2.2 Polynomial Functions of Higher Degree

Graph of Polynomial Functions:

The graph of a polynomial function is continuous, meaning it has no breaks. The graph also only has smooth, rounded turns, polynomial functions cannot have sharp turns.

A.

Continuous Graph

The Leading Coefficient Test:

Whether the graph of a polynomial eventually rises or falls can be determined by the funtion's degree (even or odd) and by its leading coefficient.

When n is odd:

the graph falls to the left and rises to the right.

the graph rises to the left and falls to the right.

When n is even:

the graph rises to the left and right.

the graph falls to the left and right.

Zeros of Polynomial Functions:

It can be shown that for a polynomial function f of degree n, the following statements are true.

1. The graph of f has at most n real zeros.

2. The function f has at most n-1 relative extrema (relative minimums or maximums).

Real Zeros of Polynomial Functions:

1. x=a is a zero of the function f

2. x=a is a solution of the polynomial equation f(x)=0

3. (x-a) is a factor of the polynomial f(x).

4. (a,0) is an x-intercept of the graph of f.

Intermediate Value Theorem:

Let a and b be real numbers such that a < b. If f is a polynomial function such that f(a)f(b), then in the interval [a,b], f takes on every falue between f(a) and f(b).