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4.3.   More Worked Examples:  Different Forms of Rational Functions

 

 

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 least 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 r  and express it as a ratio of polynomials:

                     

Rewrite the constant term with a

denominator of 1, then find the least common denominator. 

 

 

The least 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 6

 

Combine like terms.

 

 

 

 

 

 

 

Example 4.  Name the horizontal asymptote of the graph of the function s  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 3 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.

 

 

 

 

 

Example 5.  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.

 

 

 

 

 

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