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

2. Quick Review

3. Quick Review Solutions

4. 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.

5. The Vocabulary of Polynomials

6. Cubic Functions

7. Quartic Function

8. 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

9. Limit Notation

10. Leading Term Test for Polynomial End Behavior

11. Example 3
What do you notice as you zoom out?

12. Example 4 Applying Polynomial Theory

13. Example 5 Finding the Zeros of a Polynomial Function

14. 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.

15. 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

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

17. 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.

18. 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].

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

20. Example 8

21. 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?