200 CHAPTER 4 Polynomial and Rational Functions Identifying the Graph of a Polynomial Function Which of the graphs in Figure 13 could be the graph of f x x ax bx x5 6 4 3 2 ( ) = + + − − where a b 0, 0? > > EXAMPLE 8 Solution The y-intercept of f is f 0 6. ( ) = − We can eliminate the graph in Figure 13(a), whose y-intercept is positive. We are not able to solve f x 0 ( ) = to find the x-intercepts of f, so we move on to investigate the turning points of each graph. Since f is of degree 4, the graph of f has at most 3 turning points. Eliminate the graph in Figure 13(c) because that graph has 5 turning points. Now look at end behavior. For large values of x, the graph of f will behave like the graph of y x .4 = This eliminates the graph in Figure 13(d), whose end behavior is like the graph of y x .4 = − Only the graph in Figure 13(b) could be the graph of f x x ax bx x5 6 4 3 2 ( ) = + + − − where a b 0, 0. > > Figure 13 x y x y x y x y (a) (b) (c) (d) Finding a Polynomial Function from a Graph Find a polynomial function whose graph is shown in Figure 14 (use the smallest degree possible). EXAMPLE 9 Solution The x-intercepts are 2, − 0, and 2. Therefore, the polynomial must have the factors x 2 , ( ) + x x 0 , ( ) − = and x 2 , ( ) − respectively. There are three turning points, so the degree of the polynomial must be at least 4. The graph touches the x-axis at x 2, = − so 2− has an even multiplicity. The graph crosses the x-axis at x 0 = and Figure 14 x y 15 9 –9 –21 –15 3 (2, 0) (0, 0) (–2, 0) (–1, 6) –3 For example, if f x x x x x 2 4 7 1, 4 3 2 ( ) = − + + − + the graph of f will resemble the graph of the power function y x2 4 = − for large x . The graph of f will behave like Figure 12(b) for large x . Now Work PROBLEM 61(d)
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