SECTION 4.2 The Graph of a Polynomial Function; Models 209 How to Analyze the Graph of a Polynomial Function Using a Graphing Utility Analyze the graph of the polynomial function f x x x x 2.48 4.3155 2.484406 3 2 ( ) = + − + EXAMPLE 3 Figure 19 f x x x x 2.48 4.3155 2.484406 3 2 ( ) = + − + Step-by-Step Solution Step 1 Determine the end behavior of the graph of the function. The polynomial function f is of degree 3. The graph of f behaves like y x3 = for large values of x . See Figure 19 for the graph of f using Desmos. Step 2 Graph the function using a graphing utility. The y-intercept is f 0 2.484406. ( ) = Since it is not readily apparent how to factor f, click on the Desmos graph and determine the x-intercept is 3.79, − rounded to two decimal places. Step 3 Use the graphing utility to approximate the x- and y-intercepts of the graph. Table 5 shows values of x on each side of the x-intercept using Desmos. The points 4, 4.57 ( ) − − and 2, 13.04 , ( ) − rounded to two decimal places, are on the graph. Step 4 Use the graphing utility to create a TABLE to find points on the graph around each x-intercept. From the graph of f shown in Figure 19, we see that f has two turning points. After clicking on the graph near each turning point, we find that the turning points are 2.28, 13.36 ( ) − and 0.62, 1 , ( ) each rounded to two decimal places. Step 5 Approximate the turning points of the graph. Figure 20 shows a graph of f drawn by hand using the information in Steps 1 through 5. Step 6 Use the information in Steps 1 through 5 to draw a complete graph of the function by hand. Table 5 Figure 20 –5 2 –6 6 12 x y End behavior: Resembles y = x3 End behavior: Resembles y = x3 (–2.28, 13.36) (–2, 13.04) (–3.79, 0) (0, 2.484406) (0.63, 1) (–4, –4.57) (continued)
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