SECTION 4.2 The Graph of a Polynomial Function; Models 207 SUMMARY Analyzing the Graph of a Polynomial Function Step 1 Determine the end behavior of the graph of the function. 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. Step 4 Use a graphing utility to graph the function. Step 5 Approximate the turning points of the graph. Step 6 Use the information in Steps 1 through 5 to draw a complete graph of the function by hand. Step 7 From the graph, find the range of the polynomial function. Step 8 Use the graph to determine where the function is increasing and where it is decreasing. Step 7 From the graph, find the range of the polynomial function. The range of f is the set of all real numbers. Step 6 Use the information in Steps 1 to 5 to draw a complete graph of the function by hand. Figure 16 shows a graph of f using the information in Steps 1 through 5. Figure 16 f x x x 2 1 3 2 ( ) ( )( ) = + − 230 220 240 x y 22 21 2 1 10 210 20 40 30 4 5 3 (0, 9) (0.67, 12.70) (3, 0) 1 – 2 , 0 ( ) 2 End behavior: Resembles y 5 2x3 End behavior: Resembles y 5 2x3 Step 8 Use the graph to determine where the function is increasing and where it is decreasing. Based on the graph, f is increasing on the intervals , 0.67 ( ] −∞ and 3, . [ )∞ Also, f is decreasing on the interval 0.67, 3 . [ ] How to Analyze the Graph of a Polynomial Function Analyze the graph of the polynomial function f x x x x 2 3 5 . 2 ( ) ( )( ) ( ) = − + − EXAMPLE 2 f x x x x x x x x x x x x 7 10 3 7 10 6 9 23 3 90 2 2 2 2 4 3 2 ( ) ( )( ) ( ) ( ) = − + + = − + + + = − − − + The polynomial function is of degree 4. The graph of f behaves like y x4 = for large values of x . So, f x( ) →∞ as x →−∞ and as x . →∞ Step-by-Step Solution Step 1 Determine the end behavior of the graph of the function. Multiply x x 2 5 . ( )( ) − − Multiply x 3 . 2 ( ) + Multiply. (continued)
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