End behavior

Definition

End behavior describes the behavior of a function's graph as $x$ approaches positive or negative infinity. It is crucial for understanding how functions behave at extreme values.

5 Must Know Facts For Your Next Test

1. End behavior helps predict whether a function will approach infinity, negative infinity, or a finite value as $x$ goes to positive or negative infinity.
2. Polynomial functions' end behaviors depend on the leading term; specifically, the coefficient and degree of the leading term.
3. Rational functions' end behaviors can be determined by comparing the degrees of the numerator and denominator polynomials.
4. Exponential functions grow faster than polynomial functions at both positive and negative extremes.
5. Horizontal asymptotes indicate end behavior for rational functions where the degrees of numerator and denominator are equal or when the degree of the numerator is less.

Review Questions

• How does the end behavior of a polynomial function change based on its leading term?
• What does it mean if a rational function has a horizontal asymptote at $y = 0$?
• Explain why exponential functions have different end behaviors compared to polynomial functions.

Related terms

Asymptote: A line that a graph approaches but never touches. Asymptotes can be vertical, horizontal, or oblique.

Degree (of a Polynomial): The highest power of $x$ in a polynomial expression. The degree determines many properties of the polynomial, including its end behavior.

Leading Coefficient: The coefficient of the term with the highest degree in a polynomial. The sign and magnitude affect the graph's direction as $x$ approaches infinity.