Rational Functions

$f (x) = \frac{P (x)}{Q (x)}$ , where $P (x)$ and $Q (x)$ are polynomials with degrees $c$ and $d$ respectively and coefficients $a_{n}$ and $b_{n}$ respectively

note on graphing

Test around the asymptotes if not of the form $\frac{m x + n}{a x + b}$

note on solving rational inequalities

Do not multiply over the inequality, it just makes life harder. Instead, add/subtract over the inequality and make everything a single fraction

Key Features

Vertical Asymptotes

$x = roots of Q (x)$

Horizontal Asymptotes

if $d > c$ , HA is $y = 0$
if $d = c$ , HA is $y = \frac{a_{c}}{b_{d}}$
if $d < c$ , no HA

Linear Oblique Asymptote

if $c = d + 1$ , LOA is $y = \frac{P (x)}{Q (x)} - R$

Other Asymptotes

$y = \frac{P (x)}{Q (x)} - R$ , with a degree of $c - d$

Holes

If there is a root $q x - p$ in both $P (x)$ and $Q (x)$ , and the root fully cancels in $Q (x)$ , there is a hole at $\frac{p}{q}$

Intercepts

Set the other variable to 0 and solve