7. 2 Polynomial Functions and Their Graphs Objectives: Identify and describe the important features of the graph of polynomial function. Use a polynomial function to model real-world data. Use a polynomial function to model real-world data. Standard: S. Analyze properties and relationships of polynomial functions.

Graphs of Polynomial Functions When a function rises and then falls over an interval from left to right, the function has a local maximum If the function falls and then rises over an interval from left to right, it has a local minimum.

Ex 1. P(x) = x 3 + 3x 2 – x – 3 Maxima: (-2.2, 3.1) Minima: (0.2, -3.1) The function increases from __________ it decreases from ______________.

Ex 2. P(x) = -x 3 + 2x 2 + x + 4 Maxima: (1.5, 6.6) Minima: (-0.2, 3.9) The function decreases from __________ and it increases from _______________________

Polynomial functions are one type of continuous functions. The graph of a continuous function is unbroken. The graph of a discontinuous function has breaks or holes in it. Discontinuous Functions Continuous Functions

Every polynomial function y = P(x) is continuous for all values of x. What happens to a polynomial function as its domain values get very small and very large is called the end behavior of a polynomial function. If a polynomial is written in standard form, f(x) = a n x n + a n-1 x n-1 + … + a 1 x + a 0, the leading coefficient is a n. That is, the leading coefficient is the coefficient of the term of greatest degree in the polynomial. The end behavior of a polynomial function depends on the sign of its leading coefficient and whether the degree of the polynomial is odd or even.

Ex 4. Describe the end behavior of each function. (See bottom of pg. 435.) a.P(x) = -x 3 + x 2 + 3x – 1 Degree 3 – Odd – Rises on the Left & Falls on the Right b. Q(x) = -x + 3 – x 4 + 3x 2 + x 3 Degree 4 – Even – Falls on the Left & the Right c. P(x) = x 3 – 2x 2 – 5x + 3 Degree 3 – Odd – Falls on the Left & Rises on the Right

(See key stroke guide at the end of the chapter)

Ex. 6 Find a quartic regression model for the data in the table below. X Y Y = -.10x x 3 – 26.16x x –

Writing Activities

Review of Polynomial Functions and Their Graphs

Homework Integrated Algebra II- Section 7.2 Level A Academic Algebra II- Section 7.2 Level B