1. The Graphs of Polynomials
5 November 2010

2. Even vs. Odd All polynomials are either even or odd
Describes the shape of a graph
Good for generalizing graphs
Degree determines if polynomial is even or odd

3. Even Degree is an even number Roughly U-Shaped
Both ends of the graph are either increasing or decreasing
Can open either up or down
Depends on leading coefficient
Positive leading coefficient = opens up
Negative leading coefficient = opens down

4. Leading Coefficient Positive Leading Coefficient Negative
Even, cont.
Opens Up
Opens Down
Leading Coefficient Positive
Leading Coefficient Negative

5. Odd Degree is an odd number
One end of the graph points up and the other end points down
Depends on leading coefficient
Positive leading coefficient = left end points down, right end points up
Negative leading coefficient = left end points up, right end points down

6. Leading Coefficient Positive Leading Coefficient Negative
Odd, cont.
Left Down, Right Up
Left Up, Right Down
Leading Coefficient Positive
Leading Coefficient Negative

7. Your Turn:
Decide if the graphs for problems 7-8, on page in your Precalculus textbook are even or odd.

8. Extrema The vertex points of polynomials Extrema (maximum)
the maximum or minimum points
peaks and valleys
Extrema (maximum)
Extrema (minimum)

9. Extrema, cont. Limited by the degree of a polynomial
A polynomial of degree n can have at most n – 1 extrema.
Translation: The greatest number of extrema a polynomial can have is one less than the degree of the polynomial
Or: The degree of the polynomial is 1 greater than the number of extrema

10. Extrema, cont. 5th Degree Polynomial
We can use extrema to figure out the degree of a polynomial
# of Extrema = 4
Degree of Polynomial = # of Extrema + 1

11. Your Turn:
Determine the degree of each polynomial for problems 7-8, on page in your Precalculus textbook.

12. Matching Equations to Graphs
Step 1: Determine if the equation is even or odd.
Step 2: Check if the leading coefficient is positive or negative.
Step 3: Determine the maximum number of extrema for the equation.
Step 4: Match a graph to the characteristics determined in steps 1-3.

13. Matching Equations to Graphs: y = x5 + 12x
Even or Odd?
Odd
Leading Coefficient Positive or Negative?
Positive
Degree = 5
Max Extrema = 4

14. Matching Graphs to Equations: y = 2x4 – 5x2 +2 or y = -x6 + 3?
Even or Odd?
Even
Leading Coefficient Positive or Negative?
Positive
# of Extrema = 3
Degree = 4

15. Your Turn:
Answer questions on pages in your Precalculus textbook.

16. Multiplicity
Describes how often a root or factor occurs in an equation or graph
Either even or odd
If a root occurs an even number of times, then the graph touches the x-axis at the root.
If a root occurs an odd number of times, then the graph crosses the x-axis at the root.

17. -3 2 Touch -1 1 Cross 3 y = (x + 3)2(x + 1)(x – 1)3 Zero Multiplicity
x-axis -3 2
Touch -1 1
Cross 3

18. -1 3 Cross 1 2 Touch y = (x + 1)3(x – 1)(x – 3)2 Zero Multiplicity
x-axis -1 3
Cross 1 2
Touch

19. Determining Multiplicity from Graphs (Pg. 265)
Zero
Multiplicity
x-axis -1
Even
Touch 2
Odd
Cross
3.5

20. Your Turn:
Complete problems on page 270 of your Precalculus textbook.

21. Hmwk:
Handout, problems 1-8