362 CHAPTER 3 Polynomial and Rational Functions Figure 27 The graph is tangent to the x-axis at 1c, 02 if c is a zero of even multiplicity. The graph bounces, or turns, at c. The graph crosses and is tangent to the x-axis at 1c, 02 if c is a zero of odd multiplicity greater than 1. The graph wiggles at c. The graph crosses the x-axis at 1c, 02 if c is a zero of multiplicity 1. x c c x c x c x c x x c Figure 27 generalizes the behavior of such graphs at their zeros. Turning Points and End Behavior The graphs in Figures 25 and 26 show that polynomial functions often have turning points where the function changes from increasing to decreasing or from decreasing to increasing. Turning Points A polynomial function of degree n has at most n - 1 turning points, with at least one turning point between each pair of successive zeros. The end behavior of a polynomial graph is determined by the dominating term—that is, the term of greatest degree. A polynomial of the form ƒ1x2 = anx n + a n-1x n-1 + g+ a 0 has the same end behavior as ƒ1x2 = anx n. For example, ƒ1x2 = 2x3 + 8x2 + 2x - 12 has the same end behavior as ƒ1x2 = 2x3. It is large and positive for large positive values of x, while it is large and negative for negative values of x with large absolute value. That is, it rises to the right and falls to the left. Figure 25(a) shows that as x increases without bound, y does also. For the same graph, as x decreases without bound, y does also. As x S∞, y S∞ and as x S-∞, y S-∞. x 1 0 –3 –2 –6 –12 6 y f(x) = 2x3 + 8x2 + 2x – 12 f(x) = 2(x – 1)(x + 2)(x + 3) As x `, y ` (rises to the right) As x –`, y –` (falls to the left) Figure 25(a) (repeated) As x –`, y ` (rises to the left) As x `, y –` (falls to the right) x 2 0 –4 6 2 y f(x) = –x3 + 2x2 – x + 2 f(x) = –(x – 2)(x – i)(x + i) Figure 25(b) (repeated) The graph in Figure 25(b) has the same end behavior as ƒ1x2 = -x3. As x S∞, y S-∞ and as x S-∞, y S∞. The graph of a polynomial function with a dominating term of even degree will show end behavior in the same direction. See Figure 26. LOOKING AHEAD TO CALCULUS To find the x-coordinates of the two turning points of the graph of ƒ1x2 = 2x3 + 8x2 + 2x - 12, we can use the “maximum” and “minimum” capabilities of a graphing calculator and determine that, to the nearest thousandth, they are -0.131 and -2.535. In calculus, their exact values can be found by determining the zeros of the derivative function of ƒ1x2, ƒ′1x2 = 6x2 + 16x + 2, because the turning points occur precisely where the tangent line has slope 0. Using the quadratic formula would show that the zeros are -4 {213 3 , which agree with the calculator approximations.
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