Rational Functions: Asymptotes, Holes and End Behavior

Rational functions close Unit 1 of AP Precalculus, Polynomial and Rational Functions. A rational function is a ratio of two polynomials, r(x) = p(x)/q(x), and almost every question about it asks what happens where the denominator is zero and what happens far out on the x-axis.

Factor first

Every question starts the same way: factor the numerator and the denominator completely. The factors tell you everything else.

Vertical asymptotes and holes

For each zero of the denominator:

• If the factor cancels with the numerator, the graph has a hole there. Find its y-value by plugging the x-value into the simplified function.
• If the factor does not cancel, the graph has a vertical asymptote there.

A factor that appears more times in the denominator than in the numerator still leaves an asymptote after cancelling.

Horizontal and slant asymptotes

Compare the degree of the numerator n with the degree of the denominator m:

The graph can cross a horizontal asymptote in the middle; the asymptote describes only the ends.

Worked example

Analyze r(x) = (x² − x − 6) / (x² − 9).

1. Factor: r(x) = (x − 3)(x + 2) / ((x − 3)(x + 3)).
2. x − 3 cancels, so there is a hole at x = 3. Simplified: (x + 2)/(x + 3), which gives 5/6 at x = 3. Hole at (3, 5/6).
3. x + 3 does not cancel: vertical asymptote at x = −3.
4. Equal degrees, leading coefficients 1 and 1: horizontal asymptote y = 1.
5. Zero of the simplified numerator: x-intercept at x = −2.

A common mistake is to call x = 3 an asymptote because the original denominator is zero there. Always cancel first.

The free response approach

• Show the factored form. It is the evidence for every claim that follows.
• Use limit language when asked about behavior: as x → −3⁺, r(x) → −∞.
• Give the hole as a point, with both coordinates, not only its x-value.