Family Overview

In Plain Language

End behavior describes what a polynomial does as inputs go to the far left or far right—as $x\to\infty$ or $x\to-\infty$. For a nonconstant polynomial, outputs also go to $\infty$ or $-\infty$ on each side.

Those limits are controlled by the leading term $a_n x^n$: for large $|x|$, it dominates every lower-degree term. So you only need:

• the degree $n$ (even or odd), and
• the sign of the leading coefficient $a_n$.

Notation examples: $\lim_{x\to\infty} p(x)=\infty$ or $\lim_{x\to-\infty} p(x)=-\infty$.

Why It Matters

AP Precalculus asks students to describe long-run behavior with limit language and to match graphs/equations to the four end-behavior cases. Later rational end behavior (1.7) builds on the same “what dominates for large $|x|$?” idea.

How Families Can Help

• Ask: “What’s the highest power? What’s its coefficient’s sign?”
• Practice the four cases with $x^2$, $-x^2$, $x^3$, $-x^3$.
• Have them say both limits out loud: “as $x$ goes to infinity…” and “as $x$ goes to negative infinity…”

What Not to Do

• Do not use middle bumps or zeros to decide end behavior.
• Do not forget the left-hand limit ($x\to-\infty$).
• Do not mix up even degree (same ends) with odd degree (opposite ends).
• Do not treat constant polynomials as the focus of 1.6 end-behavior items.