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
$\text{\hspace{0.17em}}x=12.$
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?