The cotangent graph has vertical asymptotes at each value of
$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ where
$\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}x=0;\text{\hspace{0.17em}}$ we show these in the graph below with dashed lines. Since the cotangent is the reciprocal of the tangent,
$\text{\hspace{0.17em}}\mathrm{cot}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ has vertical asymptotes at all values of
$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ where
$\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}x=0,\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}\mathrm{cot}\text{\hspace{0.17em}}x=0\text{\hspace{0.17em}}$ at all values of
$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ where
$\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ has its vertical asymptotes.
Features of the graph of
y =
A Cot(
Bx )
The stretching factor is
$\text{\hspace{0.17em}}\left|A\right|.$
The period is
$\text{\hspace{0.17em}}P=\frac{\pi}{\left|B\right|}.$
The domain is
$\text{\hspace{0.17em}}x\ne \frac{\pi}{\left|B\right|}k,\text{\hspace{0.17em}}$ where
$\text{\hspace{0.17em}}k\text{\hspace{0.17em}}$ is an integer.
The range is
$\text{\hspace{0.17em}}(-\infty ,\infty ).$
The asymptotes occur at
$\text{\hspace{0.17em}}x=\frac{\pi}{\left|B\right|}k,\text{\hspace{0.17em}}$ where
$\text{\hspace{0.17em}}k\text{\hspace{0.17em}}$ is an integer.
$y=A\mathrm{cot}\left(Bx\right)\text{\hspace{0.17em}}$ 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.
$$y=A\mathrm{cot}\left(Bx-C\right)+D$$
Properties of the graph of
y =
A Cot(
Bx −c)+
D
The stretching factor is
$\text{\hspace{0.17em}}\left|A\right|.$
The period is
$\text{\hspace{0.17em}}\frac{\pi}{\left|B\right|}.$
The domain is
$\text{\hspace{0.17em}}x\ne \frac{C}{B}+\frac{\pi}{\left|B\right|}k,$ where
$\text{\hspace{0.17em}}k\text{\hspace{0.17em}}$ is an integer.
The range is
$\text{\hspace{0.17em}}(\mathrm{-\infty},-\left|A\right|]\cup [\left|A\right|,\infty ).$
The vertical asymptotes occur at
$\text{\hspace{0.17em}}x=\frac{C}{B}+\frac{\pi}{\left|B\right|}k,$ where
$\text{\hspace{0.17em}}k\text{\hspace{0.17em}}$ is an integer.
There is no amplitude.
$y=A\mathrm{cot}(Bx)\text{\hspace{0.17em}}$ 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
$\text{\hspace{0.17em}}f\left(x\right)=A\mathrm{cot}\left(Bx\right),$ graph one period.
Express the function in the form
$\text{\hspace{0.17em}}f\left(x\right)=A\mathrm{cot}\left(Bx\right).$
Identify the stretching factor,
$\text{\hspace{0.17em}}\left|A\right|.$
Identify the period,
$\text{\hspace{0.17em}}P=\frac{\pi}{\left|B\right|}.$
Draw the graph of
$\text{\hspace{0.17em}}y=A\mathrm{tan}(Bx).$
Plot any two reference points.
Use the reciprocal relationship between tangent and cotangent to draw the graph of
$\text{\hspace{0.17em}}y=A\mathrm{cot}\left(Bx\right).$
Sketch the asymptotes.
Graphing variations of the cotangent function
Determine the stretching factor, period, and phase shift of
$\text{\hspace{0.17em}}y=3\mathrm{cot}(4x),\text{\hspace{0.17em}}$ and then sketch a graph.
Step 1. Expressing the function in the form
$\text{\hspace{0.17em}}f\left(x\right)=A\mathrm{cot}\left(Bx\right)\text{\hspace{0.17em}}$ gives
$\text{\hspace{0.17em}}f\left(x\right)=3\mathrm{cot}\left(4x\right).$
Step 2. The stretching factor is
$\text{\hspace{0.17em}}\left|A\right|=3.$
Step 3. The period is
$\text{\hspace{0.17em}}P=\frac{\pi}{4}.$
Step 4. Sketch the graph of
$\text{\hspace{0.17em}}y=3\mathrm{tan}(4x).$
Step 5. Plot two reference points. Two such points are
$\text{\hspace{0.17em}}\left(\frac{\pi}{16},3\right)\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}\left(\frac{3\pi}{16},\mathrm{-3}\right).$
Step 6. Use the reciprocal relationship to draw
$\text{\hspace{0.17em}}y=3\mathrm{cot}(4x).$
Step 7. Sketch the asymptotes,
$\text{\hspace{0.17em}}x=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}x=\frac{\pi}{4}.$
The orange graph in
[link] shows
$\text{\hspace{0.17em}}y=3\mathrm{tan}\left(4x\right)\text{\hspace{0.17em}}$ and the blue graph shows
$\text{\hspace{0.17em}}y=3\mathrm{cot}\left(4x\right).$
Given a modified cotangent function of the form
$\text{\hspace{0.17em}}f\left(x\right)=A\mathrm{cot}\left(Bx-C\right)+D,\text{\hspace{0.17em}}$ graph one period.
Express the function in the form
$\text{\hspace{0.17em}}f\left(x\right)=A\mathrm{cot}\left(Bx-C\right)+D.$
Identify the stretching factor,
$\text{\hspace{0.17em}}\left|A\right|.$
Identify the period,
$\text{\hspace{0.17em}}P=\frac{\pi}{\left|B\right|}.$
Identify the phase shift,
$\text{\hspace{0.17em}}\frac{C}{B}.$
Draw the graph of
$\text{\hspace{0.17em}}y=A\mathrm{tan}(Bx)\text{\hspace{0.17em}}$ shifted to the right by
$\text{\hspace{0.17em}}\frac{C}{B}\text{\hspace{0.17em}}$ and up by
$\text{\hspace{0.17em}}D.$
Sketch the asymptotes
$\text{\hspace{0.17em}}x=\frac{C}{B}+\frac{\pi}{\left|B\right|}k,$ where
$\text{\hspace{0.17em}}k\text{\hspace{0.17em}}$ is an integer.
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
$\text{\hspace{0.17em}}f\left(x\right)=4\mathrm{cot}\left(\frac{\pi}{8}x-\frac{\pi}{2}\right)-2.$
Step 1. The function is already written in the general form
$\text{\hspace{0.17em}}f\left(x\right)=A\mathrm{cot}\left(Bx-C\right)+D.$
Step 2.$\text{\hspace{0.17em}}A=4,$ so the stretching factor is 4.
Step 3.$\text{\hspace{0.17em}}B=\frac{\pi}{8},$ so the period is
$\text{\hspace{0.17em}}P=\frac{\pi}{\left|B\right|}=\frac{\pi}{\frac{\pi}{8}}=8.$
Step 4.$\text{\hspace{0.17em}}C=\frac{\pi}{2},$ so the phase shift is
$\text{\hspace{0.17em}}\frac{C}{B}=\frac{\frac{\pi}{2}}{\frac{\pi}{8}}=4.$
Step 5. We draw
$\text{\hspace{0.17em}}f\left(x\right)=4\mathrm{tan}\left(\frac{\pi}{8}x-\frac{\pi}{2}\right)-2.$
Step 6-7. Three points we can use to guide the graph are
$\text{\hspace{0.17em}}(6,2),(8,-2),\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}(10,-6).\text{\hspace{0.17em}}$ We use the reciprocal relationship of tangent and cotangent to draw
$\text{\hspace{0.17em}}f\left(x\right)=4\mathrm{cot}\left(\frac{\pi}{8}x-\frac{\pi}{2}\right)-2.$
Step 8. The vertical asymptotes are
$\text{\hspace{0.17em}}x=4\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}x=12.$
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