The 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 .

and

and .

The domain of function is .

The rational function .

The function is 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.

To find -intercept equate the numerator to zero.

.

-intercept is .

Determine the behaviour of the graph of near each -intercept.

Near : .

Plot the point and indicate a line with negative slope.

Find the intercept by substituting in the rational function.

- intercept is .

Vertical asymptote can be found by making denominator.

or

or

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 less than the degree of denominator,

Horizontal asymptote is .

 The real zeros of numerator is and the real zeros of denominator are and .

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

Choosing a number for in each interval and evaluating . 

Behavior of the graph :

and , hence the graph of is will approach the vertical asymptote, at .

and , hence the graph of is will approach the vertical asymptote,at  .

and , hence the graph of is will approach the horizontal asymptote, at  . 

Graph:

Draw the coordinate plane.

Next dash the horizontal and vertical asymptotes.

Plot the , intercepts and coordinate pairs found in the table.

Connect the plotted points.

When you draw your graph, use smooth curves complete the graph.

The domain of function is .

The rational function .

-intercept is .

- intercept is .

Vertical asymptote are or .

Horizontal asymptote is .

Graph of the rational function is

.