1. Before Find the vertex and zeros and then graph Copyright © by Houghton Mifflin Company, Inc. All rights reserved.1

2. Question to be answered: How do we graph any polynomial function?

3. Basic Characteristics of Polynomial Functions Continuous Graphs Only smooth rounded curves Leading Coefficient Test Zeros Max and Min Increasing and Decreasing Copyright © by Houghton Mifflin Company, Inc. All rights reserved.3 y x –2 2

4. Polynomial Function Copyright © by Houghton Mifflin Company, Inc. All rights reserved.4 Leading coefficient – the number in front of the highest exponent. Examples: Find the leading coefficient and degree of each polynomial function. Polynomial FunctionLeading Coefficient Degree – 2 5 1 3 14 0 Degree – the highest exponent.

5. Graphing! With your partner come up with some polynomial equations of higher degree and graph them with your calculator. Do a graph for degree 3, 4, 5, 6, 7 and 8. What patterns do you see? Copyright © by Houghton Mifflin Company, Inc. All rights reserved.5

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.6

7. End Behavior 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! Copyright © by Houghton Mifflin Company, Inc. All rights reserved.7

8. Leading Coefficient Test 4 cases Copyright © by Houghton Mifflin Company, Inc. All rights reserved.8 Even Exponent Odd Exponent Positive Negative

9. End Behavior Describe the end behavior of these functions. 1. 2. 3. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.9

10. Try some… Page 148, #1-7 odd and 13-19 odd Copyright © by Houghton Mifflin Company, Inc. All rights reserved.10

11. Zeros of a Function Copyright © by Houghton Mifflin Company, Inc. All rights reserved.11 A real number a is a zero of a function y = f (x) if and only if f (a) = 0. A polynomial function of degree n has at most n zeros. Real Zeros of Polynomial Functions If y = f (x) is a polynomial function and a is a real number then the following statements are equivalent. 1. x = a is a zero of f. 2. x = a is a solution of the polynomial equation f (x) = 0. 3. (x – a) is a factor of the polynomial f (x). 4. (a, 0) is an x-intercept of the graph of y = f (x).

12. Example: Real Zeros Copyright © by Houghton Mifflin Company, Inc. All rights reserved.12 y x –2 2 Example: Find all the real zeros of f (x) = x 4 – x 3 – 2x 2. Factor completely: f (x) = x 4 – x 3 – 2x 2 The real zeros are x =, x =, and x =. These correspond to the ________________. f (x) = x 4 – x 3 – 2x 2 (–1, 0) (0, 0) (2, 0)

13. Zeros of a Function Copyright © by Houghton Mifflin Company, Inc. All rights reserved.13 The zeros of a function are the x- values for which f(x) or y = 0. They are the x-intercepts. –2–2 x y 2 (-3.33,0)(0,0) (2.66,0)

14. Find the zero’s of the following functions Copyright © by Houghton Mifflin Company, Inc. All rights reserved.14 Solve for the zeros algebraically and then check using your calculator! 1. f(x) = 3x 2 + x – 10 2. g(x) = 3. h(x) =

15. Zero’s with Calculator Input function in y= Graph the function 2 nd Calc Zero Left bound and enter Right bound and enter Guess and enter

16. Minimum and Maximum Values Copyright © by Houghton Mifflin Company, Inc. All rights reserved.16 x y Relative minimum Relative maximum Relative Maximum and Minimum Values

17. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.17 Graphing Utility: Approximating a Relative Minimum Approximate the relative minimum of the function – 6 6 6 – 0.86 – 4.79 – 1.79 2.14 0.580.76 -3.24 -3.43 Zoom In: The approximate minimum is (0.67, –3.33).

18. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.18 Graphing Utility: Exact value of a Relative Minimum – 6 6 6 – 0.86 – 4.79 – 1.79 2.14 0.580.76 -3.24 -3.43 Press: 2 nd Calc 3:minimum Left Bound? Right Bound? Guess? Minimum: The exact value is (0.67, –3.33). Find the exact value of the relative minimum of

19. Find the exact value of the relative minimum or maximum of Copyright © by Houghton Mifflin Company, Inc. All rights reserved.19

20. Warm-up Find the zeros and max/min using your calculators Y=3x 3 - 4x 2 + x - 1 Copyright © by Houghton Mifflin Company, Inc. All rights reserved.20

21. Increasing and Decreasing Increasing – When the graph is growing or going up Decreasing – When the graph is falling or going down **Always in terms of the x values! Written in interval notation (x-values!)

22. Increasing and Decreasing Copyright © by Houghton Mifflin Company, Inc. All rights reserved.22 The graph of y = f (x): increases on ( – ∞, – 3), decreases on ( – 3, 3), increases on (3, ∞). (3, – 4) x y ( – 3, 6) –2–2 2

23. Practice Problems Copyright © by Houghton Mifflin Company, Inc. All rights reserved.23 Graph the following functions and determine the zeros, the max and mins and where it is increasing and decreasing F(x)=(x-4)(x+2) F(x)=x(x-2)(x+3) F(x)=x 3 -3x 2 -x+1

24. Working Backwards Write the equation of least degree with the following roots: 1. 3, 4, -2 2. -1, 7, 2i, -2i 3. 0 multiplicity 3, 4, 2 multiplicity 2. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.24

25. Homework pg 149 #14-22 Even, 27, 29, 35, 41 Copyright © by Houghton Mifflin Company, Inc. All rights reserved.25