365 3.4 Polynomial Functions: Graphs, Applications, and Models Begin sketching at either end of the graph with the appropriate end behavior, and draw a smooth curve that crosses the x-axis at each zero, has a turning point between successive zeros, and passes through the y-intercept as shown in Figure 31. Additional points may be used to verify whether the graph is above or below the x-axis between the zeros and to add detail to the sketch of the graph. The zeros divide the x-axis into four intervals: 1-∞, -22, a-2, - 3 2b, a3 2 , 1b, and 11, ∞2. Select an x-value as a test value in each interval, and substitute it into the equation for ƒ1x2 to determine additional points on the graph. A typical selection of test values and the results of the tests are shown in the table. x y 6 –6 f(x) = 2x3 + 5x2 – x – 6 f(x) = (x – 1)(2x + 3)(x + 2) 0 1 2 –2 –1 Figure 31 Interval Test Value x Value of ƒ1x2 Sign of ƒ1x2 Graph Above or Below x-Axis 1-∞, -22 -3 -12 Negative Below A -2, -3 2B - 7 4 11 32 Positive Above A -3 2, 1B 0 -6 Negative Below 11, ∞2 2 28 Positive Above S Now Try Exercise 29. We emphasize the important relationships among the following concepts. • the x-intercepts of the graph of y = ƒ1x2 • the zeros of the function ƒ • the solutions of the equation ƒ1x2 = 0 • the factors of ƒ1x2 For instance, look again at the graph of the function in Example 3. ƒ1x2 = 2x3 + 5x2 - x - 6 ƒ1x2 = 1x - 1212x + 321x + 22 Factored form It has x-intercepts 11, 02, A -3 2, 0B, and 1-2, 0 as shown in Figure 31. Because 1, -3 2, and -2 are the x-values where the function is 0, they are the zeros of ƒ. Also, 1, -3 2, and -2 are the solutions of the polynomial equation 2x3 + 5x2 - x - 6 = 0. This discussion is summarized as follows. Relationships among x-Intercepts, Zeros, Solutions, and Factors If ƒ is a polynomial function and 1c, 02 is an x-intercept of the graph of y = ƒ1x2, then c is a zero of ƒ, c is a solution of ƒ1x2 =0, and x −c is a factor of ƒ1x2.
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