361 3.4 Polynomial Functions: Graphs, Applications, and Models A polynomial function of even degree has a range of the form 1-∞, k4 or 3k, ∞2, for some real number k. Figure 26 shows two typical graphs. The graphs in Figure 25 suggest that for every polynomial function ƒ of odd degree there is at least one real value of x that satisfies ƒ1x2 = 0. The real zeros correspond to the x-intercepts of the graph and can be determined by inspecting the factored form of each polynomial. x 1 0 –3 –2 –6 –12 6 y f(x) = 2x3 + 8x2 + 2x – 12 f(x) = 2(x – 1)(x + 2)(x + 3) Three real zeros x 2 0 –4 6 2 y f(x) = –x3 + 2x2 – x + 2 f(x) = –(x – 2)(x – i)(x + i) One real zero (b) Figure 25 x 1 –2 0 –4 8 2 y f(x) = x5 + 4x4 + x3 – 10x2 – 4x + 8 f(x) = (x – 1)2(x + 2)3 Two real zeros (c) (a) Odd Degree Even Degree x 1 –2 –1 0 2 y f(x) = –x6 – x5 + 4x4 + 2x3 – 5x2 – x + 2 f(x) = –(x + 2)(x + 1)2(x – 1)3 The graph touches the x-axis and then turns at (–1, 0). The graph crosses the x-axis and changes shape at (1, 0). The graph crosses the x-axis at (–2, 0). k (b) Figure 26 (a) x 1 2 –2 4 y f(x) = x4 – 5x2 + 4 f(x) = (x – 1)(x + 1)(x – 2)(x + 2) Four real zeros k –1 Behavior at Zeros Figure 26(b) shows a sixth-degree polynomial function with three distinct zeros, yet the behavior of the graph at each zero is different. This behavior depends on the multiplicity of the zero as determined by the exponent on the corresponding factor. The factored form of the polynomial function ƒ1x2 is -1x + 2211x + 1221x - 123. • 1x + 22 is a factor of multiplicity 1. Therefore, the graph crosses the x-axis at 1-2, 02. • 1x + 12 is a factor of multiplicity 2. Therefore, the graph is tangent to the x-axis at 1-1, 02. This means that it touches the x-axis, then turns and changes behavior from decreasing to increasing similar to that of the squaring function ƒ1x2 = x2 at its zero. • 1x - 12 is a factor of multiplicity 3. Therefore, the graph crosses the x-axis and is tangent to the x-axis at 11, 02. This causes a change in concavity (that is, how the graph opens upward or downward) at this x-intercept with behavior similar to that of the cubing function ƒ1x2 = x3 at its zero.
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