Graphing Rational Functions
Identify vertical asymptotes (zeros of the denominator after full cancellation), horizontal asymptotes (end behavior from degree comparison), oblique asymptotes, and holes. Sketch graphs incorporating all features.
A lesson of Algebra 2, in Rational Functions, about 10 minutes.
By the end of this lesson, you will be able to find vertical, horizontal, and oblique asymptotes; identify holes (removable discontinuities); and sketch the graph of a rational function using asymptotes, intercepts, and sign analysis.
The Big Idea
Rational functions have rich graphical behavior — they can have vertical walls (asymptotes), horizontal ceilings (asymptotes), diagonal trends (oblique asymptotes), and invisible holes (where a factor canceled). Understanding where all these features are and why they exist makes graphing rational functions systematic rather than mysterious.
Vocabulary
Vertical asymptote (VA): a vertical line $x = a$ where the function grows without bound. Occurs where the denominator = 0 and the numerator ≠ 0 (after factoring and canceling).
Hole (removable discontinuity): a point where a common factor cancels; the function is undefined there but the graph has only a missing point.
Horizontal asymptote (HA): a horizontal line $y = b$ that the function approaches as $x \to \pm\infty$.
Oblique asymptote: a slanted line $y = mx + b$ that the function approaches; occurs when the degree of the numerator is exactly 1 more than the denominator.
Finding Asymptotes
Step 1: Factor numerator and denominator fully.
Step 2: Cancel common factors → these are holes at the $x$-values that make those factors zero.
Step 3: Remaining denominator factors → vertical asymptotes.
Step 4: Horizontal or oblique asymptote from the degrees:
| $\deg(\text{num})$ vs. $\deg(\text{den})$ | Type | Rule |
|---|---|---|
| num $<$ den | Horizontal | $y = 0$ |
| num $=$ den | Horizontal | $y = \dfrac{\text{leading coeff. num}}{\text{leading coeff. den}}$ |
| num $=$ den $+ 1$ | Oblique | Divide to find $y = mx + b$ |
| num $>$ den $+ 1$ | None | No horizontal or oblique asymptote |
Example 1 — Finding All Features
$f(x) = \dfrac{2x^2 - 8}{x^2 - x - 6}$
Factor: $\dfrac{2(x+2)(x-2)}{(x-3)(x+2)}$
Hole: $x = -2$ (common factor $(x+2)$ cancels); hole at $\left(-2, f_{\text{simplified}}(-2)\right)$
After canceling: $\dfrac{2(x-2)}{x-3}$; $f_{\text{simplified}}(-2) = \dfrac{2(-4)}{-5} = \dfrac{8}{5}$