Math 130

3.5 – Rational Functions: Graphs, Applications, and Models

A rational function is of the form

( ) ( )

( )

where ( ) and ( ) are polynomials, and ( ) .

ex: Here are some examples of rational functions:

( )

( )

( )

Reciprocal Function ( ( )

)

Asymptotes are lines that a graph gets closer and closer to.

What is the vertical asymptote of

?

What is the horizontal asymptote of

?

Ex 1.

Graph ( )

. Give the domain and range, and any vertical and/or horizontal

asymptotes.

Graph of

What is the vertical asymptote of

?

What is the horizontal asymptote of

?

Ex 2.

Graph ( )

( ) . Give the domain and range, and any vertical and/or horizontal

asymptotes.

Asymptotes

A rational function is in lowest terms if there are no more factors to cancel.

ex:

( )( ) is not in lowest terms since you could factor the top and cancel an

For a rational function ( ) ( )

( ) in lowest terms, here’s how to find its asymptotes:

1. Vertical asymptotes where ( )

2. If degree of top smaller than degree of bottom, then horizontal asymptote

3. If degree of top equal to degree of bottom, then divide leading coefficients to get

horizontal asymptote

4. If degree of top larger than degree of bottom by exactly 1, then divide top by bottom and

ignore the remainder to get equation for oblique (slanted) asymptote

Ex 3.

Find all asymptotes of ( )

Ex 4.

Find all asymptotes of ( )

Ex 5.

Find all asymptotes of ( )

Graphing Rational Functions

To graph a rational function:

1. Find vertical asymptote(s)

2. Find horizontal or oblique asymptote(s)

3. Find -intercept and -intercept(s)

4. Find places where graph intersects horizontal or oblique asymptote(s)

5. If necessary, plot more points, and then draw the graph

Ex 6.

Graph ( )

.

Ex 7.

Graph ( )

.

Ex 8.

Graph ( )

.

Ex 9.

Graph ( )

.

Ex 10.

Graph ( )

.

Q: The more you take, the more you leave

behind. What are they?