SECTION 4.6 The Graph of a Rational Function 249 SUMMARY Analyzing the Graph of a Rational Function Step 1 Factor the numerator and denominator of R. Find the domain of the rational function. Step 2 Write R in lowest terms. Step 3 Find the intercepts of the graph. The y-intercept, if there is one, is ( ) R 0 . The x-intercepts, if any, of ( ) ( ) ( ) = R x p x q x in lowest terms satisfy the equation ( ) = p x 0. Use multiplicity to determine the behavior of the graph of R at each x-intercept. Step 4 Find the vertical asymptotes. The vertical asymptotes, if any, of ( ) ( ) ( ) = R x p x q x in lowest terms are found by identifying the real zeros of ( ) q x . Each zero of the denominator gives rise to a vertical asymptote. Use multiplicity to determine the behavior of the graph of R on either side of each vertical asymptote. Step 5 Find the horizontal or oblique asymptote, if one exists, using the procedure given in Section 4.5. Find points, if any, at which the graph of R intersects this asymptote. Step 6 Graph R using a graphing utility. Step 7 Use the results obtained in Steps 1 through 6 to graph R by hand. Figure 44 ( ) = − − R x x x 1 4 2 x 3 –3 y 3 –3 (1, 0) 0, x = –2 x = 2 1 – 4 ( ) Figure 44 shows the graph of R. Step 7 Use the results obtained in Steps 1 through 6 to graph R by hand. Now Work PROBLEM 7 Analyzing the Graph of a Rational Function Analyze the graph of the rational function: ( ) = − R x x x 1 2 EXAMPLE 2 Solution Step 1 ( ) ( )( ) = − = + − R x x x x x x 1 1 1 2 The domain of R is { } ≠ x x 0 . Step 2 R is in lowest terms. Step 3 There is no y-intercept because x cannot equal 0. The graph has two x-intercepts, −1 and 1. Because each x-intercept is of odd multiplicity, the graph of R will cross the x-axis at each x-intercept. Step 4 The real zero of the denominator with R in lowest terms is 0, so the graph of ( ) R x has the line = x 0 (the y-axis) as a vertical asymptote.The multiplicity is odd, so the graph approaches ∞ on one side of the asymptote and −∞ on the other. (continued)
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