Polynomial Functions

We often refer to polynomial functions simply as polynomials. The following polynomial has degree 5, leading coefficient 3, and constant term -6.

The table lists some more examples of polynomials.

If a polynomial consists of just a single term, then it is called a monomial. For example, P(x) = x3 and Q(x) = -6x5 are monomials.

Graphing Basic Polynomial Functions

Examples of monomial polynomials:

Transformations of Monomials:

Graphs of Polynomial Functions: End Behavior

The graph of a polynomial function is continuous. This means that the graph has no breaks or holes.

The domain of a polynomial function is the set of all real numbers, so we can sketch only a small portion of the graph. However, for values of x outside the portion of the graph we have drawn, we can describe the behavior of the graph.

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

For example, the monomial y = x2 has the following end behavior.

y -> ∞ as x ->∞ and y ->∞ as x -> -∞

The monomial y =x3 has the following end behavior.

y -> ∞ as x -> -∞ and y ->∞ as x -> -∞

For any polynomial the end behavior is determined by the term that contains the highest power of x, because when x is large, the other terms are relatively insignificant in size. The following box shows the four possible types of end behavior, based on the highest power and the sign of its coefficient.

Check the End Behavior of a Polynomial

Another example of End Behavior of a Polynomial

 Using Zeros to Graph Polynomials

If P is a polynomial function, then c is called a zero of P if P(c) = 0. In other words, the zeros of P are the solutions of the polynomial equation P(x) = 0. Note that if P(c) = 0, then the graph of P has an x-intercept at x - c, so the x-intercepts of the graph are the zeros of the function.

Example of Using Zeros to Graph a Polynomial Function

Example of Finding Zeros and Graphing a Polynomial Function

Shape of the Graph Near a Zero

Example of Graphing a Polynomial Function Using Its Zeros

 Local Maxima and Minima of Polynomials

A point (a, f(a)) is a local maximum point on the graph and that (b, f(b)) is a local minimum point. The local maximum and minimum points on the graph of a function are called its local extrema.

 

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Practice