1. Section 2.2 Polynomial Functions Of Higher Degree

2. Polynomial Functions of Higher Degree If the power is odd: If the coefficient is positive, the graph falls to the left and rises to the right If the coefficient is negative, the graph falls to the right and rises to the left The Leading Coefficient Test

3. Polynomial Functions of Higher Degree Describe the right-hand and left-hand behavior of the graph of the polynomial function. Try this Rises to the left Falls to the right

4. Polynomial Functions of Higher Degree If the power is even: If the leading coefficient is positive, the graph rises to the left and right. If the leading coefficient is negative, the graph falls to the left and right. The Leading Coefficient Test (cont.):

5. Polynomial Functions of Higher Degree Describe the right-hand and left-hand behavior of the graph of the polynomial function. Try this: Rises to the left Rises to the right

6. Polynomial Functions of Higher Degree Describe the right-hand and left-hand behavior of the graph of the polynomial function. Try this Rises to the left Falls to the right

7. Polynomial Functions of Higher Degree Describe the right-hand and left-hand behavior of the graph of the polynomial function. Try this Rises to the left Rises to the right

8. Polynomial Functions of Higher Degree Describe the right-hand and left-hand behavior of the graph of the polynomial function. Try this Falls to the left Rises to the right

9. Polynomial Functions of Higher Degree For a polynomial function f with degree n A function has at most n real zeros A graph has at most n-1 relative min or max Zeros of Polynomial Function Example: For it has at most 3 real zeros it has at most 2 relative minimas or maximas

10. Polynomial Functions of Higher Degree Real Zeros of Polynomial Functions If f is a polynomial function and a is a real number, the following statements are equivalent. 1. x = a is a zero of the function f. 2. x = a is a solution of the polynomial equation f(x) = 0. 3. (x – a) is a factor of the polynomial f(x). 4. (a, 0) is an x-intercept of the graph of f. Example: Find all real zeros of Zeros are 0, -1, and 2.

11. Polynomial Functions of Higher Degree Find a polynomial function with the following zeros. Zeros are Work backwards -

12. Polynomial Functions of Higher Degree Try these. 1. Find all real zeros for 2. Find the polynomial function given the zeros are -3, 0, 3, and 4. Zeros are 0, -2, -1, 1, and 2. f(x) = x 4 – 4x 3 – 9x 2 + 36x

13. Polynomial Functions of Higher Degree The Intermediate Value Theorem Let a and b be real numbers such that a < b. If f is a polynomial function such that f(a)  f(b), then in the interval [a, b], f takes on every value between f(a) and f(b). What does this mean? When finding the zeros of a function, if you can find a value x = b which is positive, and then find another value x = b which is negative, then you can conclude that the function has at least one zero between these two values. Example: Given the polynomial function If the function is evaluated at –2, the result is if the function is evaluated at –1, the result is then this tells one that there must be a zero between the interval [-2, -1]., a negative and, a positive,

14. Polynomial Functions of Higher Degree The graph verifies that there is a zero between the interval (-2, -1).

15. Polynomial Functions of Higher Degree Try this. Find three intervals of length 1 in which the polynomial function is guaranteed to have a zero. [Note: “… intervals of length 1 means finding consecutive integers between which one knows a zero occurs.] x -3 -2 0 1 2 3 f(x) = 12x 3 – 32x 2 + 3x + 5 f(x)f(x) f(-3) = 12(-3) 3 – 32(-3) 2 + 3(-3) +5 f(-2) = 12(-2) 3 – 32(-2) 2 + 3(-2) +5 f(-1) = 12(-1) 3 – 32(-1) 2 + 3(-1) +5 f(0) = 12(0) 3 – 32(0) 2 + 3(0) +5 f(1) = 12(1) 3 – 32(1) 2 + 3(1) +5 f(2) = 12(2) 3 – 32(2) 2 + 3(2) +5 f(3) = 12(3) 3 – 32(3) 2 + 3(3) +5 - 616 - 225 - 42 5 - 12 -21 50 Graph Intervals are (-1, 0), (0, 1), and (2, 3).

16. Polynomial Functions of Higher Degree What you should know: 1.How to use the Leading Coefficient Test to determine the end behavior of graphs of polynomial functions. 2.How to determine the zeros of polynomial functions and write a polynomial function knowing the zeros. 3.How to use the Intermediate Value Theorem to help locate the zeros of polynomial functions.