SECTION 4.6 The Graph of a Rational Function 251 Figure 48 x 3 23 y 6 (1, 2) (21, 2) y 5 x2 x 5 0 Figure 47 = + Y x x 1 1 4 2 6 0 23 3 Step 6 See Figure 47 for the graph using a TI-84 Plus CE. Notice the vertical asymptote at = x 0, and note that there are no intercepts. Step 7 Since the graph of R is above the x-axis, and the multiplicity of the zero that gives rise to the vertical asymptote, = x 0, is even, the graph of R approaches ∞ from both the left and the right of = x 0. Figure 48 shows the graph drawn by hand. NOTE Notice that R in Example 3 is an even function. Do you see the symmetry about the y-axis in the graph of R? j Now Work PROBLEM 13 Analyzing the Graph of a Rational Function Analyze the graph of the rational function: ( ) = − + − R x x x x x 3 3 12 2 2 EXAMPLE 4 Step 1 Factor R to get ( ) ( ) ( )( ) = − + − R x x x x x 3 1 4 3 The domain of R is { } ≠ − ≠ x x x 4, 3 . Step 2 R is in lowest terms. Step 3 The y-intercept is ( ) = R 0 0. The real zeros of the numerator that satisfy ( ) − = x x 3 1 0 are = x 0 and = x 1. Therefore, the graph has two x-intercepts, 0 and 1, each of odd multiplicity. The graph of R will cross the x-axis at = x 0 and = x 1. Step 4 The real zeros of the denominator of R with R in lowest terms are −4 and 3. So the graph of R has two vertical asymptotes: = − x 4 and = x 3. The multiplicities of the values that give rise to the asymptotes are both odd, so the graph approaches ∞ on one side of each 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, 3, and the leading coefficient of the denominator, 1. The graph of R has the horizontal asymptote = y 3. To find out whether the graph of R intersects the asymptote, solve the equation ( ) = R x 3. ( ) = − + − = − = + − − = − = R x x x x x x x x x x x 3 3 12 3 3 3 3 3 36 6 36 6 2 2 2 2 The graph intersects the line = y 3 at = x 6, and ( ) 6, 3 is a point on the graph of R. Solution (continued)
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