In a previous post, I listed the steps you need to follow to analyze and graph a rational function. In this and subsequent posts, I will walk you through each of those steps. Usually the first step in plotting a rational function is finding the 
– and 
-intercepts. You have done this before for other functions and the process here is no different.
- To find the -intercept, set
and solve for 
.
- To find the -intercept(s), set
and solve for . Recall that a rational function is a fraction, and a fraction can only be equal to zero when its numerator is equal to zero (as long as the denominator is not also equal to zero!). So in practice, you find the -intercept(s) by setting the numerator equal to zero and solving for .
Example: Find the – and -intercepts, if any, for the following rational function:
Finding the -intercept is very easy if the polynomials are written in standard form as above. 
is simply the ratio of the constant terms: 
. If the rational function has already been factored, you need to plug zero in to each term and evaluate.
To find the -intercept (and all the other parts of the rational function), the first thing you need to do is fully factor the numerator and denominator.
From here, it is easy to see that the roots of the numerator are 
, 
and 
. When 
, the denominator is also zero, so this is not an -intercept; it is a hole. The ‑intercepts are at and .
Is it possible that a rational function does not have a -intercept? Yes, if the function is not defined for . In other words, if a rational function has either a hole or a vertical asymptote at , there is no -intercept.
[Check out my other posts in this section that describe how to perform the other steps when analyzing a rational function.]
.
and solve for 
