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f ( x ) = A tan ( B x C ) + D

The graph of a transformed tangent function is different from the basic tangent function tan x in several ways:

Features of the graph of y = A Tan( Bx C )+ D

  • The stretching factor is | A | .
  • The period is π | B | .
  • The domain is x C B + π | B | k , where k is an integer.
  • The range is ( −∞ , | A | ] [ | A | , ) .
  • The vertical asymptotes occur at x = C B + π 2 | B | k , where k is an odd integer.
  • There is no amplitude.
  • y = A tan ( B x ) is and odd function because it is the qoutient of odd and even functions(sin and cosine perspectively).

Given the function y = A tan ( B x C ) + D , sketch the graph of one period.

  1. Express the function given in the form y = A tan ( B x C ) + D .
  2. Identify the stretching/compressing factor , | A | .
  3. Identify B and determine the period, P = π | B | .
  4. Identify C and determine the phase shift, C B .
  5. Draw the graph of y = A tan ( B x ) shifted to the right by C B and up by D .
  6. Sketch the vertical asymptotes, which occur at   x = C B + π 2 | B | k , where   k   is an odd integer.
  7. 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 y = −2 tan ( π x + π ) −1.

  • Step 1. The function is already written in the form y = A tan ( B x C ) + D .
  • Step 2. A = −2 , so the stretching factor is | A | = 2.
  • Step 3. B = π , so the period is P = π | B | = π π = 1.
  • Step 4. C = π , so the phase shift is C B = π π = −1.
  • Step 5-7. The asymptotes are at x = 3 2 and x = 1 2 and the three recommended reference points are ( −1.25 , 1 ) , ( −1, −1 ) , and ( −0.75, −3 ) . The graph is shown in [link] .
    A graph of one period of a shifted tangent function, with vertical asymptotes at x=-1.5 and x=-0.5.

How would the graph in [link] look different if we made A = 2 instead of −2 ?

It would be reflected across the line y = 1 , becoming an increasing function.

Given the graph of a tangent function, identify horizontal and vertical stretches.

  1. Find the period P from the spacing between successive vertical asymptotes or x -intercepts.
  2. Write f ( x ) = A tan ( π P x ) .
  3. Determine a convenient point ( x , f ( x ) ) on the given graph and use it to determine A .

Identifying the graph of a stretched tangent

Find a formula for the function graphed in [link] .

A graph of two periods of a modified tangent function, with asymptotes at x=-4 and x=4.
A stretched tangent function

The graph has the shape of a tangent function.

  • Step 1. One cycle extends from –4 to 4, so the period is P = 8. Since P = π | B | , we have B = π P = π 8 .
  • Step 2. The equation must have the form f ( x ) = A tan ( π 8 x ) .
  • Step 3. To find the vertical stretch A , we can use the point ( 2 , 2 ) .
    2 = A tan ( π 8 2 ) = A tan ( π 4 )

Because tan ( π 4 ) = 1 , A = 2.

This function would have a formula f ( x ) = 2 tan ( π 8 x ) .

Find a formula for the function in [link] .

A graph of four periods of a modified tangent function, Vertical asymptotes at -3pi/4, -pi/4, pi/4, and 3pi/4.

g ( x ) = 4 tan ( 2 x )

Analyzing the graphs of y = sec x And y = csc x

The secant    was defined by the reciprocal identity sec x = 1 cos x . Notice that the function is undefined when the cosine is 0, leading to vertical asymptotes at π 2 , 3 π 2 , 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 y = sec x 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?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Essential precalculus, part 2. OpenStax CNX. Aug 20, 2015 Download for free at http://legacy.cnx.org/content/col11845/1.2
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