392 CHAPTER 3 Polynomial and Rational Functions EXAMPLE 9 Graphing a Rational Function Defined by an Expression That Is Not in LowestTerms Graph ƒ1x2 = x2 - 4 x - 2 . GRAPHING CALCULATOR SOLUTION If we set the window of a graphing calculator so that an x-value of 2 is displayed, then we can see that the calculator cannot determine a value for y. We define y1 = x2 - 4 x - 2 and graph it in such a window, as in Figure 57. The error message in the table further supports the existence of a discontinuity at x = 2. 1For the table, y2 = x + 2.2 ALGEBRAIC SOLUTION The domain of this function cannot include 2. The expression x 2 - 4 x - 2 should be written in lowest terms. ƒ1x2 = x2 - 4 x - 2 ƒ1x2 = 1x + 22 1x - 22 x - 2 Factor. ƒ1x2 = x + 2, x ≠2 Therefore, the graph of this function will be the same as the graph of y = x + 2 (a straight line), with the exception of the point with x-value 2. A hole appears in the graph at 12, 42. See Figure 56. Factor and then divide. y 4 2 –2 –2 x 0 f(x) = x2 – 4 x – 2 Figure 56 −2.1 −6.6 6.1 6.6 y1 = x2 − 4 x − 2 Figure 57 Notice the visible discontinuity at x = 2 in the graph for the chosen window. (If the standard viewing window is chosen, the discontinuity is not visible.) S Now Try Exercise 91. Rational Models Rational functions have a variety of applications. EXAMPLE 10 Modeling Traffic Intensity with a Rational Function Vehicles arrive randomly at a parking ramp at an average rate of 2.6 vehicles per minute. The parking attendant can admit 3.2 vehicles per minute. However, since arrivals are random, lines form at various times. (Data from Mannering, F., and W. Kilareski, Principles of Highway Engineering and Traffic Analysis, 2nd ed., John Wiley & Sons.) (a) The traffic intensity x is defined as the ratio of the average arrival rate to the average admittance rate. Determine x for this parking ramp. (b) The average number of vehicles waiting in line to enter the ramp is given by the rational function ƒ1x2 = x2 211 - x2 , where 0 … x 61 is the traffic intensity. Graph ƒ1x2 and compute ƒ10.81252 for this parking ramp.
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