Math / Functions

Rational Functions Calculator

Analyze a rational function from its numerator and denominator: domain, holes, vertical and horizontal asymptotes, intercepts, and sample points.

DomainHolesAsymptotesIntercepts

Coefficients

Numerator: a·x² + b·x + c

Denominator: d·x² + e·x + f

x² − 4x − 2

Analysis

Domainx ≠ 2
Holes(2, 4)
Vertical asymptotesNone
Horizontal asymptoteNone
Oblique asymptotey = 1x + 2
x-intercepts-2
y-intercept2

Sample points

Sample points
xy
-6-4
-5-3
-4-2
-3-1
-2-0
-11
-0.51.5
0.52.5
13
35
46
57
68

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About this calculator

Method, formulas, and limits.

What this does

Takes a rational function as the quotient of two linear or quadratic polynomials and reports the domain, holes, vertical and horizontal or oblique asymptotes, intercepts, and sample points.

Who it is for

Precalculus and algebra students checking homework, and anyone graphing or reasoning about rational functions.

How it works

The tool factors each polynomial, cancels common factors (which become holes), and classifies the remaining denominator zeros as vertical asymptotes. Degree comparison between numerator and denominator determines the end behavior.

Limitations

The analysis covers linear and quadratic numerator/denominator inputs only; higher-degree polynomials are not analyzed. The horizontal or oblique asymptote is derived from leading terms only.

Key calculations

Domain
All real x except the zeros of the denominator.
Vertical asymptotes
Occur at denominator zeros that do not cancel with the numerator.
Holes
Occur at denominator zeros that also zero the numerator (a common factor cancels).
Horizontal asymptote
y = leadingNum / leadingDen when the degrees are equal; y = 0 when the denominator degree is higher.

How to use it

  1. 1.Enter the numeratorProvide the coefficients of ax² + bx + c (linear is fine — leave the x² coefficient at 0).
  2. 2.Enter the denominatorProvide the coefficients of dx² + ex + f. The denominator cannot be all zeros.
  3. 3.Review the analysisThe tool lists the domain, any holes with their y-limits, vertical and horizontal or oblique asymptotes, and intercepts.
  4. 4.Use the sample pointsThe sample table gives x, y pairs for sketching the graph by hand.

A rational function is a ratio of two polynomials, such as f(x) = (x² − 1) / (x − 1). Its domain excludes any x where the denominator is zero.

Both happen where the denominator is zero. If the numerator is also zero at that x, the factor cancels and the graph has a hole (a missing point). If only the denominator is zero, the graph has a vertical asymptote.

Compare the degrees: if the numerator degree is lower, the asymptote is y = 0; if equal, it is the ratio of leading coefficients; if the numerator degree is one higher, there is an oblique asymptote instead.

Linear and quadratic polynomials for both the numerator and the denominator (up to x² in each). Higher-degree polynomials are not analyzed.

Sample points are computed from the entered coefficients and rounded for display. Values near excluded x coordinates are skipped so the table stays finite.

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