Warmup Determine the left and righthand behavior of

Warm-up: Determine the left and right-hand behavior of the graph of the polynomial function, then find the x-intercepts (zeros). y = x 3 + 2 x 2 – 8 x HW: page 227( 1 -8, 27 -41 ODD, 47 -55 ODD, 63 -66, 70, 79 a-d)

HW Answers: page 227(9 -11, 13 -22) 9) a. b. c. d. 10) a. b. c. d. 11) a. b. c. d. 2

HW Answers: page 227(9 -11, 13 -22) 13) Falls left Rises right 14) Rises left Rises right 15) Falls left Falls right 16) Falls left Falls right 17) Rises left Falls right 18) Falls left Rises right 19) Rises left Falls right 20) Rises left Rises right 21) Falls left Falls right 22) Rises left Falls right 3

Polynomial Functions of a Higher Degree Objective: • Use the Leading Coefficient Test to determine the end behavior of graphs of polynomial functions • Find the Zeros of a Function • Determine the number of turning points. • Use the Intermediate Value Theorem • Sketch Polynomials of a higher degree

Real Zeros of Polynomial Functions If y = f (x) is a polynomial function and a is a real number then the following statements are equivalent. 1. a is a zero of f. 2. 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 y = f (x). A turning point of a graph of a function is a point at which the graph changes from increasing to decreasing or vice versa. A polynomial function of degree n has at most n – 1 turning points and at most n zeros. 5

Example: Find all the real zeros and number of turning points of the graph of f (x) = x 4 – x 3 – 2 x 2. Factor completely: f (x) = x 4 – x 3 – 2 x 2 = x 2(x 2 – x – 2) = x 2(x + 1)(x – 2). y The real zeros are x = – 1, x = 0, and x = 2. These correspond to the x-intercepts (– 1, 0), (0, 0) and (2, 0). Leading coefficient is (+) so it rises to the right and is even so left side also rise to the left Turning point Since the degree is four, the graph shows that there are three turning points. Turning point x Turning point f (x) = x 4 – x 3 – 2 x 2

Repeated Zeros If k is the largest integer for which (x – a) k is a factor of f (x) and k > 1, then a is a repeated zero of multiplicity k. 1. If k is odd the graph of f (x) crosses the x-axis at (a, 0). 2. If k is even the graph of f (x) touches, but does not cross through, the x-axis at (a, 0). Example: Determine the multiplicity of the zeros of f (x) = (x – 2)3(x +1)4. y Zero 2 Multiplicity Behavior 3 odd crosses x-axis at (2, 0) – 1 4 even touches x-axis at (– 1, 0) 7 x

Example: Sketch the graph of f (x) = 4 x 2 – x 4. 1. Write the polynomial function in standard form: f (x) = –x 4 + 4 x 2 The leading coefficient is negative and the degree is even. falling to the right, falling towards the left 2. Find the zeros of the polynomial by factoring. f (x) = –x 4 + 4 x 2 = –x 2(x 2 – 4) = – x 2(x + 2)(x – 2) y Zeros: x = – 2, 2 multiplicity 1 x = 0 multiplicity 2 (– 2, 0) x-intercepts: (– 2, 0), (2, 0) crosses through (0, 0) touches only 8 (2, 0) x (0, 0) Example continued

Example continued: Sketch the graph of f (x) = 4 x 2 – x 4. 3. Since f (–x) = 4(–x)2 – (–x)4 = 4 x 2 – x 4 = f (x), the graph is even and symmetrical about the y-axis. 4. Plot additional points and their reflections in the y-axis: (1. 5, 3. 9) and (– 1. 5, 3. 9 ), ( 0. 5, 0. 94 ) and (– 0. 5, 0. 94) y 5. Draw the graph. (– 1. 5, 3. 9 ) (– 0. 5, 0. 94 ) (1. 5, 3. 9) (0. 5, 0. 94) x

Example: Sketch a graph showing all zeros, end behavior, and important test points.

Try: Sketch a graph showing all zeros, end behavior, and important test points.

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).

The idea behind the Intermediate Value Theorem is this: If you have one point below a line and one point above a line…. . And they are part of a continuous function . . . then there will be at least one place where the function crosses the line! Could possibly cross more than once!

Use the Intermediate Value Theorem to approximate the real zero of f(x) = x 3 – x 2 + 1 x f(x) -2 -11 (-2)3 – (-2)2 + 1 -1 -1 (-1)3 – (-1)2 + 1 0 1 1 1 2 5 Graph crosses xaxis somewhere between here! Since f(x) is a polynomial function, it is continuous everywhere Change of sign indicates the graph must cross the x-axis! By the IVT there must be some value “c” between -1 and 0 such that f(c) = 0

Polynomial Functions of a Higher Degree Summary: • Use the Leading Coefficient Test to determine the end behavior of graphs of polynomial functions • Find the Zeros of a Function • Determine the number of turning points. • Use the Intermediate Value Theorem • Sketch Polynomials of a higher degree

Sneedlegrit: Sketch the graph of f(x) = 3 x 4 – 4 x 3 HW: page 227( 1 -8, 27 -41 ODD, 47 -55 ODD, 63 -66, 70, 79 a-d)