1. 3.1 Polynomial Functions and their Graphs

2. f(x) = 3x 5 + 6x 4 – 2x 3 + 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 = x 4 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. 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) = -3x 3 + 20x 2 + 60x + 2 f(x) = -2x 4 + 5x 3 + 4x - 7

8. 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) = x 2 + 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 Zeros A point in between each zero A point in between each zero Y-intercept Y-intercept Need to know end behavior

12. P(x) = (x + 2)(x – 1)(x – 3) Find the zeros and graph. XF(x) -20 10 30

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. XF(x) -20 10 30

15. P(x) = 3x 4 – 5x 3 – 12x 2 XF(x) -4/30 00 30 Find the zeros, y-int, and graph.

16. P(x) = x 3 + 3x 2 – 9x - 27 XF(x) -30 30 Find the zeros, y-int, and graph.

17. Local Extrema The number of local extrema must be less than the degree.

18. P(x) = 3x 4 – 5x 3 – 12x 2 XF(x) -4/30 00 30 Find the zeros, y-int, and graph.

19. P(x) = (x + 2)(x – 1)(x – 3) Find the zeros and graph. XF(x) -20 10 30

20. Homework pg 262 #1, 3, 5 - 10, 13, 15, 16, 21, 25-45 odd