Polynomial Functions of Higher Degree

2. Polynomial Functions of Higher Degree Section 2.2

3. Objective Identify End Behavior Recognize Continuity Find Zeros Plot Additional Points

4. Relevance Learn how to evaluate data from real world applications that fit into a quadratic model.

5. Explore – Look at the relationship between the degree & sign of the leading coefficient and the right- and left-hand behavior of the graph of the function.

6. Explore – Look at the relationship between the degree & sign of the leading coefficient and the right- and left-hand behavior of the graph of the function.

7. Explore – Look at the relationship between the degree & sign of the leading coefficient and the right- and left-hand behavior of the graph of the function.

8. Continuous Function • A function is continuous if its graph can be drawn with a pencil without lifting the pencil from the paper. Continuous Not Continuous

9. Polynomial Function • Polynomial Functions have continuous graphs with smooth rounded turns. • Written: • Example:

10. Explore using graphing CalculatorDescribe graph as S or W shaped.

11. Generalizations? • The number of turns is one less than the degree. • Even degree → “W” Shape • Odd degree → “S” Shape

12. Describe the Shape and Number of Turns.

13. Let’s explore some more….we might need to revise our generalization. • Take a look at the following graph and tell me if your conjecture is correct.

14. Lead Coefficient Test When n is odd Lead Coefficient is Positive: (an >0), the graph falls to the left and rises to the right Lead Coefficient is Negative: (an <0), the graph rises to the left and falls to the right

15. Lead Coefficient Test When n is even Lead Coefficient is Positive: (an >0), the graph rises to the left and rises to the right Lead Coefficient is Negative: (an <0), the graph falls to the left and falls to the right

16. Leading Coefficient: an a > 0 a < 0

17. Use the Leading Coeffiicent Test to describe the right-hand and left-hand behavior of the graph of each polynomial function:

18. Use the Leading Coeffiicent Test to describe the right-hand and left-hand behavior of the graph of each polynomial function:

19. Use the Leading Coeffiicent Test to describe the right-hand and left-hand behavior of the graph of each polynomial function:

20. Use the Leading Coeffiicent Test to describe the right-hand and left-hand behavior of the graph of each polynomial function:

21. A polynomial function (f) of degree n , the following are true • The function has at most n real zeros • The graph has at most (n-1) relative extrema (relative max/min)

22. Local Max / Min (in terms of y)Increasing / Decreasing (in terms of x)

23. Local Max / Min (in terms of y)Increasing / Decreasing (in terms of x)

24. Approximate any local maxima or minima to the nearest tenth.Find the intervals over which the function is increasing and decreasing.

25. Find the Zeros of the polynomial function below and sketch on the graph:

26. Find the Zeros of the polynomial function below and sketch on the graph: Multiplicity of 2 – EVEN - Touches

27. Find the Zeros of the polynomial function below and sketch on the graph:

28. Find the Zeros of the polynomial function below and sketch on the graph: NO X-INTERCEPTS!

29. Find the Zeros of the polynomial function below and sketch on the graph: