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The domain of csc x was given to be all x such that x k π for any integer k . Would the domain of y = A csc ( B x C ) + D be x C + k π 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 y = A csc ( B x ) , graph one period.

  1. Express the function given in the form y = A csc ( B x ) .
  2. | A | .
  3. Identify B and determine the period, P = 2 π | B | .
  4. Draw the graph of y = A sin ( B x ) .
  5. Use the reciprocal relationship between y = sin x and y = csc x to draw the graph of y = A csc ( B x ) .
  6. Sketch the asymptotes.
  7. Plot any two reference points and draw the graph through these points.

Graphing a variation of the cosecant function

Graph one period of f ( x ) = −3 csc ( 4 x ) .

  • Step 1. The given function is already written in the general form, y = A csc ( B x ) .
  • Step 2. | A | = | 3 | = 3 , so the stretching factor is 3.
  • Step 3. B = 4 , so P = 2 π 4 = π 2 . The period is π 2 units.
  • Step 4. Sketch the graph of the function g ( x ) = −3 sin ( 4 x ) .
  • Step 5. Use the reciprocal relationship of the sine and cosecant functions to draw the cosecant function .
  • Steps 6–7. Sketch three asymptotes at x = 0 , x = π 4 , and x = π 2 . We can use two reference points, the local maximum at ( π 8 , −3 ) and the local minimum at ( 3 π 8 , 3 ) . [link] shows the graph.
    A graph of one period of a cosecant function. There are vertical asymptotes at x=0, x=pi/4, and x=pi/2.

Graph one period of f ( x ) = 0.5 csc ( 2 x ) .

A graph of one period of a modified secant function, which looks like an downward facing prarbola and a upward facing parabola.

Given a function of the form f ( x ) = A csc ( B x C ) + D , graph one period.

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

Graphing a vertically stretched, horizontally compressed, and vertically shifted cosecant

Sketch a graph of y = 2 csc ( π 2 x ) + 1. What are the domain and range of this function?

  • Step 1. Express the function given in the form y = 2 csc ( π 2 x ) + 1.
  • Step 2. Identify the stretching/compressing factor, | A | = 2.
  • Step 3. The period is 2 π | B | = 2 π π 2 = 2 π 1 2 π = 4.
  • Step 4. The phase shift is 0 π 2 = 0.
  • Step 5. Draw the graph of y = A csc ( B x ) but shift it up D = 1.
  • Step 6. Sketch the vertical asymptotes, which occur at x = 0 , x = 2 , x = 4.

The graph for this function is shown in [link] .

A graph of 3 periods of a modified cosecant function, with 3 vertical asymptotes, and a dotted sinusoidal function that has local maximums where the cosecant function has local minimums and local minimums where the cosecant function has local maximums.
A transformed cosecant function

Given the graph of f ( x ) = 2 cos ( π 2 x ) + 1 shown in [link] , sketch the graph of g ( x ) = 2 sec ( π 2 x ) + 1 on the same axes.

A graph of two periods of a modified cosine function. Range is [-1,3], graphed from x=-4 to x=4.
A graph of two periods of both a secant and consine function. Grpah shows that cosine function has local maximums where secant function has local minimums and vice versa.

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 cot x = 1 tan x . Notice that the function is undefined when the tangent function is 0, leading to a vertical asymptote in the graph at 0 , π , 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 y = cot x 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.

Questions & Answers

what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
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?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
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Adin Reply
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Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
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Adin
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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.
Bharti
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Damian Reply
absolutely yes
Daniel
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Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
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
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
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
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
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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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