The graph of a transformed tangent function is different from the basic tangent function
$\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ in several ways:
Features of the graph of
y =
A Tan(
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
$\text{\hspace{0.17em}}x=\frac{C}{B}+\frac{\pi}{2\left|B\right|}k,$ where
$\text{\hspace{0.17em}}k\text{\hspace{0.17em}}$ is an odd integer.
There is no amplitude.
$y=A\text{\hspace{0.17em}}\mathrm{tan}(Bx)\text{\hspace{0.17em}}$ is and odd function because it is the qoutient of odd and even functions(sin and cosine perspectively).
Given the function
$\text{\hspace{0.17em}}y=A\mathrm{tan}(Bx-C)+D,\text{\hspace{0.17em}}$ sketch the graph of one period.
Express the function given in the form
$\text{\hspace{0.17em}}y=A\mathrm{tan}\left(Bx-C\right)+D.$
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,
$\text{\hspace{0.17em}}P=\frac{\pi}{\left|B\right|}.$
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
$\text{\hspace{0.17em}}y=A\mathrm{tan}(Bx)\text{\hspace{0.17em}}$ 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 vertical asymptotes, which occur at
$\text{}x=\frac{C}{B}+\frac{\pi}{2\left|B\right|}k,$ where
$\text{}k\text{}$ is an odd integer.
Plot any three reference points and draw the graph through these points.
Graphing one period of a shifted tangent function
Graph one period of the function
$\text{\hspace{0.17em}}y=\mathrm{-2}\mathrm{tan}(\pi x+\pi )\text{\hspace{0.17em}}\mathrm{-1.}$
Step 1. The function is already written in the form
$\text{\hspace{0.17em}}y=A\mathrm{tan}\left(Bx-C\right)+D.$
Step 2.$\text{\hspace{0.17em}}A=\mathrm{-2},\text{\hspace{0.17em}}$ so the stretching factor is
$\text{\hspace{0.17em}}\left|A\right|=2.$
Step 3.$\text{\hspace{0.17em}}B=\pi ,\text{\hspace{0.17em}}$ so the period is
$\text{\hspace{0.17em}}P=\frac{\pi}{\left|B\right|}=\frac{\pi}{\pi}=1.$
Step 4.$\text{\hspace{0.17em}}C=-\pi ,\text{\hspace{0.17em}}$ so the phase shift is
$\text{\hspace{0.17em}}\frac{C}{B}=\frac{-\pi}{\pi}=\mathrm{-1.}$
Step 5-7. The asymptotes are at
$\text{\hspace{0.17em}}x=-\frac{3}{2}\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}x=-\frac{1}{2}\text{\hspace{0.17em}}$ and the three recommended reference points are
$\text{\hspace{0.17em}}\left(\mathrm{-1.25},1\right),\text{\hspace{0.17em}}$$\left(\mathrm{-1,}\mathrm{-1}\right),\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}\left(\mathrm{-0.75,}\mathrm{-3}\right).\text{\hspace{0.17em}}$ The graph is shown in
[link] .
How would the graph in
[link] look different if we made
$\text{\hspace{0.17em}}A=2\text{\hspace{0.17em}}$ instead of
$\text{\hspace{0.17em}}\mathrm{-2}?$
It would be reflected across the line
$\text{\hspace{0.17em}}y=-1,\text{\hspace{0.17em}}$ becoming an increasing function.
Given the graph of a tangent function, identify horizontal and vertical stretches.
Find the period
$\text{\hspace{0.17em}}P\text{\hspace{0.17em}}$ from the spacing between successive vertical asymptotes or
x -intercepts.
Determine a convenient point
$\text{\hspace{0.17em}}(x,f(x))\text{\hspace{0.17em}}$ on the given graph and use it to determine
$\text{\hspace{0.17em}}A.$
Identifying the graph of a stretched tangent
Find a formula for the function graphed in
[link] .
The graph has the shape of a tangent function.
Step 1. One cycle extends from –4 to 4, so the period is
$\text{\hspace{0.17em}}P=8.\text{\hspace{0.17em}}$ Since
$\text{\hspace{0.17em}}P=\frac{\pi}{\left|B\right|},\text{\hspace{0.17em}}$ we have
$\text{\hspace{0.17em}}B=\frac{\pi}{P}=\frac{\pi}{8}.$
Step 2. The equation must have the form
$f(x)=A\mathrm{tan}\left(\frac{\pi}{8}x\right).$
Step 3. To find the vertical stretch
$\text{\hspace{0.17em}}A,$ we can use the point
$\text{\hspace{0.17em}}\left(2,2\right).$
The
secant was defined by the
reciprocal identity$\text{\hspace{0.17em}}\mathrm{sec}\text{\hspace{0.17em}}x=\frac{1}{\mathrm{cos}\text{\hspace{0.17em}}x}.\text{\hspace{0.17em}}$ Notice that the function is undefined when the cosine is 0, leading to vertical asymptotes at
$\text{\hspace{0.17em}}\frac{\pi}{2},\text{\hspace{0.17em}}$$\frac{3\pi}{2},\text{\hspace{0.17em}}$ etc. Because the cosine is never more than 1 in absolute value, the secant, being the reciprocal, will never be less than 1 in absolute value.
We can graph
$\text{\hspace{0.17em}}y=\mathrm{sec}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ by observing the graph of the cosine function because these two functions are reciprocals of one another. See
[link] . The graph of the cosine is shown as a dashed orange wave so we can see the relationship. Where the graph of the cosine function decreases, the graph of the
secant function increases. Where the graph of the cosine function increases, the graph of the secant function decreases. When the cosine function is zero, the secant is undefined.
Questions & Answers
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?