SECTION 4.1 Polynomial Functions 201 x 2, = so 0 and 2 have odd multiplicities. Using the smallest degree possible (1 for odd multiplicity and 2 for even multiplicity), we write f x ax x x 2 2 2 ( ) ( ) ( ) = + − All that remains is to find the leading coefficient, a. From Figure 14 on the previous page, the point 1, 6 ( ) − is on the graph of f. a a a 6 1 1 2 1 2 6 3 2 2 ( )( ) ( ) = − − + − − = = f 1 6 ( ) − = The polynomial function f x x x x 2 2 2 2 ( ) ( ) ( ) = + − has the graph in Figure 14. Check: Graph Y x x x 2 2 2 1 2 ( ) ( ) = + − using a graphing utility to verify this result. Now Work PROBLEMS 77 AND 81 SUMMARY Graph of a Polynomial Function = + + + + ( ) ≠ − − 0 1 1 1 0 f x a x a x a x a a n n n n n • The domain of a polynomial function is the set of all real numbers. • Degree of the polynomial function f: n • y -intercept: f a 0 0 ( ) = • Graph is smooth and continuous. • Maximum number of turning points: n 1 − • At a real zero of even multiplicity: The graph of f touches the x -axis. • At a real zero of odd multiplicity: The graph of f crosses the x -axis. • Between real zeros, the graph of f is either above or below the x -axis. • End behavior: For large x , the graph of f resembles the graph of y a x . n n = ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 4.1 Assess Your Understanding 1. The intercepts of the graph of x y 9 4 36 2 + = are . (pp. 20–21) 2. Is the expression x x 4 3.6 2 3 2 − − a polynomial? If so, what is its degree? (pp. A23–A24) 3. To graph y x 4, 2 = − you would shift the graph of y x2 = a distance of units. (pp. 112–114) 4. True or False The x -intercepts of the graph of a function y f x( ) = are the real solutions of the equation f x 0. ( ) = (pp. 78–81) 5. Multiple Choice The cube function f x x3 ( ) = is . (a) even (b) odd (c) neither The graph of the cube function . (a) has no symmetry (b) is symmetric about the y -axis (c) is symmetric about the origin (d) is symmetric about the line y x = (pp. 10 0–105) Concepts and Vocabulary 6. Interactive Figure Exercise Exploring Multiplicity Open the “Multiplicity” interactive figure, which is available in the Video & Resource Library of MyLab Math (under Sullivan Interactive Figures). (a) Use the slider to set a to 1, b to 2, and c to 1.What are the zeros of f ? (b) (i) Leave a set to 1, b to 2, and c to 1. Note the behavior of the graph of the polynomial function near each zero. (ii) Leave the slider for b at 2 and c at 1. Now use the slider to change a from 1 to 2 to 3 to 4.As you do this, notice the behavior of the graph of the polynomial function at x 3. = 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure (continued)
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