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Transcript
Chapter 2 Polynomial and Rational Functions Warm Up 2.2 A rancher has 200 feet of fencing to enclose two adjacent rectangular corrals. Write the area A of the corral as a function of x. Find the dimensions that will produce the maximum area. y x x 2 2.2 Polynomial Functions of Higher Degree Objectives: Use transformations to sketch graphs of polynomial functions. Use the Leading Coefficient Test to determine the end behavior of graphs of polynomial functions. Find and use zeros of polynomial functions as sketching aids. Use the Intermediate Value Theorem to help locate zeros of polynomial functions. 3 Vocabulary Continuous Function Polynomial Function Monomial, Binomial, Trinomial Functions Leading Coefficient Test Zeros of Polynomial Functions Repeated Zero Multiplicity Intermediate Value Theorem 4 Continuous Function No breaks, holes or gaps. Continuous or not? 5 Polynomial Function Characteristics Continuous Smooth, rounded turns. No sharp corners. Polynomial or not? 6 Definition of a Polynomial Function Polynomial Function of x with degree n f (x) = anxn + an – 1 xn – 1 + … + a2x2 + a1x + a0 where: n is a non-negative integer and an, an – 1, … , a2, a1, a0 are real numbers an ≠ 0 7 Monomial Functions Simplest type of polynomial. Monomial form: f (x) = xn, where n is an integer greater than zero. n = Even Integer Graph similar to f (x) = x2 Touches x-axis at x-intercept. n = Odd Integer Graph similar to f (x) = x3 Crosses x-axis at x-intercept. As n increases, the graph gets flatter at the origin. 8 Transformations of Monomials Sketch the graph of each function. 1. f ( x) x 5 2. g ( x) x 4 1 3. h( x) x 1 4 9 Leading Coefficient Test Using the degree of the function (n) and the leading coefficient (an), we can determine the end behavior of a function. n is even n is odd an > 0 an < 0 10 Use the Leading Coefficient Test Describe the end behavior of each function. 1. f ( x) x 4 x 2. f ( x) x 4 5 x 2 4 3. f ( x) x x 3 5 11 Zeros of Polynomial Functions Zero = Root = x-intercept = Solution A number x for which f (x) = 0. For a polynomial function f of degree n: The function f has at most n real zeros. The graph of f has at most n – 1 relative extrema (relative minima or maxima). 12 Real Zeros of Polynomials If f is a polynomial function and a is a real number, then the following statements are equivalent. 1. x = a is a zero of the function. 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. 13 Finding Zeros of Polynomials Find all real zeros and relative extrema of the polynomial functions. 1. f ( x) x x 2 x 2. f ( x) 2 x 4 2 x 2 3 2 14 Warm Up 2.2.2 Use your graphing calculator to find the zeros and relative extrema of the function f ( x) 2 x 2 x 4 2 15 Repeated Zeros A repeated zero of multiplicity k occurs if a function has a factor of the form (x – a)k, where k > 1. If k is odd If k is even The graph crosses the x-axis at x = a. The graph touches (but does not cross) the x-axis at x = a. 16 Example - Multiplicity Find all the real zeros of each function and determine the multiplicity of each zero. Solve algebraically and graphically. 1. f ( x) x 2 10 x 25 2. g ( x) 2 x 3 6 x 2 17 Find a Function Given the Zeros Find a polynomial function that has the given zeros. 1. 0, 2, 5 2. 4, 2 7 , 2 7 18 Basic Curve Sketching 1. Apply the Leading Coefficient Test. 2. Find the zeros of the polynomial. 3. Plot a few additional points. 4. Draw the graph. 19 Curve Sketching Example 1 Sketch the graph f ( x) 3 x 4 4 x 3 of by hand. 20 Curve Sketching Example 2 Sketch the graph 9 f ( x) 2 x 6 x x 2 3 2 of by hand. 21 Homework 2.2 Worksheet 2.2 # 47, 49, 53, 59, 61, 65, 67, 69, 71, 85 22