Recall that a rational function is a quotient of two polynomial functions.

r(x) = p(x) divided by q(x) p(x) q(x) , q(x) ≠0, where p(x) and q(x) are polynomial expressions.

The graphs of rational functions can be quite interesting and challenging.

Some rational functions have vertical asymptotes:

While some rational functions do not:

Some rational functions have horizontal asymptotes at y = 0 (the x-axis)

While others may have a horizontal asymptote elsewhere:

And still other rational function graphs may have slant (oblique) asymptotes, or asymptotes that are parabolic (or some other polynomial function):

And some rational functions may have a hole, or removable discontinuity, on their graph: