SECTION 4.6 The Graph of a Rational Function 253 ( ) ( ) = − + = − = + − = + − = R x x x x x x x 2 1 2 2 2 1 2 2 2 1 2 4 1 4 Impossible The graph does not intersect the line = y 2. Step 6 Figure 51 shows the graph of ( ) R x using Desmos. Notice that the graph has one vertical asymptote at = − x 2. Also, the function appears to be continuous at = x 2. Step 7 The analysis presented thus far does not explain the behavior of the graph at = x 2. We create a table using Desmos to determine the behavior of the graph of R as →x 2. See Table 18. From the table, we conclude that →R 0.75 as →x 2. This result is further verified by evaluating R in lowest terms at = x 2. We conclude that there is a hole in the graph at ( ) 2, 0.75 . Using the information gathered in Steps 1 through 6, we obtain the graph of R shown in Figure 52. Figure 51 ( ) = − + − R x x x x 2 5 2 4 2 2 Figure 52 ( ) = − + − R x x x x 2 5 2 4 2 2 x y = 2 x = –2 y (2, 0.75) (0, –0.5) (0.5, 0) –4 2 –4 8 As Example 5 shows, the values excluded from the domain of a rational function give rise to either vertical asymptotes or holes. Now Work PROBLEM 33 Constructing a Rational Function from Its Graph Find a rational function that might have the graph shown in Figure 53. EXAMPLE 6 Figure 53 x y 215 15 210 10 25 5 x 5 2 y 5 2 x 5 25 210 10 25 5 Table 18 (continued)
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