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A graph of y=tangent of x. Asymptotes at -pi over 2 and pi over 2.
Graph of the tangent function

Graphing variations of y = tan x

As with the sine and cosine functions, the tangent    function can be described by a general equation.

y = A tan ( B x )

We can identify horizontal and vertical stretches and compressions using values of A and B . The horizontal stretch can typically be determined from the period of the graph. With tangent graphs, it is often necessary to determine a vertical stretch using a point on the graph.

Because there are no maximum or minimum values of a tangent function, the term amplitude cannot be interpreted as it is for the sine and cosine functions. Instead, we will use the phrase stretching/compressing factor when referring to the constant A .

Features of the graph of y = A Tan( Bx )

  • The stretching factor is | A | .
  • The period is P = π | B | .
  • The domain is all real numbers x , where x π 2 | B | + π | B | k such that k is an integer.
  • The range is ( −∞ , ) .
  • The asymptotes occur at x = π 2 | B | + π | B | k , where k is an integer.
  • y = A tan ( B x ) is an odd function.

Graphing one period of a stretched or compressed tangent function

We can use what we know about the properties of the tangent function to quickly sketch a graph of any stretched and/or compressed tangent function of the form f ( x ) = A tan ( B x ) . We focus on a single period    of the function including the origin, because the periodic property enables us to extend the graph to the rest of the function’s domain if we wish. Our limited domain is then the interval ( P 2 , P 2 ) and the graph has vertical asymptotes at ± P 2 where P = π B . On ( π 2 , π 2 ) , the graph will come up from the left asymptote at x = π 2 , cross through the origin, and continue to increase as it approaches the right asymptote at x = π 2 . To make the function approach the asymptotes at the correct rate, we also need to set the vertical scale by actually evaluating the function for at least one point that the graph will pass through. For example, we can use

f ( P 4 ) = A tan ( B P 4 ) = A tan ( B π 4 B ) = A

because tan ( π 4 ) = 1.

Given the function f ( x ) = A tan ( B x ) , graph one period.

  1. Identify the stretching factor, | A | .
  2. Identify B and determine the period, P = π | B | .
  3. Draw vertical asymptotes at x = P 2 and x = P 2 .
  4. For A > 0 , the graph approaches the left asymptote at negative output values and the right asymptote at positive output values (reverse for A < 0 ).
  5. Plot reference points at ( P 4 , A ) , ( 0 , 0 ) , and ( P 4 ,− A ) , and draw the graph through these points.

Sketching a compressed tangent

Sketch a graph of one period of the function y = 0.5 tan ( π 2 x ) .

First, we identify A and B .

An illustration of equations showing that A is the coefficient of tangent and B is the coefficient of x, which is within the tangent function.

Because A = 0.5 and B = π 2 , we can find the stretching/compressing factor and period. The period is π π 2 = 2 , so the asymptotes are at x = ± 1. At a quarter period from the origin, we have

f ( 0.5 ) = 0.5 tan ( 0.5 π 2 ) = 0.5 tan ( π 4 ) = 0.5

This means the curve must pass through the points ( 0.5 , 0.5 ) , ( 0 , 0 ) , and ( 0.5 , −0.5 ) . The only inflection point is at the origin. [link] shows the graph of one period of the function.

A graph of one period of a modified tangent function, with asymptotes at x=-1 and x=1.

Sketch a graph of f ( x ) = 3 tan ( π 6 x ) .

A graph of two periods of a modified tangent function, with asymptotes at x=-3 and x=3.

Graphing one period of a shifted tangent function

Now that we can graph a tangent function that is stretched or compressed, we will add a vertical and/or horizontal (or phase) shift. In this case, we add C and D to the general form of the tangent function.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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.
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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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