Rational Function Grapher & Analyzer

Define Rational Function

Numerator P(x) — emerald coefficients

a (x³)

b (x²)

c (x)

d (const)

Denominator Q(x) — gold coefficients

e (x³)

f (x²)

g (x)

h (const)

f(x) =

x² − 1

Quick Examples

Analysis Results

Enter numerator and denominator coefficients, then press Graph & Analyze to find asymptotes, holes, intercepts, and domain.

Domain

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Vert. Asymptotes

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Horiz./Oblique Asymptote

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Holes

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x-Intercepts

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y-Intercept

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Asymptote Rules by Degree Comparison

f(x) = P(x)/Q(x), deg(P) = n, deg(Q) = m

The relationship between the degrees of numerator and denominator determines the horizontal or oblique asymptote:

n < m: Horizontal asymptote y = 0 (the x-axis). The denominator "wins."
n = m: Horizontal asymptote y = (leading coeff of P) / (leading coeff of Q).
n = m + 1: Oblique (slant) asymptote — perform polynomial long division; the quotient is the asymptote line.
n > m + 1: No horizontal or oblique asymptote (behavior is polynomial-like).

Vertical asymptotes occur at zeros of Q(x) that are not cancelled by a common factor with P(x).

A rational function can cross a horizontal asymptote in the middle of its domain — but it cannot cross a vertical asymptote.

Holes vs. Vertical Asymptotes

f(x) = (x−a)·g(x) / [(x−a)·h(x)]

When both P(x) and Q(x) share a common factor (x − a), that factor cancels. The result:

Hole (removable discontinuity): x = a is not in the domain, but the limit exists. Plot an open circle at (a, lim_x→a f(x)).
Vertical asymptote: x = b where Q(b) = 0 but (x − b) did NOT cancel — the function blows up to ±∞.

Example: f(x) = (x²−1)/(x−1) = (x+1)(x−1)/(x−1). Cancel (x−1) → simplified form is x+1, with a hole at x = 1. No vertical asymptote.

Always factor both numerator and denominator fully before identifying asymptotes and holes.