Polynomial & Rational Functions

Analyzing Rational Graphs

Determining vertical/horizontal asymptotes and end behavior of a rational function.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

The Asymptote That Isn't Flat

Recall from Algebra II: a rational function's horizontal asymptote depends on comparing the numerator's and denominator's degrees, and there's no horizontal asymptote when the numerator's degree is larger. Before reading on, think about the specific case where the numerator's degree is exactly one more than the denominator's — could there still be some kind of asymptote, just not a flat, horizontal one?

Definition — Slant (Oblique) Asymptote

When a rational function's numerator has degree exactly one more than its denominator, the function approaches a slanted line — found by performing polynomial long division and keeping only the quotient (the linear part), ignoring the remainder.

This connects directly back to Algebra II's polynomial division: dividing the numerator by the denominator rewrites the rational function as (quotient) + (remainder)/(denominator). As x grows very large in either direction, that remainder-over-denominator piece shrinks toward 0, since the denominator keeps growing while the remainder stays fixed — leaving the function behaving more and more like just the quotient alone, a straight line.

Worked Example — Finding a Slant Asymptote

Find the slant asymptote of f(x) = (x² + 1)/(x − 2). Divide x² + 1 by x − 2: x² ÷ x = x, multiply x(x − 2) = x² − 2x, subtract: (x² + 1) − (x² − 2x) = 2x + 1. Continue: 2x ÷ x = 2, multiply 2(x − 2) = 2x − 4, subtract: (2x + 1) − (2x − 4) = 5. The quotient is x + 2, with remainder 5: f(x) = x + 2 + 5/(x − 2). The slant asymptote is y = x + 2.

Worked Example — Full Rational Function Analysis

Using f(x) = (x² + 1)/(x − 2) from the previous example, find the vertical asymptote. The denominator is 0 at x = 2, and the numerator at x = 2 is 4 + 1 = 5, which doesn't cancel it — so x = 2 is a genuine vertical asymptote, not a hole. The function has a slant asymptote y = x + 2 and a vertical asymptote at x = 2, with no horizontal asymptote (since a slant asymptote replaces it whenever one exists).

Function Explorer

f(x) =

sin(x)

Transform: g(x) = a·f(b(x − h)) + k

a — vertical stretch/reflect1

b — horizontal stretch/reflect1

h — horizontal shift0

k — vertical shift0

Composition

Compose with another function

Trace mode

Show inverse view

Analysis (of the transformed function, in view)

y-intercept: (0, 0)
x-intercepts: (-9.42, 0), (-6.28, 0), (-3.14, 0), (0, 0), (3.14, 0), (6.28, 0), (9.42, 0)
Extrema: local min at (-7.85, -1); local max at (-4.71, 1); local min at (-1.57, -1); local max at (1.57, 1); local min at (4.71, -1); local max at (7.85, 1)
Inflection points: (-9.42, 0), (-6.28, 0), (-3.14, 0), (0, 0), (3.14, 0), (6.28, 0), (9.42, 0)
Vertical asymptotes: none found in view
Horizontal asymptotes: none found
Domain: all real numbers in view
Range (estimated): approximately [-1, 1]

Tip

A rational function has at most one of: a horizontal asymptote or a slant asymptote — never both, since a slant asymptote only appears when the numerator's degree exceeds the denominator's by exactly one, which is precisely the case that rules out a horizontal asymptote.

Common Mistakes

Looking for a slant asymptote whenever the numerator's degree is larger than the denominator's, regardless of by how much.
A slant asymptote only exists when the numerator's degree exceeds the denominator's by exactly 1 — a larger gap (like degree 3 over degree 1) produces a curved, non-linear asymptote instead.
Including the remainder term as part of the asymptote's equation.
The slant asymptote is only the quotient from the division — the remainder-over-divisor part shrinks to 0 as x grows large and isn't part of the asymptote line itself.

Key Takeaways

A slant asymptote occurs when a rational function's numerator degree exceeds its denominator's degree by exactly 1.
It's found by polynomial long division, keeping only the quotient and discarding the remainder.
A rational function never has both a horizontal and a slant asymptote at the same time.

Summary

Slant asymptotes complete the rational-function analysis toolkit, connecting directly back to polynomial division. The next unit shifts to trigonometric functions, graphing them as continuous, periodic curves rather than triangle ratios.

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