Determine the end behavior of a polynomial function.

Use multiplicities of zeros and end behavior to graph a polynomial function.

End behavior, zeros, and polynomial graphs

Consider the polynomial , where the coefficients , , , are real numbers with .

The general shape of the graph of can be easily determined from the polynomial's factored form.

The graph crosses the -axis at every zero of multiplicity 1.

The graph flattens and crosses the -axis at every zero of odd multiplicity.

The graph flattens, touches, and bounces off the -axis at every zero of even multiplicity.

The end behavior (the behavior as ) of the graph of is identical to that of , where is the degree and is the leading coefficient:

even and positive up left and up right

even and negative down left and down right

odd and positive down left and up right

odd and negative up left and down right

Example

Here is the graph of .

Discuss the zeros and the features of the graph of .

Imagine you were given the graph of a polynomial function, but you were not given the polynomial itself. Could you make good predictions about the zeros and their multiplicities?