End behavior

Math

Definition

End behavior describes what happens to the values of a function as x approaches positive infinity or negative infinity. For polynomials, end behavior is determined by the leading term's degree and coefficient.

Examples

  • f(x) = x³: as x → ∞, f(x) → ∞; as x → -∞, f(x) → -∞
  • f(x) = -2x⁴: both ends go to -∞ because the degree is even and the leading coefficient is negative
  • Exponential functions like 2ˣ grow to ∞ as x → ∞ and approach 0 as x → -∞
Key Fact

For axⁿ: if n is even, both ends go the same direction; if n is odd, ends go opposite directions.

Related Terms

Zeros and end behavior Asymptotes Zeros and multiplicity Polynomial operations

Study This Concept

Practice end behavior with free review games in these units:

Pre-Calculus Unit 2: Polynomial and Rational Functions