How to Graph Rational Functions

In order to graph a rational function you must determine four things:
1. x-intercepts
2. y-intercepts
3. vertical asymptote
4. horizontal asymptote

Lets start with something easy...

First, we find the x-intercepts:

(x-2) (x+3)=0

x=2      x= -3

Then, the y-intercepts:

y=  

y=1

When you find the vertical asymptote, you want to find where the fraction is undefined.

 

The denominator:

=0

(x+2) (x-3)=0

x= -2   x=3

Lastly, we find the horizontal asymptotes. Since both leading coefficients have the same degree, we take the ratio of the leading coefficients.

y=1

Shown in the graph above:

• the x-intercepts are at x= -3 and x=2
• the y-intercept is at y=1
• the asymptotes are labeled correctly and are shown with dotted lines

Graphing is a little more complicated...

If there is a hole in the graph.

For instance, lets take the equation:

Since the factor (x-3) is found in both the numerator and denominator, it can be canceled out, creating a whole in the graph.

Therefore, x=3 is not considered a y-intercept

To graph this:

• x-intercepts: x= -1  x=4
• y-intercept: y=2
• V.A.: x= -2   x=2
• H.A.: y=2
• hole in graph: x=3

The arrow is pointing to the hole in the graph where x=3. You singnify the hole with an open circle.

Lastly, to know whether the graph is going up or down, you look where the x-intercepts are. If there is no x-intercept where you are graphing it goes the opposite direction.

For example:
In the graph above, there is no x-intercept to the left of the vertical asymptote x= -2, so the graph would be going up.  The graph in the middle of the two asymptotes needs to touch the x- and y-intercept. The graph on thr right of the vertical asymptote x=2 is going down because it also needs to go through the x-intercept x=4.