2. Polynomial Functions of Higher Degree
Section 2.2

3. Plot Additional Points
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 Calculator Describe graph as S or W shaped.
Function
Degree
# of U turns

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
End Behavior -
left
right
n - even
n - odd
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
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: