<< Chapter < Page Chapter >> Page >
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

what is math number
Tric Reply
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply
12, 17, 22.... 25th term
Alexandra Reply
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Shirleen Reply
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Adu
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
kinnecy Reply
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
hi
Ayuba
Hello
opoku
hi
Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Essential precalculus, part 2. OpenStax CNX. Aug 20, 2015 Download for free at http://legacy.cnx.org/content/col11845/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Essential precalculus, part 2' conversation and receive update notifications?

Ask