1. 2.2 Polynomial Functions of Higher Degree

2. Features of the Graphs
Graphs of polynomial functions are continuous and have smooth, rounded turns.

3. Degree of a polynomial
The degree of a polynomial is the highest value of an exponent of the variable.

4. Even / Odd behavior
Even
Odd

5. Left and Right Hand Behavior
Basic form of a polynomial f(x) = xn, where n is a positive integer.
If n is even, the left and right behaviors are …
The same
If n is odd, the left and right behaviors are …
opposite
As n approaches infinity, the graph gets flatter at the origin

6. Even / Odd behavior
Odd
f(x) = x2 + 2
Even

7. Even
Even / Odd behavior
Odd
2. f(x) = x6-3
Even

8. Even
Even / Odd behavior
Odd
3. f(x) = x3-2
Odd

9. Even
Even / Odd behavior
Odd
4. f(x) = x7+1
Odd

10. Even
Even / Odd behavior
Odd
5. f(x) = -x3-2
Odd

11. Leading Coefficient Test
f(x) = axn + bx n-1 + …+ cx + d
When n is odd and a > 0, falls left rises right.
a < 0, rises left falls right.
When n is even and a > 0, rises left and right.
a < 0, falls left and right.

12. Number of turns Degree of polynomial “n”
Max # of turns for the graph is n-1

13. Describe the left and right hand behavior and maximum number of turns of the graph of the given functions:
f(x) = 7x3 + 2
f(x) = 2x4 – 3x + 5
Falls left and rises right (2 turns max)
Rises left and right (3 turns max)

14. Describe the left and right hand behavior of the graph of the given functions and the maximum number of turns:
3. f(x) = -5x5 + 9
4. f(x) = -2x4 + 16x – 12
3. Rises left and falls right (4 turns max)
4. Falls left and right (3 turns max)

15. HW/Practice
Pg 227 (1-8)
p. 228 #13-20

16. Zeros of Polynomial Functions
What are three other terms for zeros?
Roots, Solutions, x – intercepts
For a polynomial function, f, of degree n…
The graph of f has at most n-1 turns
The function f has at most n real zeros.

17. Real Zeros Let f be a polynomial function and a € R
x = a is a zero of f
x = a is a solution of f(x) = 0
(x – a) is a factor of f(x).
(a, 0) is an x-intercept of the graph of f.

18. Find the zeros of the given function
1. f(x) = 2x3 + 8x2 – 42x
2. f(x) = x4 + x3 – 6x2

19. Now we have a better idea about the graph of these functions
1. f(x) = 2x3 + 8x2 – 42x f(x) = x4 + x3 – 6x2

20. Multiplicity
The number of times the factor appears is its multiplicity.
Factors 3, 4, and 7 (multiplicity 2), implies that there are 2 factors of 7
(x-3)(x-4)(x-7)(x-7)
Generally this means the graph will be tangent to this zero rather than crossing it.

21. Write a polynomial function with the following solutions:
3.) 4, -5, 0
4.) 1, 1, -3

22. HW/Practice
p. 228 #27-41 odd …when solving find exact values (no decimals); 49, 51, 53