238 CHAPTER 4 Polynomial and Rational Functions Now Work PROBLEMS 35(a) AND 35(b) Figure 37 x y 22 3 Replace x by x 2 2; shift right 2 units. 1 Add 1; shift up 1 unit. (1, 1) (21, 1) x y 5 3 (3, 1) (1, 1) x y 5 3 (3, 2) (1, 2) x 5 0 x 5 2 x 5 2 y 5 1 y 5 0 y 5 0 (b) y 5 1 (x – 2)2 (a) y 5 1 x2 (c) y 5 1 1 1 (x 2 2)2 Using Transformations to Graph a Rational Function Graph the rational function: ( ) ( ) = − + R x x 1 2 1 2 Solution EXAMPLE 3 The domain of R is the set of all real numbers except = x 2. To graph R, start with the graph of = y x 1 . 2 See Figure 37 for the steps. Check: Graph ( ) = − + Y x 1 2 1 1 2 using a graphing utility to verify the graph obtained in Figure 37(c). Asymptotes Notice that the y-axis in Figure 37(a) is transformed into the vertical line = x 2 in Figure 37(c), and the x-axis in Figure 37(a) is transformed into the horizontal line = y 1 in Figure 37(c). The Exploration that follows will help us analyze the role of these lines. Exploration (a) Using a graphing utility and the TABLE feature, evaluate the function ( ) ( ) = − + H x x 1 2 1 2 at = x 10, 100, 1000, and 10,000. What happens to the values of H as x becomes unbounded in the positive direction, expressed as ( ) →∞ H x lim ? x (b) Evaluate H at =− − − x 10, 100, 1000, and −10,000. What happens to the values of H as x becomes unbounded in the negative direction, expressed as ( ) →−∞ H x lim ? x (c) Evaluate H at = x 1.5, 1.9, 1.99, 1.999, and 1.9999. What happens to the values of H as x approaches 2, < x 2, expressed as ( ) → − H x lim ? x 2 (d) Evaluate H at = x 2.5, 2.1, 2.01, 2.001, and 2.0001. What happens to the values of H as x approaches 2, > x 2, expressed as ( ) → + H x lim ? x 2 Result (a) Table 12 on the next page shows the values of ( ) = Y H x 1 as x approaches ∞. Notice that the values of H are approaching 1, so ( ) = →∞ H x lim 1. x (b) Table 13 on the next page shows the values of ( ) = Y H x 1 as x approaches −∞. Again the values of H are approaching 1, so ( ) = →−∞ H x lim 1. x (c) From Table 14 on the next page we see that, as x approaches < x 2, 2, the values of H are increasing without bound, so ( ) =∞ → − H x lim . x 2 (d) Finally, Table 15 on the next page reveals that, as x approaches > x 2, 2, the values of H are increasing without bound, so ( ) =∞ → + H x lim . x 2
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