Section 3.2 Polynomial Functions and Their Graphs

Graphs of Polynomials

The graph of a polynomial function is always a smooth curve; that is, it has no breaks or corners.

The simplest polynomial functions are the monomials whose graphs are shown in the Figure below.

End Behavior and the Leading Term

The end behavior of a polynomial is a description of what happens as becomes large in the positive or negative direction. To describe end behavior, we use the following notation:

For example, the monomial has the following end behavior:

The monomial has the following end behavior:

For any polynomial, the end behavior is determined by the term that contains the highest power of , because when is large, the other terms are relatively insignificant in size.

COMPARE: Here are the graphs of the monomials and

Using Zeros to Graph Polynomials

If is a polynomial function, then is called a zero of if . In other words, the zeros of are the solutions of the polynomial equation . Note that if , then the graph of has an -intercept at so the -intercepts of the graph are the zeros of the function.

The following theorem has many important consequences.

One important consequence of this theorem is that between any two successive zeros, the values of a polynomial are either all positive or all negative. This observation allows us to use the following guidelines to graph polynomial functions.

Shape of the Graph Near a Zero

If is a zero of and the corresponding factor occurs exactly times in the factorization of then we say that is a zero of multiplicity One can show that the graph of crosses the -axis at if the multiplicity is odd and does not cross the -axis if is even. Moreover, it can be shown that near the graph has the same general shape as

EXAMPLE: Graph the polynomial

Local Maxima and Minima of Polynomials

If the point is the highest point on the graph of within some viewing rectangle, then is a local maximum point on the graph and if is the lowest point on the graph of within some viewing rectangle, then is a local minimum point. The set of all local maximum and minimum points on the graph of a function is called its local extrema.

For a polynomial function the number of local extrema must be less than the degree, as the following principle indicates.

A polynomial of degree may in fact have less than local extrema. For example, has no local extrema, even though it is of degree 3.