1.6 Polynomial Functions and End Behavior

End behavior depends on degree and leading coefficient

For a polynomial function, as inputs approach positive or negative infinity, outputs are governed by the leading term. If the degree is even, both ends go the same direction (up if leading coefficient positive, down if negative). If the degree is odd, the ends go in opposite directions. This is called end behavior.

Using limit notation

End behavior is described with limits: as x → ∞, f(x) → ∞ or −∞; as x → −∞, f(x) → ∞ or −∞. These outcomes depend only on the sign and parity (even/odd) of the leading term, since lower-degree terms become negligible for very large |x|.

Average rate of change reveals concavity trends

Polynomial functions of degree n ≥ 2 show changing rates of increase/decrease. Comparing average rates of change over successive equal-length intervals helps identify where a graph is concave up or down, and confirms behavior consistent with the polynomial's degree and turning points.

Key terms

End behavior

The trend of a function's output values as x approaches positive or negative infinity.

Leading term

The term of a polynomial with the highest degree; it determines end behavior for large |x|.

Even-degree polynomial end behavior

Both ends go in the same direction: up if leading coefficient is positive, down if negative.

Odd-degree polynomial end behavior

Ends go in opposite directions: right end matches sign of leading coefficient, left end is opposite.

Why lower-degree terms don't affect end behavior

As |x| → ∞, higher-degree terms grow much faster, making lower-degree terms negligible in comparison.

Try an AP-style question

What is the end behavior of f(x) = −2x⁵ + x?

1. As x → ±∞, f → ∞
2. As x → ∞, f → ∞; as x → −∞, f → −∞
3. As x → ±∞, f → −∞
4. As x → ∞, f → −∞; as x → −∞, f → ∞