390 CHAPTER 3 Polynomial and Rational Functions –10 –5 2 5 10 15 x y y = 3 x = –4 (–2, 3) (–10, 9) f(x) = 3x2 – 3x – 6 x2 + 8x + 16 (5, )2 3 (–8, 13 )1 8 Figure 54 and 7 Step 5 We set ƒ1x2 = 3 and solve to locate the point where the graph intersects the horizontal asymptote. 3x2 - 3x - 6 x2 + 8x + 16 = 3 Set ƒ1x2 = 3. 3x2 - 3x - 6 = 3x2 + 24x + 48 Multiply each side by x2 + 8x + 16. -27x = 54 Subtract 3x2 and 24x. Add 6. x = -2 Divide by -27. The graph intersects its horizontal asymptote at 1-2, 32. Steps 6 Some other points that lie on the graph are 1-10, 92, A -8, 13 1 8B, and A5, 2 3B. These are used to complete the graph, as shown in Figure 54. S Now Try Exercise 83. We have observed that the behavior of the graph of a polynomial function near its zeros is dependent on the multiplicity of the zero. The same statement can be made for rational functions. Suppose that ƒ1x2 is defined by a rational expression in lowest terms. If n is the greatest positive integer such that 1x - c2n is a factor of the numerator of ƒ1x2, then the graph will behave in the manner illustrated. –8–6–4–2 2 4 6 8 –8 –6 –4 4 6 8 x y x = 3 y = 2 (–4, 1) (6, ) f(x) = 2x + 1 x – 3 (1, – ) 3 2 13 3 Figure 53 (repeated) Behavior of Graphs of Rational Functions near Vertical Asymptotes Suppose that ƒ1x2 is a rational expression in lowest terms. If n is the largest positive integer such that 1x - a2n is a factor of the denominator of ƒ1x2, then the graph will behave in the manner illustrated. If n is even: y = f(x) a x or y = f(x) a x If n is odd: y = f(x) a x or y = f(x) a x If n =1: x c or x c If n is even: x c or x c If n is an odd integer greater than 1: x c or x c Notice the behavior of the graph of the function in Figure 54 near the line x = -4. As x S-4 from either side, ƒ1x2 S∞. If we examine the behavior of the graph of the function in Figure 53 (repeated in the margin) near the line x = 3, we find that ƒ1x2 S-∞ as x approaches 3 from the left, while ƒ1x2 S∞ as x approaches 3 from the right. The behavior of the graph of a rational function near a vertical asymptote x = a partially depends on the exponent on x - a in the denominator.
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