Leading Coefficient Test

What can be said about the graph of a general polynomial

+ ...+ ?

If n, the degree of the polynomial, is odd then

- the range is the set of all real numbers.

- if then y starts out negative and eventually remains positive.

- if then y starts out positive and eventually remains negative.

If n, the degree of the polynomial, is even then

- the range is limited.

- the function will have a minimum value or a maximum value.

- if then y has a minimum value.

- if then y has a maximum value.

The following graphs will illustrate this summary.

Example 1

Graph

Solution .

Example 2

Graph

Solution .

Example 3

Graph

Solution .

Example 4

Graph

Solution .