Algebra 2 Advanced

Horizontal & Oblique Asymptotes

Determine end behavior of rational functions — horizontal asymptotes (y = k) when degrees match or numerator is lower, oblique (slant) when numerator degree is exactly one higher.

Live Calculator · Step-by-Step · Algebra 2

Rational Function f(x) = p(x) / q(x)

Numerator p(x)

— divided by —

Denominator q(x)

Accept: x^3, x³, implied coefficients (x = 1x, −x = −1x), constants

Examples

Result

Enter numerator and denominator, then press Find Asymptote.

Horizontal (y = a/b)

Asymptote

y = 2

—

deg(numerator) m

vs

—

deg(denominator) n

Step-by-Step Solution

Graph

Teal curve: f(x) · Dashed gold line: asymptote · Red dashed lines: vertical asymptotes

End Behavior Rules for f(x) = p(x) / q(x)

Let m = deg(p), n = deg(q), and let a, b be the leading coefficients of p and q respectively.

m < n → y = 0 (horizontal, x-axis)

m = n → y = a/b (horizontal, ratio of leading coefficients)

m = n+1 → y = mx + b (oblique/slant — use polynomial long division)

m > n+1 → none (no horizontal or oblique asymptote)

For oblique asymptotes, divide p(x) ÷ q(x) by polynomial long division. The quotient (ignoring the remainder) is the asymptote line.

Why Does the Curve Approach the Asymptote?

f(x) = (oblique) + remainder/q(x)

After long division: f(x) = (quotient line) + r(x)/q(x).

As x → ±∞, the remainder term r(x)/q(x) → 0, so f(x) gets arbitrarily close to the quotient line — that is the asymptote.

For horizontal asymptotes the same idea applies: the lower-degree terms become negligible, leaving only the ratio of leading terms.

A rational curve can cross its horizontal or oblique asymptote at finite x-values — it only must approach it as x → ±∞.

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