TABLE OF CONTENTS

4.3  Different Forms of Rational Functions

Goals:   Students will be able to:

• Combine terms and express a function in rational form as a ratio of polynomials

• Name the horizontal asymptote of the graph of a rational function given as the ratio of 2 polynomials with like degree

Terms to know:

·         asymptote

·         leading coefficient

·         domain of a function

·         least common denominator

·         function

·         rational function

Prep assignment: Combining Fractions

Prep assignment: Finding the LCD

Changing the Form of a Rational Function 

In this section we review the technique of combining terms of a rational function.

Example 1.  Combine terms in the function g and express it as a ratio of polynomials:   

Rewrite the constant term with a

denominator of 1, then find the common

denominator. 

The common denominator is  

Multiply the constant term by 1, that is

multiply the constant term by

Expand using the distributive property,

write using single denominator, and combine

like terms.

Example 2.  Combine terms in the function h  and express it as a ratio of polynomials:    

Rewrite the constant term with a

denominator of 1, then find the common

denominator. 

The common denominator is  

Multiply the constant term by 1, that is,

multiply the constant term by the

Write using a single denominator

Example 3.  Combine terms in the function f  and express it as a ratio of polynomials:

Rewrite the constant term with a

denominator of 1, then find the common

denominator. 

The common denominator is  

Multiply the constant term by 1, that is, 

multiply the constant term by the

Expand the binomial, write using a single

denominator, and combine like terms.

Factoring out the negative is optional.

Example 4.  Combine terms in the function h  and express it as a ratio of polynomials:

Rewrite the constant term with a

denominator of 1, then find the common

denominator. 

The common denominator is  

Multiply the constant term by 1, that is, 

multiply the constant term by the

Expand the binomial and write using a single

denominator

Distribute the -4

Combine like terms.

Factoring out the negative is optional.

Checkpoint 4.3A

Horizontal Asymptotes Revisited

Consider the horizontal asymptotes of the functions in the examples above.

Now consider the combined form of these same functions:

Take a close look at the combined form and the horizontal asymptotes.  Do you see a pattern?  Would you be able to determine the horizontal asymptote if given only the combined form of a rational function?

Try it!  Find the horizontal asymptote of the following.  Then check on your graphing calculator.

The horizontal asymptote of these functions is the leading coefficient of the numerator divided by the leading coefficient of the denominator.  Looking back at the steps for combining two terms of a rational function will help us understand why this is the case.  Consider the procedure in Example 1:

Multiply the constant term by 1, that is multiply

the constant term by the

We multiplied the least common denominator  which happens to be the denominator of the first term  by the constant term, which happens to be the value of the asymptote.  The resulting numerator has the SAME DEGREE as the denominator and the SAME LEADING COEFFICIENT as the asymptote.  

The horizontal asymptote of the graph of a rational function in combined form WITH POLYNOMIALS OF EQUAL DEGREE IN THE NUMERATOR AND DENOMINATORis the leading coefficient of the numerator divided by the leading coefficient of the denominator.

If the degree of the polynomials are not the same, this rule does not apply.        

Example 5.  Name the horizontal asymptote of the graph of the function r  given by

.  Then find the x-intercept(s) and the y-intercept.

Since the degree of the numerator =

the degree of the denominator, the

horizontal asymptote is the leading 

coefficient over the leading coefficient.

The leading coefficient of the numerator is 8 and the leading coefficient of the denominator is 4, so the horizontal asymptote is  or  

To find the y-intercept, set  

The point  is on the graph

To find the x-intercept, set  

We could cross multiply or just set

the numerator  and solve.

 So the point  is on the graph

Example 6.  Name the horizontal asymptote of the graph of the function r  given by

.  Then find the x-intercept(s) and the y-intercept.

The horizontal asymptote is the leading 

coefficient of the numerator over the

leading coefficient of the denominator

The leading coefficient of the numerator is 2 and the leading coefficient of the denominator is 1, so the horizontal asymptote is  or  

To find the y-intercept, set  

The point  is on the graph

To find the x-intercept(s), set  

We could cross multiply or just set the

numerator  and solve.

So the points  and  are on the

Graph.

Checkpoint 4.3B

Separating a rational function into two terms

Now suppose we needed to go in the reverse direction.  That is, suppose we start with rational function in combined form and want to write it as a function with two terms.

Example 7.   Separate the function h given by  into two terms such that the second term is a constant.

First we need to identify the horizontal

asymptote.

The horizontal asymptote is  

The separated form of h will look like

this: , since the  

denominator will be the same and the

 horizontal asymptote indicates  a  

vertical shift up 4.

Set the two forms equal to each other.

Solve for the “Something.”  Abbreviate

“Something” with an S.

Multiply both sides by the LCD.

Solve for S

Write the separated form.

More worked examples

Homework problems

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