End Behavior of Functions

In Precalculus, end behavior describes how a function behaves as the input values become very large or very small.
Instead of focusing on exact values, end behavior helps us understand the overall direction and long-term trend of a graph.

This concept is essential for:

• Analyzing polynomial and rational functions

• Sketching graphs efficiently

• Predicting function behavior without full computation

• Preparing for limits in Calculus

What Is End Behavior?

End behavior examines what happens to a function as:

x→∞ and x→−∞

We describe end behavior using statements such as:

• “As $x$ increases without bound, f(x) increases without bound.”

• “As $x$ decreases without bound, f(x) decreases without bound.”

End Behavior Notation

End behavior is written using limit notation:

$$\lim_{x\to\infty}f(x) and \lim_{x\to-\infty}f(x)$$

Examples:

• $$\lim_{x\to\infty}f(x)=\infty$$

• $$\lim_{x\to-\infty}f(x)=-\infty$$

• $$\lim_{x\to\infty}f(x)=c$$ (approaches a constant)

End Behavior of Polynomial Functions

For polynomial functions, end behavior depends only on the leading term:

$$f(x)=a_nx^n$$ + lower degree terms

Key Rule

• Degree n (even or odd)

• Sign of leading coefficient​

Lower-degree terms become insignificant as ∣x∣ becomes large.

Degree and End Behavior

Even Degree Polynomials

Positive Leading Coefficient

$$f(x)=x^4$$

• $$x \longrightarrow \infty then f(x) \longrightarrow \infty$$

• $$x \longrightarrow -\infty then f(x) \longrightarrow \infty$$

Graph opens up on both ends.

Negative Leading Coefficient

$$f(x)=-2x^6$$

• $$x \longrightarrow \infty then f(x) \longrightarrow -\infty$$

• $$x \longrightarrow \infty then f(x) \longrightarrow -\infty$$

Graph opens down on both ends.

Odd Degree Polynomials

Positive Leading Coefficient

$$f(x)=x^3$$

• $$x \longrightarrow \infty then f(x) \longrightarrow \infty$$

• $$x \longrightarrow -\infty then f(x) \longrightarrow -\infty$$

Graph goes down left, up right.

Negative Leading Coefficient

$$f(x)=-x^5$$

• $$x \longrightarrow \infty then f(x) \longrightarrow -\infty$$

• $$x \longrightarrow -\infty then f(x) \longrightarrow \infty$$

Graph goes up left, down right.

Summary Table (Polynomials)