1.7 Rational Functions and End Behavior

Rational functions and end behavior

A rational function is a ratio of two polynomials, f(x) = p(x)/q(x). As x approaches positive or negative infinity, end behavior depends on the degrees of p and q. If the degree of the numerator is less than the denominator's, the function's end behavior approaches 0 (a horizontal asymptote at y=0). If degrees are equal, end behavior approaches the ratio of leading coefficients.

When numerator degree exceeds denominator

If the numerator's degree is exactly one more than the denominator's, the function has a slant (oblique) asymptote, and end behavior mimics that line. If the numerator's degree exceeds the denominator's by two or more, end behavior resembles a polynomial found via division, not a line — the graph increases/decreases without bound following that quotient.

Modeling end behavior with quotients

Using polynomial long division, p(x)/q(x) = quotient + remainder/q(x). As |x| grows large, the remainder term shrinks toward 0, so f(x) behaves like the quotient polynomial. This quotient reveals horizontal, slant, or curved end behavior — a key tool for predicting a rational function's long-run trend.

Key terms

Rational function

A function of the form f(x) = p(x)/q(x), where p and q are polynomials and q(x) ≠ 0.

End behavior when numerator degree < denominator degree

f(x) approaches 0; horizontal asymptote at y = 0.

End behavior when degrees are equal

f(x) approaches the ratio of the leading coefficients.

Slant (oblique) asymptote

Occurs when numerator's degree is exactly one more than denominator's; end behavior follows a linear function.

Using division to find end behavior

Divide p(x) by q(x) to get quotient + remainder/q(x); as |x|→∞, remainder/q(x)→0, so f(x) behaves like the quotient.

Try an AP-style question

What is the horizontal asymptote of (2x² + 1)/(x² − 4)?

1. y = 2
2. None
3. y = 0
4. y = −1/4