1. 3.1 Polynomial Functions and their Graphs

2. f(x) = 3x5 + 6x4 – 2x3 + 7x - 6

3. The graph of a polynomial function is always a smooth curve – no breaks, holes, or corners.
Recall section 2.4 about stretches, shifts, etc.
EX: Graph y = x4 and y = (x – 2)4

4. Polynomial?

5. End Behavior & the Leading Coefficient
Tells what happens as x becomes large in the positive or negative direction.
x gets bigger
x gets smaller

6. Always opposite Always the same
End behavior is determined by the term that contains the highest power of x. (leading coefficient)
Always opposite
Always the same

7. EX Determine the end behavior:
f(x) = -3x3 + 20x2 + 60x + 2
f(x) = -7x4 + 5x3 + 4x - 7

8. Find factors Find zeros
Using Zeros to Graph
If P is a polynomial and c is a real number, then the following are equivalent:
‘c’ is a zero if P(c) = 0
X = c is an x-intercept of the graph of P
X = c is a solution of the equation P(x) = 0
X – c is a factor of P(x)
Find factors Find zeros

9. Find the zeros by factoring:
P(x) = x2 + x - 6

10. If you have positive and negative y-values, your polynomial has to have at least one zero.

11. Table must include:
Zeros
A point in between each zero
Y-intercept
Need to know end behavior

12. P(x) = (x + 2)(x – 1)(x – 3)
Find the zeros and graph. X
F(x) -2 1 3

13. Multiplicity
m is the exponent
Passes through
Bounces off

14. P(x) = (x + 2)(x – 1)(x – 3)2
Find the zeros and graph. X
F(x) -2 1 3

15. Phone number
Key in the 1st three digits of your phone number (not the area code)
Multiply by 80
Add 1
Multiply by 250
Add the last four digits of your phone #
Add the last four digits of your phone # AGAIN
Subtract 250
Divide by 2
SEE YOUR PHONE NUMBER ON YOUR CALCULATOR???

16. P(x) = 3x4 – 5x3 – 12x2 Find the zeros, y-int, and graph. -4/3 3 X
F(x)
-4/3 3

17. P(x) = x3 + 3x2 – 9x - 27 Find the zeros, y-int, and graph. -3 3 X
F(x) -3 3

18. The number of local extrema must be less than the degree.

19. P(x) = 3x4 – 5x3 – 12x2 Find the zeros, y-int, and graph. -4/3 3 X
F(x)
-4/3 3

20. P(x) = (x + 2)(x – 1)(x – 3)
Find the zeros and graph. X
F(x) -2 1 3

21. Homework pg 262 #1, 3, , 13, 15, 16, 21, odd