AP Exam KitAP Precalculus

中文 Open Learn

Unit 11.6 Polynomial Functions and End BehaviorNo calculator

The graph of a polynomial is shown. Which end-behavior statement is consistent with the graph?

A smooth quartic-like curve rises on both ends and has turning points near x=-1.5, 0, and 1.5.

As $x\to-\infty$ , $p(x)\to-\infty$ ; as $x\to\infty$ , $p(x)\to\infty$
As $x\to\pm\infty$ , $p(x)\to\infty$
As $x\to\pm\infty$ , $p(x)\to-\infty$
As $x\to-\infty$ , $p(x)\to\infty$ ; as $x\to\infty$ , $p(x)\to-\infty$

Answer and derivation

Correct answer: B

Both ends rise, so $p(x)$ approaches positive infinity in both directions.

On the far left, decreasing $x$ corresponds to increasing $p(x)$ .
On the far right, increasing $x$ also corresponds to increasing $p(x)$ .
This is the even-degree, positive-leading-coefficient end pattern.

Why the other options fail

A: It sends the left tail downward.
B: Correct description of both rising tails.
C: It reverses both tails.
D: It reverses the right tail.

Common misconception: Turning points in the middle do not determine end behavior.

China/AP bridge: Connect the graph to the sign and parity of the leading term.