Factor the numerator and denominator of . Find the domain of the rational function :

The rational function is .

The domain of a rational function is the set of all real numbers except those for which the denominator is .

To find which number make the fraction undefined create an equation where the denominator is not equal to .

The domain of is the set of all real numbers of except .

The domain of function is .

Write in lowest terms :

The function is .

The function is in lowest terms.

Locate the intercepts of the graph and determine the behavior of the graph of near each - intercept :

The rational function is  .

Change to .

Find the intercepts :

To find -intercept equate the numerator to zero. 

The -intercept is .

Find the  -intercept by substituting in the rational function.

There is no -intercepts.

Determine the vertical asymptotes :  

Find the vertical asymptote by equating denominator to zero.   

So the function has vertical asymptote at .

Determine the horizantal asymptotes / oblique asymptotes :

The function is .

To find horizontal asymptote, first find the degree of the numerator and degree of the denominator.

Degree of the numerator and degree of the denominator .

Since the degree of the numerator is greater  than degree of denominator, there is no horizontal asymptote.

Find oblique asymptote :

Oblique asymptote is found by long division.

The function is .

Since the quotient is a squared term, there is no oblique asymptote.

No oblique asymptote.

Use the zeros of the numerator and denominator of to divide the -axis into intervals :

The numerator has one real zero at and the denominator has one real zero at .

Use these values to divide the -axis into three intervals. 

, and .

The solid circle represent the real zeros of the numerator and the denominator.

Interval	\ \		\ \
\ Number chosen \	\ \		\ \
\ Value of \	\ \	\ \	\ \ \
\ Location of graph \	\ \ Above -axis \ \	Below -axis	\ Above -axis   \
\ Point of graph \	\ \	\ \	\ \

End behavior of the graph :

Since the graph of is below the -axis for and is above for   and and since the graph of does not intersect  .

Since the graph of is below the -axis for , the graph of will approach the vertical asymptote, at the top to the left of ().

Since the graph of is above the -axis for   and , the graph of will approach the vertical asymptote, at the bottom to the right of ().

Use the results obtained in Steps 1 through 7 to graph the function :

The graph of :

.

1.

; Domain : .

2.

The function is in lowest terms.

3.

The -intercept is .

No -intercept.

4.

The function is in lowest terms; vertical asymptote : .

5.

No horizontal or oblique asymptote.

6.

Interval	\ \		\ \
\ Number chosen \	\ \		\ \
\ Value of \	\ \	\ \	\ \ \
\ Location of graph \	\ \ Above -axis \ \	Below -axis	\ Above -axis   \
\ Point of graph \	\ \	\ \	\ \

7/8.

The graph of  the rational function :

.