Polynomial Functions and Models Section 5.1 Polynomial Functions and Models 1

Polynomial Functions Three of the families of functions studied thus far: constant, linear, and quadratic, belong to a much larger group of functions called polynomials. We begin our formal study of general polynomials with a definition and some examples.

f (x)  an xn + an1 xn1 + … + a2 x2 + a1 x + a0 Polynomial Functions A polynomial function is a function of the form f (x)  an xn + an1 xn1 + … + a2 x2 + a1 x + a0 where a0, a1, . . . , an are real numbers and n  1 is a natural number. The domain of a polynomial function is ( , ).

f (x)  an xn + an1 xn1 + … + a2 x2 + a1 x + a0 Polynomial Functions Suppose f is the polynomial function f (x)  an xn + an1 xn1 + … + a2 x2 + a1 x + a0 where an  0. We say that, The natural number n is the degree of the polynomial f. The term anxn is the leading term of the polynomial f. The real number an is the leading coefficient of the polynomial f. The real number a0 is the constant term of the polynomial f. If f (x)  a0, and a0  0, we say f has degree 0. If f (x)  0, we say f has no degree. 4

Identifying Polynomial Functions Determine which of the following functions are polynomials. For those that are, state the degree.

Identifying Polynomial Functions Determine which of the following functions are polynomials. For those that are, state the degree. 6

Polynomial Functions: Example A box with no top is to be built from a 10 inch by 12 inch piece of cardboard by cutting out congruent squares from each corner of the cardboard and then folding the resulting tabs. Let x denote the length of the side of the square which is removed from each corner.

Polynomial Functions: Example A diagram representing the situation is,

Polynomial Functions: Example 1. Find the volume V of the box as a function of x. Include an appropriate applied domain. 2. Use a graphing calculator to graph y  V (x) on the domain you found in part 1 and approximate the dimensions of the box with maximum volume to two decimal places. What is the maximum volume?

Summary of the Properties of the Graphs of Polynomial Functions

Graphs of Polynomial Functions

Power Functions A power function of degree n is a function of the form f (x)  axn where a  0 is a real number and n  1 is an integer.

Power Functions: a  1, n even

Power Functions: a  1, n even

Power Functions: a  1, n even

Power Functions: a  1, n odd

Power Functions: a  1, n odd

Power Functions: a  1, n odd

Identifying the Real Zeros of a Polynomial Function and Their Multiplicity

Graphs of Polynomial Functions

Definition: Real Zero

Finding a Polynomial Function from Its Zeros Find a polynomial of degree 3 whose zeros are  4,  2, and 3. The value of the leading coefficient a is, at this point, arbitrary. The next slide shows the graph of three polynomial functions for different values of a.

Finding a Polynomial Function from Its Zeros 23

Definition: Multiplicity For the polynomial, list all zeros and their multiplicities. 2 is a zero of multiplicity 1 because the exponent on the factor x – 2 is 1. 1 is a zero of multiplicity 3 because the exponent on the factor x + 1 is 3. 3 is a zero of multiplicity 4 because the exponent on the factor x – 3 is 4.

Graphing a Polynomial Using Its x-Intercepts

Behavior Near a Zero

Example

Example y = 4(x - 2)

y = 4(x - 2)

Turning Points: Theorem

End Behavior

End Behavior: Example 34

End Behavior: Example 35

Summary

Analyze the Graph of a Polynomial Function

The polynomial is degree 3 so the graph can turn at most 2 times.

Summary: Analyzing the Graph of a Polynomial Function

The domain and the range of f are the set of all real numbers.