Section 3.7

Rational Functions

Rational functions show up on the final in graphing and domain problems, and they tie together everything from polynomial factoring to sign analysis — skills you'll keep using in the sections ahead.

⚡ Quick Summary

A rational function is simply one polynomial divided by another — a fraction with algebra on top and bottom. The interesting part is what happens when the bottom equals zero: the function either shoots off to infinity (a vertical asymptote) or has a single missing point (a hole). These are what make rational functions behave differently from regular polynomials.

General Form

f(x) = \frac{p(x)}{q(x)}

where p(x) and q(x) are polynomials and q(x) ≠ 0

Vertical Asymptotes

\text{Set } q(x) = 0 \text{ (after canceling common factors)}

The x-values where the denominator is zero (and doesn't cancel) give vertical asymptotes

📋 Before You Start

Tap any item if you need a refresher:

Plain-English version

Think of a rational function like a highway. Vertical asymptotes are walls — the function blows up to infinity and you can never cross them. Holes are like potholes — one tiny missing point, but the road is fine on both sides. The horizontal asymptote is the speed limit the curve approaches over the long run: as $x$ gets very large, the function levels off toward a flat line.

The secret to every rational function problem is one step: factor the numerator and denominator completely. Once you do that, everything falls into place. Factors that cancel between top and bottom give you holes. Factors left in the bottom give you vertical asymptotes. Factors left on top give you $x$ -intercepts. And comparing the degrees of top and bottom tells you the horizontal asymptote.

For the horizontal asymptote, the rule is simple: if the top has a smaller degree than the bottom, the asymptote is $y = 0$ . If the degrees are equal, divide the leading coefficients. If the top has a bigger degree, there's no horizontal asymptote at all.

Key Formulas

General Form

f(x) = \frac{p(x)}{q(x)}

where p(x) and q(x) are polynomials and q(x) ≠ 0

Vertical Asymptotes

\text{Set } q(x) = 0 \text{ (after canceling common factors)}

The x-values where the denominator is zero (and doesn't cancel) give vertical asymptotes

Horizontal Asymptote (degrees equal)

y = \frac{a_n}{b_m}

When deg(p) = deg(q), the HA is the ratio of leading coefficients

Horizontal Asymptote (numerator smaller)

y = 0

When deg(p) < deg(q)

Hole Location

\text{If } (x - c) \text{ cancels from both } p(x) \text{ and } q(x), \text{ hole at } x = c

Key Takeaways

✓Factor the numerator and denominator completely — this single step reveals holes, vertical asymptotes, and x-intercepts
✓Hole vs. vertical asymptote: if a factor cancels from top and bottom, it's a hole; if it stays in the denominator, it's a VA
✓Horizontal asymptote rule: $\deg(p) < \deg(q) \to y = 0$ ; equal degrees $\to y = a_n/b_m$ ; $\deg(p) > \deg(q) \to$ no HA
✓Domain of a rational function = all real numbers except where the original denominator is zero

⚠️ Common Mistakes

✗Calling every denominator zero a vertical asymptote — if the factor ALSO cancels from the numerator, it's a hole, not a VA
✗Forgetting to find the $y$ -coordinate of a hole — plug the $x$ -value into the SIMPLIFIED function (after canceling) to get the actual point
✗Comparing degrees after canceling common factors for the horizontal asymptote — always compare the degrees of the ORIGINAL numerator and denominator
✗Thinking the domain only excludes vertical asymptotes — the domain excludes ALL $x$ -values where the original denominator is zero, including holes

Next: Section sz.ineq

sz.ineq Polynomial & Rational Inequalities