Rational Functions

A rational function is simply one polynomial divided by another, like an average cost per item, $\frac{500 + 3x}{x}$. ("Rational" comes from "ratio.") The general formula looks heavy, but only four pieces matter: $a_n$ and $b_m$ (the numbers in front of the biggest powers, top and bottom) and $n$ and $m$ (those biggest powers). The asymptote rule just compares $n$ with $m$. These functions are good at describing things that blow up or level off, and that's mostly what the graph is about.

Why it blows up. Dividing by a number close to zero gives something huge: $\frac{1}{0.001} = 1000$. So wherever the denominator heads to zero and the numerator doesn't, the graph shoots off toward $\pm\infty$. That's a vertical asymptote.

Why it levels off. For large $x$, the highest power in a polynomial swamps everything else: at $x = 1000$, $2x$ is 2000 while the $-1$ hardly matters. That's true on top and on bottom, so $\frac{2x - 1}{x + 2} \approx \frac{2x}{x} = 2$. If the bottom's degree is bigger, the bottom wins and the fraction shrinks to $0$; if the top's is bigger, there's no level to settle at.

Graphing a rational function:

1. Factor top and bottom. A factor that cancels makes a hole, not an asymptote.
2. Vertical asymptotes: zeros of the remaining denominator.
3. Horizontal asymptote: compare degrees, as above.
4. Intercepts: x-intercepts from the numerator's zeros, y-intercept $f(0)$.
5. Test a point in each region between asymptotes and zeros to see whether the graph is above or below the axis.

A common mistake. It's tempting to put a vertical asymptote at every zero of the original denominator. In $\frac{x^2 - 1}{x - 1}$, the factor $x - 1$ cancels, leaving $x + 1$. The graph is just that line with a single point missing at $x = 1$: a hole, not an asymptote.

Average cost per item, drug concentration over time and lens equations are rational functions, and their asymptotes describe the long-run behavior.