The rational function is .

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

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 :

Find the -intercept by equating the numerator to zero.

There is no real solutions.

There is no -intercepts.

Find the  -intercept by substituting in .

There is no -intercept.

Determine the vertical asymptotes :

Find the vertical asymptote by equating denominator to zero.

Vertical asymptote is at .

The function not defined at .

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Since the function is in lowest terms, there is no hole.

Determine the horizontal 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 .

Quotient is oblique asymptote.

Oblique asymptote is .

The graph does not intersect the line .

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

The numerator has no real zeros and the denominator has one real zero at .

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

and .

End behavior of the graph :

Since the graph of is below the -axis for and is above for and since the graph of does not intersect the oblique asymptote, , the graph of will approach the line .

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 , the graph of will approach the vertical asymptote, at the top to the right of ().

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

The graph of :

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1:

; Domain : .

2: The function is in lowest terms.

3:  No intercepts.

4: The function is in lowest terms.

Vertical asymptote : .

5:

Oblique asymptote : , not intersected.

6:

7 and  8:

The graph of  the rational function :

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