Polynomial Functions of Higher Degree with Modeling 2.3 Polynomial Functions of Higher Degree with Modeling

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What you’ll learn about Graphs of Polynomial Functions End Behavior of Polynomial Functions Zeros of Polynomial Functions Intermediate Value Theorem Modeling … and why These topics are important in modeling and can be used to provide approximations to more complicated functions, as you will see if you study calculus.

The Vocabulary of Polynomials

Cubic Functions

Quartic Function

Local Extrema and Zeros of Polynomial Functions A polynomial function of degree n has at most n – 1 local extrema and at most n zeros. Example – Degree 3 polynomial has at most 2 local extrema and 3 zeros

Limit Notation

Leading Term Test for Polynomial End Behavior

Example 3 What do you notice as you zoom out?

Example 4 Applying Polynomial Theory

Example 5 Finding the Zeros of a Polynomial Function

Example 6A -Create Polynomial from Zeros If a polynomial has zeros of -2, 1, 3. Write its equation with lead coefficient of 1. Then sketch its graph.

Multiplicity of a Zero of a Polynomial Function (REPEATED Zeros) Odd multiplicity: the graph crosses the x-axis at the zero Even mutiplicity: the graph touches the x-axis at the zero

Example 6B What degree is this polynomial? What is its leading coefficient?: State the multiplicity of it’ zeros. Sketch its graphs.

Example 6C Sketching the Graph of a Factored Polynomial What degree is this polynomial? What is its leading coefficient?: State the multiplicity of it’ zeros. Sketch its graphs.

Intermediate Value Theorem If a and b are real numbers with a < b and if f is continuous on the interval [a,b], then f takes on every value between f(a) and f(b). In other words, if y0 is between f(a) and f(b), then y0=f(c) for some number c in [a,b].

Example 7 Intermediate Value Theorem Explain why a polynomial function of odd degree has at least one real zero.

Example 8

Example 9 Dixie Packaging Company has contracted to make boxes with of volume Of 484 inches cubed. Squares are to be cut from the corners of a 20” x 25” Piece of cardboard to make an open box. What size squares should be cut? How does the answer change if you ask for volume of No More than 484 inches cubed?