The function is .

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

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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 .

Write in lowest terms:

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

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Change to .

Find the intercepts :

Find the -intercept by equating the numerator to zero.

-intercept is .

Behavior at -intercept:

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Plot the point and draw a line through with a negative slope.

Find the -intercept by substituting in the .

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-intercept is .

Determine the vertical asymptotes :

Find the vertical asymptote by equating denominator to zero.   

Vertical asymptote is at .

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The function not defined at .

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Hole at .

Determine the horizontal asymptotes / oblique asymptotes:

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 equal to the degree of the denominator, horizontal asymptote is the ratio of leading coefficient of numerator and denominator.

Horizontal asymptote is at .

The graph does not intersect the line .

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

Real zeros of the numerator is at  and .

Real zeros of the denominator is at and .

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

, , and .

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

The graph of is above the -axis for .

The graph of does not intersect the graph of horizontal asymptote at . Therefore the graph of will approach from and will approach the vertical asymptote at at the bottom from the left.

The graph of is below the -axis for  , the graph of will approach at the bottom from the left.

Since the graph of is above the -axis for and does not intersect the graph of horizontal asymptote at , the graph of will approach from .

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

The graph of :

1:

The domain of is .

2: .

3: -intercept is and -intercept is .

4: Vertical asymptote .

Hole at .

5: Horizontal asymptote is at , not intersected.

6:

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

 7 and 8:

The graph of :

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