Rational Functions & Asymptotes

Vertical, horizontal, and slant asymptotes from a quotient of polynomials.

The idea

A rational function is a quotient of polynomials, $f(x) = \dfrac{p(x)}{q(x)}$, defined wherever the denominator is nonzero. Its graph is shaped by asymptotes: straight lines the curve approaches more and more closely without reaching them.

Near an input where $q$ is zero but $p$ is not, the denominator shrinks toward $0$ while the numerator does not, so the values grow without bound; the vertical line through that input is a vertical asymptote. If $p$ vanishes there as well, the shared factor cancels, and the graph has a hole there rather than an asymptote.

Far from the origin, the degrees of $p$ and $q$ decide the end behavior. If $p$ has smaller degree, the values shrink toward $0$, and the horizontal asymptote is the line $y = 0$. If the degrees are equal, only the leading terms matter, and the horizontal asymptote is the ratio of the leading coefficients. If the degree of $p$ is exactly one more, dividing $p$ by $q$ writes $f$ as a linear polynomial plus a remainder that shrinks toward $0$, so far out the graph runs along that line, its slant asymptote. If the degree of $p$ is larger still, the graph approaches no line at all.

Ways to work on it

• Walkthrough. Vertical asymptote, degree comparison, and the slant asymptote from division.
• Practice. Classify a rational function's end behavior from its degrees.
• Hardest. Find the slant asymptote of a top-heavy rational function.

Not sure where to start? Take the ten-question placement test.