$\"\"$

The rational function is $\"\"$ .

Factorize the numerator and denominator.

$\"\"$

$\"\"$ .

The domain of the function is all possible values of $\"\"$ .

The denominator of the function should not be zero.

$\"\"$

$\"\"$ and $\"\"$ .

The domain of the $\"\"$ is the set of all real numbers $\"\"$ except $\"\"$ and $\"\"$ .

The domain of function $\"\"$ is $\"\"$ .

$\"\"$ .

There are no common factors between the numerator and denominator,

$\"\"$ is in lowest terms.

The rational function is $\"\"$ .

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

$\"\"$ .

Find the intercepts.

Find the $\"\"$ intercepts by equating $\"\"$ .

$\"\"$ .

The polynomial equation has imaginary roots.

There is no $\"\"$ -intercepts.

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

$\"\"$

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

Find the vertical asymptote by equating denominator to zero.

$\"\"$

The function has vertical asymptotes at $\"\"$ and $\"\"$ .

Find the horizontal or oblique asymptote.

$\"\"$

Degree of the numerator is $\"\"$ and degree of the denominator is $\"\"$ .

Since the degree of the numerator is greater degree of the denominator,

then $\"\"$ is improper.

Since the degree of numerator is $\"\"$ or more greater than the degree of denominator

the function has no horizontal or oblique asymptote.

There are no zeros of the numerator: the zeros of the denominator is $\"\"$ and $\"\"$ .

Use these values to divide the $\"\"$ -axis into three intervals:

$\"\"$ , $\"\"$ and $\"\"$ .

$\"\"$

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

Interval	\ $\"\"$ \	\ $\"\"$ \	Location of graph	\ $\"\"$ \
\ $\"\"$ \	\ $\"\"$ \	\ $\"\"$ \	\ Above $\"\"$ -axis \	\ $\"\"$ \
\ $\"\"$ \	\ $\"\"$ \	\ \ $\"\"$ \	\ Below $\"\"$ -axis \	\ $\"\"$ \
\ $\"\"$ \	\ $\"\"$ \	\ $\"\"$ \	\ Above $\"\"$ -axis \	\ $\"\"$ \

$\"\"$

End behavior of the graph:

$\"\"$ and $\"\"$ .

The rational function $\"\"$ does not have horizontal asymptote.

$\"\"$ does not intersect the vertical asymptotes $\"\"$ and $\"\"$ .

$\"\"$

Graph :

The graph of $\"\"$ :

$\"\"$

Step 1: $\"\"$ ; Domain of $\"\"$ is $\"\"$ .

Step 2: $\"\"$ is in lowest terms.

Step 3: $\"\"$ intercept: $\"\"$ ; no $\"\"$ -intercept.

Step 4: $\"\"$ is in lowest terms; vertical asymptotes: $\"\"$ and $\"\"$ .

Step 5: No horizontal or oblique asymptote.

Step 6:

$\"\"$

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

Interval	\ \ $\"\"$ \	\ $\"\"$ \	Location of graph	\ \ $\"\"$ \
\ $\"\"$ \	\ $\"\"$ \	\ $\"\"$ \	\ Below $\"\"$ -axis \	\ \ $\"\"$ \
\ $\"\"$ \	\ \ $\"\"$ \	\ \ $\"\"$ \	\ Below $\"\"$ -axis \	\ \ $\"\"$ \
\ $\"\"$ \	\ \ $\"\"$ \	\ $\"\"$ \	\ Above $\"\"$ -axis \	\ $\"\"$ \

Step 7 and step 8:

Graph of $\"\"$ .

$\"\"$