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
$\text{\hspace{0.17em}}x=12.$
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.