363 3.4 Polynomial Functions: Graphs, Applications, and Models End Behavior of Graphs of Polynomial Functions Suppose that axn is the dominating term of a polynomial function ƒ of odd degree. 1. If a 70, then as x S∞, ƒ1x2 S∞, and as x S-∞, ƒ1x2 S-∞. Therefore, the end behavior of the graph is of the type shown in Figure 28(a). We symbolize it as . 2. If a 60, then as x S∞, ƒ1x2 S-∞, and as x S-∞, ƒ1x2 S∞. Therefore, the end behavior of the graph is of the type shown in Figure 28(b). We symbolize it as . Suppose that axn is the dominating term of a polynomial function ƒ of even degree. 1. If a 70, then as x S∞, ƒ1x2 S∞. Therefore, the end behavior of the graph is of the type shown in Figure 29(a). We symbolize it as . 2. If a 60, then as x S∞, ƒ1x2 S-∞. Therefore, the end behavior of the graph is of the type shown in Figure 29(b). We symbolize it as . Figure 29 a > 0 n even (a) a < 0 n even (b) a < 0 n odd (b) a > 0 n odd (a) Figure 28 EXAMPLE 2 Determining End Behavior The graphs of the polynomial functions defined as follows are shown in A–D. ƒ1x2 = x4 - x2 + 5x - 4, g1x2 = -x6 + x2 - 3x - 4, h1x2 = 3x3 - x2 + 2x - 4, and k1x2 = -x7 + x - 4 Based on the discussion of end behavior, match each function with its graph. C. –20 x y 50 –2 2 0 A. –2 x y 30 0 2 –60 B. –2 x y 30 0 2 –30 D. –40 x y 40 –2 2 0 SOLUTION • Function ƒ has even degree and a dominating term with positive leading coefficient, as in C. • Function g has even degree and a dominating term with negative leading coefficient, as in A. • Function h has odd degree and a dominating term with positive coefficient, as in B. • Function k has odd degree and a dominating term with negative coefficient, as in D. S Now Try Exercises 21, 23, 25, and 27.
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