Concepts Examples Descartes’ Rule of Signs Let ƒ1x2 define a polynomial function with real coefficients and a nonzero constant term, with terms in descending powers of x. (a) The number of positive real zeros of ƒ either equals the number of variations in sign occurring in the coefficients of ƒ1x2, or is less than the number of variations by a positive even integer. (b) The number of negative real zeros of ƒ either equals the number of variations in sign occurring in the coefficients of ƒ1-x2, or is less than the number of variations by a positive even integer. There are three sign changes for ƒ1x2 = +3x3 - 2x2 + x - 4, 1 2 3 so there will be three or one positive real zeros. Because ƒ1-x2 = -3x3 - 2x2 - x - 4 has no sign changes, there will be no negative real zeros. The table shows the possibilities for the numbers of positive, negative, and nonreal complex zeros. Positive Negative Nonreal Complex 3 0 0 1 0 2 Graph ƒ1x2 = -1x + 224 + 1. The negative sign causes the graph to be reflected across the x-axis compared to the graph of ƒ1x2 = x4. The graph is translated to the left 2 units and up 1 unit. The function is increasing on 1-∞, -22 and decreasing on 1-2, ∞2. Determine the behavior of ƒ near its zeros, and graph. ƒ1x2 = 1x - 121x - 3221x + 123 The graph will cross the x-axis at x = 1, bounce at x = 3, and wiggle through the x-axis at x = -1. Since the dominating term is x6, the end behavior is . The y-intercept is 10, -92 because ƒ102 = -9. x y 0 1 –3 –2–1 1 2 f(x) = –(x + 2)4 + 1 x y 0 1 –1 3 –10 10 f(x) = (x–1)(x– 3)2(x+ 1)3 422 CHAPTER 3 Polynomial and Rational Functions Graphing Using Translations The graph of the function ƒ1x2 =a1x −h2n +k can be found by considering the effects of the constants a, h, and k on the graph of ƒ1x2 =axn. • When 0 a0 71, the graph is stretched vertically. • When 0 6 0 a0 61, the graph is shrunk vertically. • When a 60, the graph is reflected across the x-axis. • The graph is translated to the right h units if h 70 and to the left 0 h0 units if h 60. • The graph is translated up k units if k 70 and down 0 k 0 units if k 60. Multiplicity of a Zero The behavior of the graph of a polynomial function ƒ1x2 near a zero depends on the multiplicity of the zero. If 1x - c2n is a factor of ƒ1x2, then the graph will behave in the following manner. • For n = 1, the graph will cross the x-axis at 1c, 02. • For n even, the graph will bounce, or turn, at 1c, 02. • For n an odd integer greater than 1, the graph will wiggle through the x-axis at 1c, 02. 3.4 Polynomial Functions: Graphs, Applications, and Models
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