$\"\"$

The function is $\"\"$ .

Factor the numerator and denominator of $\"\"$ . Find the domain of 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 $\"\"$ .

$\"\"$

Write $\"\"$ in lowest terms:

$\"\"$

$\"\"$ .

$\"\"$

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

$\"\"$

Change $\"\"$ to $\"\"$ .

$\"\"$

Find the intercepts :

Find the $\"\"$ -intercept by equating the numerator to zero.

$\"\"$

$\"\"$ -intercept is $\"\"$ .

Behavior at $\"\"$ -intercept:

$\"\"$ .

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

Find the $\"\"$ -intercept by substituting $\"\"$ $\"\"$ in the $\"\"$ .

$\"\"$

$\"\"$ .

$\"\"$ -intercept is $\"\"$ .

Determine the vertical asymptotes :

Find the vertical asymptote by equating denominator to zero.

$\"\"$

Vertical asymptote is at $\"\"$ .

$\"\"$

$\"\"$ .

The function not defined at $\"\"$ .

$\"\"$ .

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 \ \	\ Below $\"\"$ -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.

$\"\"$

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \

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

7 and 8:

The graph of $\"\"$ :

$\"\"$ .