Polynomial Functions and End Behavior
Identify degree, leading coefficient, and end behavior of polynomial functions. Understand why odd-degree polynomials have opposite end behaviors and even-degree polynomials have matching end behaviors.
A lesson of Algebra 2, in Polynomial Functions, about 10 minutes.
By the end of this lesson, you will be able to classify polynomial functions by degree and leading coefficient, describe end behavior using limit notation and arrow notation, and sketch the general shape of polynomial graphs.
The Big Idea
A polynomial's long-run behavior — what happens to $y$ as $x \to \pm\infty$ — is determined entirely by its leading term. Everything else (middle terms, constants) becomes negligible at extreme $x$-values. Knowing the degree and sign of the leading coefficient tells you the "tails" of the graph instantly.
Leading Term Dominance
For large $|x|$, a polynomial behaves like its leading term $a_n x^n$.
$$f(x) = a_n x^n + a_{n-1}x^{n-1} + \cdots + a_0 \approx a_n x^n \quad \text{as } x \to \pm\infty$$
End Behavior Rules
| Degree | Leading coeff. | Left tail ($x \to -\infty$) | Right tail ($x \to +\infty$) |
|---|---|---|---|
| Even | Positive ($a > 0$) | $y \to +\infty$ | $y \to +\infty$ |
| Even | Negative ($a < 0$) | $y \to -\infty$ | $y \to -\infty$ |
| Odd | Positive ($a > 0$) | $y \to -\infty$ | $y \to +\infty$ |
| Odd | Negative ($a < 0$) | $y \to +\infty$ | $y \to -\infty$ |
Memory trick: Even degree → both tails go the same direction. Odd degree → opposite directions. Positive leading coeff. → right tail goes up.
Zeros, Multiplicity, and Graph Behavior
A zero of multiplicity $m$ at $x = r$ means $(x - r)^m$ is a factor.
- Odd multiplicity: graph crosses the $x$-axis at $x = r$
- Even multiplicity: graph touches (bounces off) the $x$-axis at $x = r$
Example 1 — End Behavior
$f(x) = -3x^4 + 5x^2 - 1$
Degree: 4 (even); leading coeff.: $-3$ (negative)
Both tails: $y \to -\infty$
As $x \to -\infty$: $y \to -\infty$; as $x \to +\infty$: $y \to -\infty$