Polynomial Functions Zeros and Graphing Section 2-2

2 Objectives I can find real zeros and use them for graphing I can determine the multiplicity of a zero and use it to help graph a polynomial I can determine the maximum number of turning points to help graph a polynomial function

3 Complex Numbers Real NumbersImaginary Numbers RationalsIrrational

4 Zeros of a Function 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.

Degree (n) The degree of a polynomial tells us: 1. End behavior –(If n is Odd) Ends in opposite directions –(if n is Even) Ends in same direction 2. Maximum number of Real Zeros (n) 3. Maximum Number of Turning Points (n-1) 5

6 Example: Real Zeros Example: Find all the real zeros and turning points of the graph of f (x) = x 4 – x 3 – 2x 2. Factor completely: f (x) = x 4 – x 3 – 2x 2 = x 2 (x + 1)(x – 2). The real zeros are x = –1, x = 0, and x = 2. These correspond to the x-intercepts (–1, 0), (0, 0) and (2, 0). The graph shows that there are three turning points. Since the degree is four, this is the maximum number possible. y x f (x) = x 4 – x 3 – 2x 2 Turning point

7 Repeated Zeros Example: Determine the multiplicity of the zeros of f (x) = (x – 2) 3 (x +1) 4. Zero Multiplicity Behavior 2 –1 3 4 odd even crosses x-axis at (2, 0) touches x-axis at (–1, 0) 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). x y

8 Multiplicity Multiplicity is how many times a solution is repeated. First find all the factors to a given polynomial. The exponents on each factor determine the multiplicity. If multiplicity is ODD, the graph crosses the solution If multiplicity is EVEN, the graph just touched or bounces off the solution

9 Multiplicity (x+2)(x-3) 2 (x+1) 3 Zeros at (-2,0) (3, 0) and (-1, 0) Crosses at (-2, 0) Touches (3, 0) Crosses (-1, 0)

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

Putting it all together 1. Find degree and LC to determine end behavior, maximum number of real zeros, and maximum number of turning points 2. Find y-intercept 3. Factor and find all zeros 4. Determine multiplicity to determine if graph crosses or touches at the zeros 5. Sketch the graph 11

12

13 Homework WS 3-5