The rational function .

Replace by .

Step 1: Factor the numerator and denominator of . Find the domain of 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 behaviour 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, 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 is .

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 .   

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 , 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.

Step 8 : Use the results obtained in Steps 1 through 7 to graph .

Graph :

The rational function is .

Domain of the function is  .

-intercepts are , .

Plot the point and indicate a line with negative slope.

Plot the point and indicate a line with negative slope.

The -intercept is .

Vertical asymptote .

Oblique asymptote is .

And point on the graph are

Graph of the rational function is