📐 Polynomials & Rational Expressions
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Graphing Rational Functions

Identify vertical asymptotes, horizontal asymptotes, and holes in rational functions.

A rational function f(x) = p(x)/q(x) has interesting behavior where q(x) = 0. At these x-values, the function either has a vertical asymptote (if the factor doesn't cancel) or a hole (if the factor cancels from numerator and denominator).

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Vertical asymptote: x = a when (x − a) is a factor of q(x) but NOT p(x). Hole: x = a when (x − a) cancels from both p(x) and q(x).

Finding Asymptotes and Holes

1Fully factor both numerator p(x) and denominator q(x).
2Cancel any common factors — each cancellation creates a hole at that x.
3Set remaining denominator = 0 for vertical asymptotes.
4Compare degrees for horizontal asymptote: if deg(p) < deg(q), HA is y = 0; if deg(p) = deg(q), HA is y = leading coefficients ratio; if deg(p) > deg(q), no HA (oblique asymptote).
✏ī¸Analyze f(x) = (x² − 4) / (x² − x − 2)
Factor: (x−2)(x+2) / [(x−2)(x+1)]. Cancel (x−2): hole at x = 2. Remaining: (x+2)/(x+1). Vertical asymptote: x = −1. Horizontal asymptote: y = 1 (equal degrees, ratio of leading coefficients = 1/1).

Horizontal Asymptote Rules

deg(p) < deg(q): y = 0deg(p) = deg(q): y = leading coeff ratiodeg(p) > deg(q): none (oblique)

Vertical/Hole Rules

Factor cancels → hole at that xFactor stays in denom → vertical asymptoteAlways factor completely first
✏ī¸ Try It!

For f(x) = (x + 3) / (x² − 9), which statement is true?

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