Rational Function Analyzer

Asymptotes, holes, and behavior of p(x)/q(x)

f(x) = (x² − 1) / (x² − x − 2)

Vertical asymptotes

x = 2

Horizontal asymptote

y = 1

Holes (removable)

(-1, 0.6667)

x-intercepts

x = 1

y-intercept

y = 0.5

Domain

all real numbers except x = 2, x = -1

About this calculator

The Rational Function Analyzer takes a numerator and denominator polynomial p(x)/q(x) and finds the vertical asymptotes, removable holes, horizontal or oblique asymptote, x- and y-intercepts, and domain, then plots the curve with every feature labeled. Useful for precalculus and calculus homework on rational functions.

Common examples

(x²−1)/(x²−x−2) → hole at (−1, 2/3), vertical asymptote x = 2, horizontal asymptote y = 1
1/x → vertical asymptote x = 0, horizontal asymptote y = 0
(x²+1)/x → vertical asymptote x = 0, oblique asymptote y = x
1/(x²−4) → vertical asymptotes x = ±2, horizontal asymptote y = 0
(2x²+3)/(x²+1) → no vertical asymptotes, horizontal asymptote y = 2