1. Degrees of Polynomials; End Behavior Unit 2 (2.2 Polynomial Functions)

2. Warm-Up Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

3. Objectives  Students will be able to determine the end behavior of the graph of a polynomial function  Students will be able to, algebraically and using a calculator, find the zeros of a polynomial  Students will be able to graph a polynomial function based on its end behavior and zeros Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

4.  Continuous Graphs  Only smooth rounded curves  Leading Coefficient Test  Zeros  Max and Min  Increasing and Decreasing Basic Characteristics of Polynomial Functions y x –2 2 Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

5. Graphing Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

6. What does the degree do?  All polynomials of even degree look something like  All polynomials of odd degree look something like  The higher exponents add “bumps” to the graph. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

7.  How a function acts as x gets really big or really small.  What does the function approach as x approaches infinity?  Also known as right-hand and left-hand behavior! End Behavior Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

8. Leading Coefficient Test  Used to determine the end behavior of the graph of a polynomial function  Leading coefficient – the number in front of the highest exponent  Degree – the highest exponent  Examples: Find the leading coefficient and degree of each polynomial function. Polynomial Function Leading Coefficient Degree – 2 5 1 3 14 0 Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

9. Leading Coefficient Test  4 cases Even Exponent Odd Exponent Positive Negative Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

10. End Behavior  Describe the end behavior of these functions. 1. 2. 3. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

11. Real Zeros of Polynomial Functions  A real number a is a zero of a function y = f (x) if and only if f (a) = 0.  If y = f (x) is a polynomial function and a is a real number then the following statements are equivalent.  x = a is a zero of f.  x = a is a solution of the polynomial equation f (x) = 0.  (x – a) is a factor of the polynomial f (x).  (a, 0) is an x-intercept of the graph of y = f (x). Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

12. Multiplicity Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

13. Find the zero’s of the following functions Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

14. Example: y x –2 2 f (x) = x 4 – x 3 – 2x 2 (–1, 0) (0, 0) (2, 0) Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

15. Zero’s with Calculator  TI-84 TI-84 Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

16. Closure Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

17. Homework  Textbook  page 148, #1-8 and 13-21 odd Copyright © by Houghton Mifflin Company, Inc. All rights reserved.