252 CHAPTER 4 Polynomial and Rational Functions 3.5 2.5 4 60 (c) y = 3 Figure 49 = − + − Y x x x x 3 3 12 1 2 2 210 210 10 10 (a) 21 21 0.5 2 (b) Figure 50 ( ) = − + − R x x x x x 3 3 12 2 2 –12 8 16 8 x y –8 x = –4 x = 3 y = 3 (6, 3) (11.48, 2.75) (0.52, 0.07) Step 7 Figure 49(a) does not clearly show the graph between the two x-intercepts, 0 and 1. Because the zeros in the numerator, 0 and 1, are of odd multiplicity (both are multiplicity 1), we know that the graph of R crosses the x-axis at 0 and 1.Therefore, the graph of R is above the x-axis for < < x 0 1. To see this part better, we graph R for − ≤ ≤ x 1 2 in Figure 49(b). Using MAXIMUM, we approximate the turning point to be ( ) 0.52, 0.07 , rounded to two decimal places. Figure 49(a) also does not display the graph of R crossing the horizontal asymptote at ( ) 6, 3 . To see this part better, we graph R for ≤ ≤ x 4 60 in Figure 49(c). Using MINIMUM, we approximate the turning point to be ( ) 11.48, 2.75 , rounded to two decimal places. Using this information along with the information gathered in Steps 1 through 6, we obtain the graph of R shown in Figure 50. Step 6 Figure 49(a) shows the graph of R using a TI-84 Plus CE. Now Work PROBLEM 31 Analyzing the Graph of a Rational Function with a Hole Analyze the graph of the rational function: ( ) = − + − R x x x x 2 5 2 4 2 2 EXAMPLE 5 Solution Step 1 Factor R and obtain ( ) ( )( ) ( )( ) = − − + − R x x x x x 2 1 2 2 2 The domain of R is { } ≠ − ≠ x x x 2, 2 . Step 2 In lowest terms, ( ) = − + ≠ − ≠ R x x x x x 2 1 2 2, 2 Step 3 The graph has one x-intercept, 0.5, with odd multiplicity. The graph crosses the x-axis at = x 1 2 . The y-intercept is ( ) = − R 0 0.5. Step 4 Since +x 2 is the only factor of the denominator of ( ) R x in lowest terms, the graph has one vertical asymptote, = − x 2. However, the rational function is undefined at both = x 2 and = − x 2. The multiplicity of −2 is odd, so the graph approaches ∞ on one side of the asymptote and −∞ on the other side. Step 5 Since the degree of the numerator equals the degree of the denominator, the graph has a horizontal asymptote. To find it, form the quotient of the leading coefficient of the numerator, 2, and the leading coefficient of the denominator, 1. The graph of R has the horizontal asymptote = y 2. To find whether the graph of R intersects the asymptote, solve the equation ( ) = R x 2.
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