248 CHAPTER 4 Polynomial and Rational Functions Figure 43 ( ) ( ) = − − Y x x 1 4 1 2 4 24 24 4 ( ) ( )( ) = − − = − + − R x x x x x x 1 4 1 2 2 2 The domain of R is { } ≠ − ≠ x x x 2, 2 . Step-by-Step Solution Step 1 Factor the numerator and denominator of R. Find the domain of the rational function. Because there are no common factors between the numerator and denominator, R is in lowest terms. Step 2 Write R in lowest terms. Since 0 is in the domain of R, the y-intercept is ( ) = R 0 1 4 . The x-intercepts are found by finding the real zeros of the numerator of R written in lowest terms. By solving − = x 1 0, the only real zero of the numerator is 1. So the only x-intercept of the graph of R is 1. The multiplicity of 1 is odd, so the graph crosses the x-axis at = x 1. Step 3 Find the intercepts of the graph. Use multiplicity to determine the bahavior of the graph of R at each x-intercept. To locate the vertical asymptotes, find the zeros of the denominator with the rational function in lowest terms. With R written in lowest terms, we find that the graph of R has two vertical asymptotes: the lines = − x 2 and = x 2. The multiplicities of the zeros that give rise to the vertical asymptotes are both odd. Therefore, the graph approaches ∞ on one side of each vertical asymptote, and it approaches −∞ on the other side. Step 4 Find the vertical asymptotes. Determine the behavior of the graph on either side of each vertical asymptote. Because the degree of the numerator is less than the degree of the denominator, R is proper and the line = y 0 (the x-axis) is a horizontal asymptote of the graph. To determine if the graph of R intersects the horizontal asymptote, solve the equation ( ) = R x 0: − − = − = = x x x x 1 4 0 1 0 1 2 The only solution is = x 1, so the graph of R intersects the horizontal asymptote at ( ) 1, 0 . Step 5 Find the horizontal or oblique asymptote. Find points, if any, at which the graph of R intersects this asymptote. How to Analyze the Graph of a Rational Function Analyze the graph of the rational function: ( ) = − − R x x x 1 4 2 EXAMPLE 1 The analysis in Steps 1 through 5 helps us to determine an appropriate viewing window to obtain a complete graph. Figure 43 shows the graph of ( ) = − − R x x x 1 4 2 using a TI-84 Plus CE. You should confirm that all the algebraic conclusions that we came to in Steps 1 through 5 are part of the graph. For example, the graph has a horizontal asymptote at = y 0 and vertical asymptotes at = − = x x 2 and 2. The y-intercept is 1 4 and the x-intercept is 1. Step 6 Graph R using a graphing utility.
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