Polynomial functions right- and left-hand behavior (end behavior):

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1 Lesson 2.2 Polynomial Functions For each function: a.) Graph the function on your calculator Find an appropriate window. Draw a sketch of the graph on your paper and indicate your window. b.) Identify the degree of the function c.) Identify the leading coefficient d.) Describe the left-hand behavior (what happens to y as x goes to negative infinity) and right-hand behavior (what happens to y as x goes to positive infinity) 1.) y = x 3 deg lead coefficient left right - x fall rise 2.) y = -2x 5 - x rise fall 3.) y = - 3x 2 + x ) y = -x 4 + 2x 3 5.) y = x 4 - x fall fall 4-1 fall fall 4 1 rise rise How can you determine the left - and right - hand behavior by examining the equation? Investigate more polynomial functions on your own if necessary. Look for patterns. November 10, 2014 Polynomial functions right- and left-hand behavior (end behavior): Even degree: left- and right- behavior is the same pos. leading coefficient rises to left and right neg. leading coefficient falls to left and right pos. neg Odd degree: left- and right- behavior is different pos. leading coefficient falls to left and rises to right neg. leading coefficient rises to left and falls to right pos. neg

2 By examining the graph, what can you say about the degree of the equation and the leading coefficient? November 10, 2014 Polynomial Functions Continuous functions Domain is the set of all real numbers Zeros of Polynomial functions For a polynomial function f of degree n, 1.) f has, at most, n real zeros. 2.) The graph of f has, at most, n-1 relative extrema (relative minima or maxima) Ex. The function f(x) = 2x 3-4x+5 has, at most, 3 real zeroes and at most, 2 relative extrema

3 For each function below: a.) Find all real zeros of the function algebraically. b.) Graph the function on your calculator and approximate the relative extrema. 1.) f(x) = x 3 - x 2-2x zeroes: -1, 0, 2 relative extrema: y =.631, y = ) y = -2x 4 + 2x 2 zeroes: -1, 0, 1 relative extrema: y =.5, 0 Equivalent statements: x = a is a zero of the function x = a is an x-intercept of the function (x - a) is a factor of the polynomial f(x) (a, 0) is an x-intercept of the graph of f a is a solution to f(x) = 0 Example: x = 2 is a zero or x-intercept of the function (2, 0) is an x-intercept of the function so (x-2) is a factor of the polynomial 2 is a solution to f(x) = 0

4 1.) The x-intercepts for a function are (6,0) and (-5,0) and (2,0). a.) What are the solutions to f(x)=0? b.) What are the zeros of the function? 6,-5, 2 6, -5, 2 2.) The zeros of a function are 3 and -2. a.) State the coordinates of the x-intercepts. b.) Write an equation for a function with these zeros. a) (3,0) (-2,0) b) y = (x - 3)(x + 2) y = x 2 - x - 6 Find the equation of a quadratic function with 5 as its only zero. Sketch the graph.

5 Graph these functions on your calculator. November 10, ) y = x(x - 2) 2 (x+4) 2.) y = -.5(x-4) 2 (x+4) 2 3.) y = x 2 (x - 2) 4.) y = (x-1) 3 (x+2) 2 Use the window -4 < x < 4-10 < y < 5 Identify the zeroes for each function. Note whether the graph crosses the x-axis or touches the x-axis at each zero. Do you notice a pattern? Describe any pattern you see. Sketching the graph of a polynomial function Repeated Zeros For a polynomial function, a factor of (x-a) k, k>1, yields a repeated zero, x = a of multiplicity k. (Multiplicity is the power of the factor. The number of times the factor appears in the product.) 1.) odd multiplicity - the graph crosses the x-axis at a. 2.) even multiplicity, the graph touches the x-axis at a. Sketch the graph of f(x) = 3x 4-4x 3

6 Sketch the graph of f(x) = 3x 4-4x 3 Finding a polynomial function with given zeros, multiplicities and degree. 1.) zero: 2, multiplicity: 1 zero: -3, multiplicity 2 degree: 3 2.) zero: -.5, multiplicity: 1 zero: 0, multiplicity: 2 zero: -4, multiplicity: 3 degree: 6 f(x) = (x-2)(x+3) 2 (x-2)(x The zero x = -1/2 corresponds to either 2 +6x+9) (x+1/2) or (2x+1) use (2x+1) x 3 + 4x 2-3x - 18 f(x) = x 2 (2x+1)(x+3) 3

7 Graph f(x) = x 3 - x 2-6x Use your calculator to find the zeroes of each function. f(x) = x 3-4x + 3 f(x) = 2x 4-6x 2 + 5

8 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) does not equal f(b), then, in the interval [a,b], f takes on every value between f(a) and f(b) The intermediate value theorem helps you to locate the real zeros of a polynomial function. If f(a) is positive and f(b) is negative, you know that the function has at least one real zero between a and b. Graph y = 12x 3-32x 2 + 3x + 5

9 Name x-intervals that are I unit long in which you will find an x-intercept.