206 CHAPTER 4 Polynomial and Rational Functions 1 Analyze the Graph of a Polynomial Function Figure 15 Y x x 2 1 3 1 2 ( )( ) = + − 50 275 22 6 Step 4 Use a graphing utility to graph the function. See Figure 15 for the graph of f using a TI-84 Plus CE. Step 5 Approximate the turning points of the graph. From the graph of f shown in Figure 15, we see that f has two turning points. Using MAXIMUM, one turning point is at 0.67, 12.70 , ( ) rounded to two decimal places. Using MINIMUM, the other turning point is at 3, 0 . ( ) How to Analyze the Graph of a Polynomial Function Analyze the graph of the polynomial function f x x x 2 1 3 . 2 ( ) ( )( ) = + − EXAMPLE 1 Step-by-Step Solution Step 1 Determine the end behavior of the graph of the function. Multiply out x 3 . 2 ( ) − Multiply. Combine like terms. f x x x x x x x x x x x x x x 2 1 3 2 1 6 9 2 12 18 6 9 2 11 12 9 2 2 3 2 2 3 2 ( ) ( ) ( )( ) ( ) = + − = + − + = − + + − + = − + + The polynomial function f is of degree 3. The graph of f behaves like y x2 3 = for large values of x . Step 2 Find the x- and y-intercepts of the graph of the function. The y-intercept is f 0 9. ( ) = To find the x-intercepts, solve f x 0. ( ) = ( ) ( )( ) ( ) = + − = + = − = = − − = f x x x x x x x 0 2 1 3 0 2 1 0 or 3 0 1 2 or 3 0 2 2 = x 3 The x-intercepts are 1 2 − and 3. 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 f are 1 2 − and 3. The zero 1 2 − is a real zero of multiplicity 1, so the graph of f crosses the x-axis at x 1 2 . = − The zero 3 is a real zero of multiplicity 2, so the graph of f touches the x-axis at x 3. =
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