A polynomial function is a function whose rule is a polynomial:

$$f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,$$

where $n$ is a nonnegative integer and $a_n\ne0$.

Degree and end behavior

The degree and leading coefficient strongly influence the graph for large positive and negative values of $x$. For example, an even-degree polynomial with positive leading coefficient eventually rises on both ends, while an odd-degree polynomial with positive leading coefficient falls to the left and rises to the right.

These statements describe broad end behavior; lower-degree terms shape the graph in between.

Zeros and intercepts

A zero of $f$ is a value $r$ for which

$$f(r)=0.$$

On the graph, real zeros are the $x$-coordinates where the curve meets the $x$-axis. If

$$f(x)=(x-2)(x+1),$$

then the zeros are $2$ and $-1$.

A repeated factor can change how the graph behaves at a zero. For example, $(x-1)^2$ has a zero of multiplicity two; the graph touches the axis there rather than crossing it in the simplest case.

Connecting formulas and graphs

The polynomial formula gives exact values, while its zeros, intercepts, degree and leading term reveal important geometric features. These pieces of algebra can be combined to sketch and interpret a polynomial graph without calculating every point.