208 CHAPTER 4 Polynomial and Rational Functions The y-intercept is f 0 90. ( ) = The x-intercepts are found by solving f x 0. ( ) = Using the Zero-Product Property, solve x x 2 0, 3 0, − = + = and x 5 0. − = The x-intercepts are 2, 3, − and 5. Step 2 Find the x- and y-intercepts of the graph of the function. Step 3 Determine the real zeros of the function and their multiplicity. Use this information to determine whether the graph crosses or touches the x-axis at each x-intercept. The real zeros of the function are 2, 3, − and 5.The zeros 2 and 5 each have multiplicity 1, so the graph of f crosses the x-axis at 2 and 5. The zero 3− has multiplicity 2, so the graph of f touches the x-axis at 3. − Step 4 Use a graphing utility to graph the function. See Figure 17 for the graph of f using Geogebra. From the graph of f shown in Figure 17, we see that f has three turning points. After clicking on the graph near each turning point, we find the turning points are 3, 0 , 0.07, 90.10 , ( ) ( ) − − and 3.82, 99.89, ( ) − each rounded to two decimal places. Step 5 Approximate the turning points of the graph. Based on the graph, the range is { } ≥− y y 99.89 or the interval 99.89, . [ ) − ∞ Step 7 From the graph, find the range of the polynomial function. Figure 18 shows a graph of f using the information in Steps 1 through 5. Step 6 Use the information in Steps 1 to 5 to draw a complete graph of the function by hand. Figure 17 f x x x x 2 3 5 2 ( ) ( )( ) ( ) = − + − Figure 18 f x x x x 2 3 5 2 ( ) ( )( ) ( ) = − + − y 400 200 –100 (–3, 0) (3.82, –99.89) (0, 90) (–0.07, 90.10) (2, 0) (5, 0) x 7 –6 End behavior: Resembles y = x4 End behavior: Resembles y = x4 Now Work PROBLEM 5 For polynomial functions that have noninteger coefficients and for polynomials that are not easily factored, we use a graphing utility early in the analysis. This is because the information that can be obtained from algebraic analysis is limited. Based on the graph, f is increasing on the intervals 3, 0.07 [ ] − − and 3.82, . [ )∞ Also, f is decreasing on the intervals , 3 ( ] −∞ − and 0.07, 3.82 . [ ] − Step 8 Use the graph to determine where the function is increasing and where it is decreasing.
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