Advanced Analysis of Rational Functions

By the end of this lesson, you will be able to perform a complete analysis of rational functions including slant asymptotes, sign analysis near asymptotes, and multiple holes; sketch fully labeled graphs.

The Big Idea

Advanced rational function analysis synthesizes everything: holes (cancellation), vertical asymptotes (remaining denominator zeros), horizontal/slant asymptotes (degree comparison), intercepts, and behavior on either side of asymptotes. A complete graph requires all of these pieces.

Complete Analysis Procedure

1. Factor numerator and denominator completely
2. Identify holes (canceled factors) and compute their $y$-coordinates
3. Identify vertical asymptotes (remaining denominator zeros)
4. Determine horizontal or slant asymptote (degree comparison)
5. Find $x$- and $y$-intercepts
6. Sign analysis (pick test points in each region)
7. Sketch

Slant (Oblique) Asymptotes

When $\deg(\text{num}) = \deg(\text{den}) + 1$, divide to get:

$$f(x) = q(x) + \frac{r(x)}{d(x)}$$

As $x \to \pm\infty$, $\frac{r(x)}{d(x)} \to 0$, so $f(x) \to q(x)$ (the slant asymptote).

Example 1 — Function with Hole and VA

$f(x) = \dfrac{x^2 - x - 6}{x^2 - 4} = \dfrac{(x-3)(x+2)}{(x-2)(x+2)}$

Hole: $x = -2$; $f_{\text{simplified}}(-2) = \dfrac{-2-3}{-2-2} = \dfrac{-5}{-4} = \dfrac{5}{4}$

Hole: $\left(-2, \frac{5}{4}\right)$

VA: $x = 2$ (remaining denominator zero)

HA: equal degrees → $y = 1$ (ratio of leading coefficients)

$x$-intercept: $x - 3 = 0 \implies x = 3$; point $(3, 0)$

$y$-intercept: $f(0) = \dfrac{-6}{-4} = \dfrac{3}{2}$; point $\left(0, \frac{3}{2}\right)$

Sign near $x = 2$: test $x = 1.9$: $(1.9-3)/((1.9-2)) = (-1.1)/(-0.1) = 11 > 0$ → $f \to +\infty$ from left; test $x = 2.1$: $(-0.9)/(0.1) = -9 < 0$ → $f \to -\infty$ from right.

Example 2 — Slant Asymptote

$g(x) = \dfrac{x^2 - 3x + 5}{x - 2}$

No common factors → no holes.

VA: $x = 2$; Slant asymptote: divide:

$$\frac{x^2-3x+5}{x-2} = x - 1 + \frac{3}{x-2}$$

Slant asymptote: $y = x - 1$.

$x$-intercepts: $x^2 - 3x + 5 = 0$; $D = 9 - 20 < 0$ → no real $x$-intercepts.

$y$-intercept: $g(0) = -5/(-2) = 5/2$