$\"\"$

The rational function is $\"\"$ .

Step 1: Factor the numerator and denominator of $\"\"$ . Find the domain of the rational function.

The rational function $\"\"$ .

$\"\"$

$\"\"$ .

Domain : Domain of the function is the set of values of $\"\"$ which makes the function mathematically correct.

$\"\"$ .

Denominator of the function should not be zero.

$\"\"$

Domain of the function is defined for all values of $\"\"$ except at $\"\"$ .

Domain of the function is $\"\"$ .

$\"\"$

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

The rational function in lowest terms is $\"\"$ .

Step 3: Locate the intercepts of the graph.

Find intercepts :

The $\"\"$ -intercepts are the zeros of the numerator of $\"\"$ that are in the domain of $\"\"$ .

To find $\"\"$ -intercept, substitute $\"\"$ .

$\"\"$

$\"\"$ or $\"\"$

$\"\"$ -intercepts are $\"\"$ , $\"\"$ .

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

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

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

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

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

Find the $\"\"$ -intercept, substitute $\"\"$ .

$\"\"$

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

Step 4: Determine the vertical asymptotes.

The rational function is $\"\"$ .

The Vertical asymptote can be found equating denominator to zero.

$\"\"$

Vertical asymptote $\"\"$ .

Step 5: Determine the horizontal or oblique asymptote.

To find horizontal asymptote, first find the degree of the numerator and the degree of denominator.

Degree of numerator $\"\"$ , 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.

$\"\"$

Quotient is oblique asymptote.

Oblique asymptote is $\"\"$ .

Step 6: Use the zeros of the numerator and denominator of $\"\"$ to divide the $\"\"$ -axis into intervals

The real zero of numerator is $\"\"$ and $\"\"$ and the real zeros of denominator $\"\"$

So the real zeros are divide the $\"\"$ - axis into four intervals.

Choosing a number for $\"\"$ in each interval and evaluating $\"\"$ .

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

Step 7 : Analyze the behavior of the graph of $\"\"$ near each asymptote and indicate this behavior on the graph.

The graph of $\"\"$ is below $\"\"$ -axis for $\"\"$ and above $\"\"$ -axis for $\"\"$ , hence the graph does not intersect the oblique asymptote.

The graph of $\"\"$ is above $\"\"$ -axis in interval $\"\"$ and below $\"\"$ -axis in interval $\"\"$ , hence the graph approaches to vertical asymptote at $\"\"$ .

The graph of $\"\"$ is below $\"\"$ -axis in interval $\"\"$ and is above $\"\"$ -axis in interval $\"\"$ , hence the graph approaches the oblique asymptote.