Polynomial Functions: Division and End Behavior Analysis

1. POLYNOMIALS – DIVIDINGEX – Long division (4x³ -15x² +11x -6) / (x-3) - 3x + 2 4x² R 0 x - 3 4x³ - 15x² + 11x - 6 - ( 4x³ - 12x² ) -3x² +11x - ( -3x² + 9x ) 2x - 6 - ( 2x - 6 ) 0

3. A function is odd if the degree which is greatest is odd and even if the degree which is greatest is even Example: even Example: odd

5. End Behavior • Behavior of the graph as x approaches positive infinity (+∞) or negative infinity (-∞) • The expression x→+∞ : as x approaches positive infinity • The expression x→-∞ : as x approaches negative infinity

6. End Behavior of Graphs of Linear Equations f(x) = x f(x) = -x f(x)→-∞ as x→+∞ f(x)→+∞ as x→-∞ f(x)→+∞ as x→+∞ f(x)→-∞ as x→-∞

7. End Behavior of Graphs of Quadratic Equations f(x) = x² f(x) = -x² f(x)→-∞ as x→+∞ f(x)→-∞ as x→-∞ f(x)→+∞ as x→+∞ f(x)→+∞ as x→-∞

8. End Behavior… • Four Possibilities • Up on both ends • Down on both ends • Up on the right & Down on the left • Up on the left & Down on the right

9. End Behavior… Four Prototypes: • Up on both ends… y = x2 • Down on both ends… y = -x2 • Up on the right & Down on the left… y = x3 • Up on the left & Down on the right… y = -x3

10. End Behavior… Notation: • Up on both ends… • Down on both ends… • Up on the right & Down on the left… • Up on the left & Down on the right…

11. Investigating Graphs of Polynomial Functions • Use a Graphing Calculator to grph each function then analyze the functions end behavior by filling in this statement: f(x)→__∞ as x→+∞ and f(x)→__∞ as x→-∞ a. f(x) = x³ c. f(x) = x4 e. f(x) = x5 g. f(x) = x6 b. f(x) = -x³ d. f(x) = -x4 f. f(x) = -x5 h. f(x) = -x6

12. Investigating Graphs of Polynomial Functions • How does the sign of the leading coefficient affect the behavior of the polynomial function graph as x→+∞? • How is the behavior of a polynomial functions graph as x→+∞ related to its behavior as x→-∞ when the functions degree is odd? When it is even?

13. Using the Leading Coefficient to Describe End Behavior: Degree is EVEN • If the degree of the polynomial is even and the leading coefficient is positive, both ends ______________. • If the degree of the polynomial is even and the leading coefficient is negative, both ends ________________.

14. Using the Leading Coefficient to Describe End Behavior: Degree is ODD • If the degree of the polynomial is odd and the leading coefficient is positive, the graph falls to the __________ and rises to the ______________. • If the degree of the polynomial is odd and the leading coefficient is negative, the graph rises to the _________ and falls to the _______________.

16. Determining End Behavior Match each function with its graph. B. A. C. D.

17. When you transform a function Inside the parentheses translates left and right Outside the parentheses translates up and down

18. Transforming Cubic functions

19. For You to do

20. Graphing Polynomial Functions f(x)= -x4 –2x³ + 2x² + 4x

21. Many correct answers For example, there are an infinite number of polynomials of degree 3 whose zeros are -4, -2, and 3. They can be expressed in the form: