$\"\"$

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 $\"\"$ .

Find the intercepts.

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

$\"\"$

$\"\"$ .

$\"\"$ -intercepts: $\"\"$ .

Determine the behavior of the graph of $\"\"$ near each $\"\"$ -intercept.

Near $\"\"$ : $\"\"$ .

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

Near $\"\"$ : $\"\"$ .

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

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

$\"\"$

$\"\"$ .

$\"\"$ -intercept: $\"\"$ .

Find the vertical asymptote by equating denominator to zero.

$\"\"$

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

Find the horizontal or oblique asymptote.

$\"\"$ .

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

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

then $\"\"$ is improper.

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

the quotient obtained is polynomial of degree $\"\"$ or higher and $\"\"$ has neither

a horizontal or oblique asymptote.

$\"\"$ has no horizontal or oblique asymptote.

$\"\"$

The zeros of the numerator are $\"\"$ ; the zeros of the denominator are $\"\"$ .

Use these values to divide the $\"\"$ -axis into five intervals.

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

$\"\"$

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

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

$\"\"$

Graph :

The graph the $\"\"$ :

$\"\"$

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

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

Step 3: $\"\"$ -intercept: $\"\"$ ; $\"\"$ -intercepts: $\"\"$ .

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

Step 5: No horizontal or oblique asymptote.

Step 6:

$\"\"$

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

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

Step 7 and step 8:

The graph rational function $\"\"$ .

$\"\"$