1. 2.2(b) Notes: Polynomial Functions of Higher Degree
Date:
2.2(b) Notes: Polynomial Functions of Higher Degree
Lesson Objective: To graph polynomials and find polynomials given zeros.
CCSS:
You will need: calculator, colored pens
This is Jeopardy!!!: These are the 4 LCTs that determine the end behavior of polynomials.

2. Lesson 1: Sketching the Graph of a Polynomial
2.2(a) Recap:
1. When n is odd:
an > an < 0
Falls left, rises right Rises left, falls right
2. When n is even:
Rises left, rises right Falls left, falls right

3. Lesson 1: Sketching the Graph of a Polynomial
2.2(a) Recap:
# of Zeros: A polynomial has at most n real zeros.

4. Lesson 1: Sketching the Graph of a Polynomial
Zeros of f(x) = 2x³ – 8x:

5. Lesson 1: Sketching the Graph of a Polynomial
Zeros of f(x) = 2x³ – 8x:
Turning Points: A polynomial graph has at most n – 1 turning points, or relative maxima or minima where the graph changes direction.

6. Lesson 1: Sketching the Graph of a Polynomial
Zeros of f(x) = 2x³ – 8x:
Turning Points: A polynomial graph has at most n – 1 turning points, or relative maxima or minima where the graph changes direction.
Repeated Zeros:
If k is odd, the graph crosses the x-axis.
If k is even, the graph touches the x-axis (does NOT cross it).

7. Lesson 1: Sketching the Graph of a Polynomial
Zeros of f(x) = 2x³ – 8x:
Polynomial functions can change signs only at its zeros. Between 2 consecutive zeros, a poly-nomial must be entirely positive or entirely negative.

8. Lesson 1: Sketching the Graph of a Polynomial
Zeros of f(x) = 2x³ – 8x:
Polynomial functions can change signs only at its zeros. Between 2 consecutive zeros, a poly-nomial must be entirely positive or entirely negative.
Test Intervals: Intervals between the zeros used to test the sign

9. Lesson 1: Sketching the Graph of a Polynomial
Tips on Graphing Polynomials:
Apply LCT.
Find Zeros.
Plot a few additional points based on Turning Points.

10. Lesson 1: Sketching the Graph of a Polynomial
A. Sketch the graph of f(x) = 2x³ – 8x.
1. LCT (from 2.2(a) Lesson 2):
2. Zeros (from 2.2(a) Lesson 3):
3. Additional Points: x
f(x)

11. Lesson 1: Sketching the Graph of a Polynomial
B. Sketch the graph of f(x) = x⁴ x³ – 9 4 x².
1. LCT:
2. Zeros:
3. Additional Points: x
f(x)

12. Lesson 2: Finding Polynomial Functions
Find a polynomial function that has the given zeros.
-4, 5

13. Lesson 2: Finding Polynomial Functions
Find a polynomial function that has the given zeros.
0, 2, 5

14. Lesson 2: Finding Polynomial Functions
Find a polynomial function that has the given zeros.
-2, -1, 0, 1, 2

15. Lesson 2: Finding Polynomial Functions
Find a polynomial function that has the given zeros.
2, , 4 – 5

16. Lesson 3: Finding Polynomial Functions
Find a polynomial function of degree n that has the given zeros.
Zero(s) Degree
x = n = 2

17. Lesson 3: Finding Polynomial Functions
Find a polynomial function of degree n that has the given zeros.
Zero(s) Degree
x = -8, -4 n = 2

18. Lesson 3: Finding Polynomial Functions
Find a polynomial function of degree n that has the given zeros.
Zero(s) Degree
x = 9 n = 3

19. Lesson 3: Finding Polynomial Functions
Find a polynomial function of degree n that has the given zeros.
Zero(s) Degree
x = -4, -1, 3, 6 n = 4

20. 2.2(b): DIGI Yes or No
Apply the LCT, find the zeros, and find a few additional points within the test intervals to sketch the graph of
Find a polynomial that has and – 3 as its zeros.
Find a polynomial of degree 5 that has x = -3, 1, 5 and 6 as its zeros