Rational Function Analyzer

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

Numerator p(x)
Denominator 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
Rational Function AnalyzerVertical asymptotes: x = 2. Horizontal asymptote: y = 1. Holes (removable): (-1, 0.6667)xyHAVA

关于这个计算器

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.

常见示例

  • (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