The domain of
$\text{\hspace{0.17em}}\mathrm{csc}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ was given to be all
$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ such that
$\text{\hspace{0.17em}}x\ne k\pi \text{\hspace{0.17em}}$ for any integer
$\text{\hspace{0.17em}}k.\text{\hspace{0.17em}}$Would the domain of$\text{\hspace{0.17em}}y=A\mathrm{csc}(Bx-C)+D\text{\hspace{0.17em}}\text{be}\text{\hspace{0.17em}}x\ne \frac{C+k\pi}{B}?$
Yes. The excluded points of the domain follow the vertical asymptotes. Their locations show the horizontal shift and compression or expansion implied by the transformation to the original function’s input.
Given a function of the form
$\text{\hspace{0.17em}}y=A\mathrm{csc}\left(Bx\right),\text{\hspace{0.17em}}$ graph one period.
Express the function given in the form
$\text{\hspace{0.17em}}y=A\mathrm{csc}\left(Bx\right).$
$\text{\hspace{0.17em}}\left|A\right|.$
Identify
$\text{\hspace{0.17em}}B\text{\hspace{0.17em}}$ and determine the period,
$\text{\hspace{0.17em}}P=\frac{2\pi}{\left|B\right|}.$
Draw the graph of
$\text{\hspace{0.17em}}y=A\mathrm{sin}\left(Bx\right).$
Use the reciprocal relationship between
$\text{\hspace{0.17em}}y=\mathrm{sin}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}y=\mathrm{csc}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ to draw the graph of
$\text{\hspace{0.17em}}y=A\mathrm{csc}\left(Bx\right).$
Sketch the asymptotes.
Plot any two reference points and draw the graph through these points.
Graphing a variation of the cosecant function
Graph one period of
$\text{\hspace{0.17em}}f(x)=\mathrm{-3}\mathrm{csc}(4x).$
Step 1. The given function is already written in the general form,
$\text{\hspace{0.17em}}y=A\mathrm{csc}\left(Bx\right).$
Step 2.$\text{\hspace{0.17em}}\left|A\right|=\left|-3\right|=3,$ so the stretching factor is 3.
Step 3.$\text{\hspace{0.17em}}B=4,$ so
$\text{\hspace{0.17em}}P=\frac{2\pi}{4}=\frac{\pi}{2}.\text{\hspace{0.17em}}$ The period is
$\text{\hspace{0.17em}}\frac{\pi}{2}\text{\hspace{0.17em}}$ units.
Step 4. Sketch the graph of the function
$\text{\hspace{0.17em}}g(x)=\mathrm{-3}\mathrm{sin}(4x).$
Step 5. Use the reciprocal relationship of the sine and cosecant functions to draw the
cosecant function .
Steps 6–7. Sketch three asymptotes at
$\text{\hspace{0.17em}}x=0,\text{\hspace{0.17em}}x=\frac{\pi}{4},\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}x=\frac{\pi}{2}.\text{\hspace{0.17em}}$ We can use two reference points, the local maximum at
$\text{\hspace{0.17em}}\left(\frac{\pi}{8},\mathrm{-3}\right)\text{\hspace{0.17em}}$ and the local minimum at
$\text{\hspace{0.17em}}\left(\frac{3\pi}{8},3\right).$[link] shows the graph.
Graph one period of
$\text{\hspace{0.17em}}f(x)=0.5\mathrm{csc}(2x).$
Given a function of the form
$\text{\hspace{0.17em}}f\left(x\right)=A\mathrm{csc}\left(Bx-C\right)+D,\text{\hspace{0.17em}}$ graph one period.
Express the function given in the form
$\text{\hspace{0.17em}}y=A\mathrm{csc}(Bx-C)+D.$
Identify the stretching/compressing factor,
$\text{\hspace{0.17em}}\left|A\right|.$
Identify
$\text{\hspace{0.17em}}B\text{\hspace{0.17em}}$ and determine the period,
$\text{\hspace{0.17em}}\frac{2\pi}{\left|B\right|}.$
Identify
$\text{\hspace{0.17em}}C\text{\hspace{0.17em}}$ and determine the phase shift,
$\text{\hspace{0.17em}}\frac{C}{B}.$
Draw the graph of
$\text{\hspace{0.17em}}y=A\mathrm{csc}(Bx)\text{\hspace{0.17em}}$ but shift it to the right by and up by
$\text{\hspace{0.17em}}D.$
Sketch the vertical asymptotes, which 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.
Graphing a vertically stretched, horizontally compressed, and vertically shifted cosecant
Sketch a graph of
$\text{\hspace{0.17em}}y=2\mathrm{csc}\left(\frac{\pi}{2}x\right)+1.\text{\hspace{0.17em}}$ What are the domain and range of this function?
Step 1. Express the function given in the form
$\text{\hspace{0.17em}}y=2\mathrm{csc}\left(\frac{\pi}{2}x\right)+1.$
Step 2. Identify the stretching/compressing factor,
$\text{\hspace{0.17em}}\left|A\right|=2.$
Step 3. The period is
$\text{\hspace{0.17em}}\frac{2\pi}{\left|B\right|}=\frac{2\pi}{\frac{\pi}{2}}=\frac{2\pi}{1}\cdot \frac{2}{\pi}=4.$
Step 4. The phase shift is
$\text{\hspace{0.17em}}\frac{0}{\frac{\pi}{2}}=0.$
Step 5. Draw the graph of
$\text{\hspace{0.17em}}y=A\mathrm{csc}(Bx)\text{\hspace{0.17em}}$ but shift it up
$\text{\hspace{0.17em}}D=1.$
Step 6. Sketch the vertical asymptotes, which occur at
$\text{\hspace{0.17em}}x=0,x=2,x=4.$
Given the graph of
$\text{\hspace{0.17em}}f(x)=2\mathrm{cos}\left(\frac{\pi}{2}x\right)+1\text{\hspace{0.17em}}$ shown in
[link] , sketch the graph of
$\text{\hspace{0.17em}}g(x)=2\mathrm{sec}\left(\frac{\pi}{2}x\right)+1\text{\hspace{0.17em}}$ on the same axes.
Analyzing the graph of
y = cot
x
The last trigonometric function we need to explore is
cotangent . The cotangent is defined by the
reciprocal identity$\text{\hspace{0.17em}}\mathrm{cot}\text{\hspace{0.17em}}x=\frac{1}{\mathrm{tan}\text{\hspace{0.17em}}x}.\text{\hspace{0.17em}}$ Notice that the function is undefined when the tangent function is 0, leading to a vertical asymptote in the graph at
$\text{\hspace{0.17em}}0,\pi ,\text{\hspace{0.17em}}$ etc. Since the output of the tangent function is all real numbers, the output of the
cotangent function is also all real numbers.
We can graph
$\text{\hspace{0.17em}}y=\mathrm{cot}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ by observing the graph of the tangent function because these two functions are reciprocals of one another. See
[link] . Where the graph of the tangent function decreases, the graph of the cotangent function increases. Where the graph of the tangent function increases, the graph of the cotangent function decreases.
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
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?