# 3.7 Rational functions  (Page 9/16)

 Page 9 / 16

## Writing rational functions

Now that we have analyzed the equations for rational functions and how they relate to a graph of the function, we can use information given by a graph to write the function. A rational function written in factored form will have an x -intercept where each factor of the numerator is equal to zero. (An exception occurs in the case of a removable discontinuity.) As a result, we can form a numerator of a function whose graph will pass through a set of x -intercepts by introducing a corresponding set of factors. Likewise, because the function will have a vertical asymptote where each factor of the denominator is equal to zero, we can form a denominator that will produce the vertical asymptotes by introducing a corresponding set of factors.

## Writing rational functions from intercepts and asymptotes

If a rational function    has x -intercepts at vertical asymptotes at $\text{\hspace{0.17em}}x={v}_{1},{v}_{2},\dots ,{v}_{m},\text{\hspace{0.17em}}$ and no then the function can be written in the form:

$f\left(x\right)=a\frac{{\left(x-{x}_{1}\right)}^{{p}_{1}}{\left(x-{x}_{2}\right)}^{{p}_{2}}\cdots {\left(x-{x}_{n}\right)}^{{p}_{n}}}{{\left(x-{v}_{1}\right)}^{{q}_{1}}{\left(x-{v}_{2}\right)}^{{q}_{2}}\cdots {\left(x-{v}_{m}\right)}^{{q}_{n}}}$

where the powers $\text{\hspace{0.17em}}{p}_{i}\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}{q}_{i}\text{\hspace{0.17em}}$ on each factor can be determined by the behavior of the graph at the corresponding intercept or asymptote, and the stretch factor $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ can be determined given a value of the function other than the x -intercept or by the horizontal asymptote if it is nonzero.

Given a graph of a rational function, write the function.

1. Determine the factors of the numerator. Examine the behavior of the graph at the x -intercepts to determine the zeroes and their multiplicities. (This is easy to do when finding the “simplest” function with small multiplicities—such as 1 or 3—but may be difficult for larger multiplicities—such as 5 or 7, for example.)
2. Determine the factors of the denominator. Examine the behavior on both sides of each vertical asymptote to determine the factors and their powers.
3. Use any clear point on the graph to find the stretch factor.

## Writing a rational function from intercepts and asymptotes

Write an equation for the rational function shown in [link] .

The graph appears to have x -intercepts at $\text{\hspace{0.17em}}x=–2\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x=3.\text{\hspace{0.17em}}$ At both, the graph passes through the intercept, suggesting linear factors. The graph has two vertical asymptotes. The one at $\text{\hspace{0.17em}}x=–1\text{\hspace{0.17em}}$ seems to exhibit the basic behavior similar to $\text{\hspace{0.17em}}\frac{1}{x},\text{\hspace{0.17em}}$ with the graph heading toward positive infinity on one side and heading toward negative infinity on the other. The asymptote at $\text{\hspace{0.17em}}x=2\text{\hspace{0.17em}}$ is exhibiting a behavior similar to $\text{\hspace{0.17em}}\frac{1}{{x}^{2}},\text{\hspace{0.17em}}$ with the graph heading toward negative infinity on both sides of the asymptote. See [link] .

We can use this information to write a function of the form

$f\left(x\right)=a\frac{\left(x+2\right)\left(x-3\right)}{\left(x+1\right){\left(x-2\right)}^{2}}.$

To find the stretch factor, we can use another clear point on the graph, such as the y -intercept $\text{\hspace{0.17em}}\left(0,–2\right).$

This gives us a final function of $\text{\hspace{0.17em}}f\left(x\right)=\frac{4\left(x+2\right)\left(x-3\right)}{3\left(x+1\right){\left(x-2\right)}^{2}}.$

Access these online resources for additional instruction and practice with rational functions.

#### Questions & Answers

difference between calculus and pre calculus?
Asma Reply
give me an example of a problem so that I can practice answering
Jenefa Reply
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
CJ Reply
I want to learn about the law of exponent
Quera Reply
explain this
Hinderson Reply
what is functions?
Angel Reply
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
Feemark Reply
can you not take the square root of a negative number
Sharon Reply
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
Elaine Reply
can I get some pretty basic questions
Ama Reply
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
Amara Reply
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
Mars Reply
what is the domain of f(x)=x-4/x^2-2x-15 then
Conney Reply
x is different from -5&3
Seid
All real x except 5 and - 3
Spiro
***youtu.be/ESxOXfh2Poc
Loree
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
jeric Reply
Don't think that you can.
Elliott
By using some imaginary no.
Tanmay
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
jeric Reply
What are the question marks for?
Elliott

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