End Behavior of Polynomials

You’ll be able to

Predict end behavior from the degree and leading coefficient of a polynomial
Explain why the leading term dominates for large |x|
Match a polynomial to the correct “arrows” description of its long-run behavior

The leading term runs the show

For very large positive or negative inputs, a polynomial behaves almost exactly like its leading term — the term with the highest power of x. In f(x) = −2x³ + 5x − 1, the −2x³ term grows so much faster than 5x or 1 that the smaller terms become irrelevant when |x| is huge. So to describe end behavior — what happens as x → +∞ and as x → −∞ — you only need the leading term.

Degree parity and the sign of the lead

Two facts decide the end behavior. The degree (even or odd) says whether the two ends point the same way or opposite ways, and the sign of the leading coefficient says which way the right end points. An even degree gives arms that agree — both up or both down. An odd degree gives arms that disagree — one up and one down. A positive lead sends the right arm up; a negative lead sends the right arm down.

End-behavior rules

even degree → arms agree · odd degree → arms disagree · sign of lead sets the right arm

Positive leading coefficient: right arm rises (f → +∞ as x → +∞). Negative leading coefficient: right arm falls.

Worked example

Describe the end behavior of f(x) = −2x³ + 5x − 1.

1.Identify the leading term: −2x³. Everything else is negligible for large |x|.
2.The degree is 3, which is odd, so the two arms point in opposite directions.
3.The leading coefficient is −2, which is negative, so the right arm falls: as x → +∞, f(x) → −∞.
4.Because the arms disagree, the left arm does the opposite: as x → −∞, f(x) → +∞.

Answer: As x → −∞, f(x) → +∞ (up on the left); as x → +∞, f(x) → −∞ (down on the right). The odd degree makes the arms disagree, and the negative lead pulls the right side down.

Checkpoint

What is the end behavior of f(x) = −2x³ + 5x − 1?

Tip

A quick mental picture: odd + positive looks like the line y = x (down-left, up-right); odd + negative flips it. Even + positive looks like a smile (up on both ends); even + negative looks like a frown.

Checkpoint

A polynomial has even degree and a positive leading coefficient. What is its end behavior?

On the exam

On the AP exam, state end behavior with limit-style language: “as x → ∞, f(x) → ∞.” Naming the degree parity and the sign of the leading coefficient is the reasoning that earns the point.

Answer the 2 checkpoints as you read.