{0.17em}}$ [link] shows the graph.

Graph one period of $\text{\hspace{0.17em}}f\left(x\right)=-6\mathrm{sec}\left(4x+2\right)-8.$

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