SECTION 4.1 Polynomial Functions 197 Need to Review? Local maximum and local minimum are discussed in Section 2.3, pp. 90–91. We can use a table to further analyze the graph. See Table 3. The sign of f x( ) is the same on each side of x 1, = − and the graph of f just touches the x -axis at x 1 = − (a zero of even multiplicity). The sign of f x( ) changes from one side of x 2 = to the other, and the graph of f crosses the x -axis at x 2 = (a zero of odd multiplicity). These observations suggest the following result: If r Is a Real Zero of Even Multiplicity • Numerically: The sign of f x( ) does not change from one side to the other side of r. • Graphically: The graph of f touches the x -axis at r. If r Is a Real Zero of Odd Multiplicity • Numerically: The sign of f x( ) changes from one side to the other side of r. • Graphically: The graph of f crosses the x -axis at r. Now Work PROBLEM 61(b) Turning Points Points on the graph where the graph changes from an increasing function to a decreasing function, or vice versa, are called turning points .* Each turning point yields a local maximum or a local minimum. Consider the function f x x x x x 1 – 2 – 3 – 2. 2 3 ( ) ( ) ( ) = + = The graph is shown in Figure 9(a) using a TI-84 Plus CE graphing calculator and in Figure 9(b) using Desmos. The graph of f has turning points at 1, 0 ( ) − andat 1, 4 . ( ) − Note: To find the turning points using a TI-84 Plus graphing calculator, use the MAXIMUM or MINIMUM feature as shown in Section 1.3. To find the turning points using Desmos, click on the graph near the turning point. Exploration Graph Y x Y x x , , 1 3 2 3 = = − and Y x x3 4. 3 3 2 = + + How many turning points do you see? Graph Y x Y x x , 4 3 , 1 4 2 4 3 = = − and Y x x2 . 3 4 2 = − How many turning points do you see? How does the number of turning points compare to the degree? THEOREM Turning Points • If f is a polynomial function of degree n, then the graph of f has at most n 1 − turning points. • If the graph of a polynomial function f has n 1 − turning points, then the degree of f is at least n. The following theorem from calculus supplies the answer to the question posed in the Exploration. Table 3 (a) TI-84 Plus CE 8 28 23 3 (b) Desmos Figure 9
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