Asymptotes

Math

Definition

Lines that a graph approaches but never reaches (or crosses only in limited cases). Vertical asymptotes occur where a function is undefined, horizontal asymptotes describe end behavior as x → ±∞, and oblique (slant) asymptotes occur when the degree of the numerator is exactly one more than the denominator.

Examples

  • f(x) = 1/x has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0
  • f(x) = (2x + 1)/(x − 3) has a vertical asymptote at x = 3 and a horizontal asymptote at y = 2
Key Fact

Horizontal asymptote: compare degrees — same degree → ratio of leading coefficients

Related Terms

End behavior Domain and range Simplifying rational expressions Hyperbolas

Study This Concept

Practice asymptotes with free review games in these units:

Algebra 2 Unit 5: Rational Functions Pre-Calculus Unit 2: Polynomial and Rational Functions