Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1 3 Polynomial and Rational Functions Copyright © 2013, 2009, 2005 Pearson Education, Inc.

2 3.5 Rational Functions: Graphs, Applications, and Models The Reciprocal Function The Function Asymptotes Steps for Graphing Rational Functions Rational Function Models

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 3 Rational function A function  of the form where p(x) and q(x) are polynomials, with q(x) ≠ 0, is a rational function.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 4 Rational Function Some examples of rational functions are Since any values of x such that q(x) = 0 are excluded from the domain of a rational function, this type of function often has a discontinuous graph—that is, a graph that has one or more breaks in it.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 5 The Reciprocal Function The simplest rational function with a variable denominator is the reciprocal function.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 6 The Reciprocal Function The domain of this function is the set of all real numbers except 0. The number 0 cannot be used as a value of x, but it is helpful to find values of  (x) for some values of x very close to 0. We use the table feature of a graphing calculator to do this. The tables suggest that  (x)  increases without bound as x gets closer and closer to 0, which is written in symbols as

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 7 The Reciprocal Function (The symbol x  0 means that x approaches 0, without necessarily ever being equal to 0.) Since x cannot equal 0, the graph of will never intersect the vertical line x = 0.This line is called a vertical asymptote. As  x  increases without bound, the values of get closer and closer to 0, as shown in the tables. Letting  x  increase without bound (written  x    ) causes the graph of to move closer and closer to the horizontal line y = 0. This line is a horizontal asymptote.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 8 Domain: (– , 0)  (0,  ) Range: (– , 0)  (0,  ) RECIPROCAL FUNCTION xy – 2– 2– ½– ½ – 1– 1– 1– 1 – ½– ½ – 2– 2 0 undefined ½2 11 2½  decreases on the intervals (– ,0) and (0,  ).

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 9 Domain: (– , 0)  (0,  ) Range: (– , 0)  (0,  ) RECIPROCAL FUNCTION xy – 2– 2– ½– ½ – 1– 1– 1– 1 – ½– ½ – 2– 2 0 undefined ½2 11 2½  It is discontinuous at x = 0.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 10 Domain: (– , 0)  (0,  ) Range: (– , 0)  (0,  ) RECIPROCAL FUNCTION xy – 2– 2– ½– ½ – 1– 1– 1– 1 – ½– ½ – 2– 2 0 undefined ½2 11 2½  The y-axis is a vertical asymptote, and the x-axis is a horizontal asymptote.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 11 Domain: (– , 0)  (0,  ) Range: (– , 0)  (0,  ) RECIPROCAL FUNCTION xy – 2– 2– ½– ½ – 1– 1– 1– 1 – ½– ½ – 2– 2 0 undefined ½2 11 2½  It is an odd function, and its graph is symmetric with respect to the origin.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 12 Example 1 GRAPHING A RATIONAL FUNCTION Solution Graph Give the domain and range and the intervals of the domain for which the function is increasing or decreasing. The expression can be written as or indicating that the graph may be obtained by stretching the graph of vertically by a factor of 2 and reflecting it across either the x-axis or y-axis. The x- and y-axes remain the horizontal and vertical asymptotes. The domain and range are both still (– , 0)  (0,  ).

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 13 Example 1 GRAPHING A RATIONAL FUNCTION Solution Graph Give the domain and range and the intervals of the domain for which the function is increasing or decreasing. The graph shows that f (x) is increasing on both sides of its vertical asymptote. Thus, it is increasing on (– , 0) and on (0,  ).

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 14 Example 2 GRAPHING A RATIONAL FUNCTION Solution The expression can be written as indicating that the graph may be obtained by shifting the graph of to the left 1 unit and stretching it vertically by a factor of 2. Graph Give the domain and range and the intervals of the domain for which the function is increasing or decreasing.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 15 Example 2 GRAPHING A RATIONAL FUNCTION Solution The horizontal shift affects the domain, which is now (– , –1)  (–1,  ). The line x = –1 is the vertical asymptote, and the line y = 0 (the x-axis) remains the horizontal asymptote. The range is still (– , 0)  (0,  ). The graph shows that f (x) is decreasing on both sides of its vertical asymptote. Thus it is decreasing on (– , –1) and on (–1,  ). Graph Give the domain and range and the intervals of the domain for which the function is increasing or decreasing.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 16 The Function The rational function defined by also has domain (– , 0)  (0,  ). We can use the table feature of a graphing calculator to examine values of  (x) for some x-values close to 0.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 17 The Function The tables suggest that  (x) increases without bound as x gets closer and closer to 0. Notice that as x approaches 0 from either side, function values are all positive and there is symmetry with respect to the y-axis. Thus,  (x)   as x  0. The y-axis (x = 0) is the vertical asymptote.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 18 The Function As │x│ increases without bound, f (x) approaches 0, as suggested by the tables. Again, function values are all positive. The x-axis is the horizontal asymptote of the graph.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 19 Domain: (– , 0)  (0,  ) Range: (0,  ) RECIPROCAL FUNCTION xy  3  2 ¼  1 1  ½ 4  ¼ 16 0undefined  increases on the interval (– ,0) and decreases on the interval (0,  ).

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 20 Domain: (– , 0)  (0,  ) Range: (0,  ) RECIPROCAL FUNCTION xy  3  2 ¼  1 1  ½ 4  ¼ 16 0undefined  It is discontinuous at x = 0.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 21 Domain: (– , 0)  (0,  ) Range: (0,  ) RECIPROCAL FUNCTION xy  3  2 ¼  1 1  ½ 4  ¼ 16 0undefined  The y-axis is a vertical asymptote, and the x-axis is a horizontal asymptote.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 22 Domain: (– , 0)  (0,  ) Range: (0,  ) RECIPROCAL FUNCTION xy  3  2 ¼  1 1  ½ 4  ¼ 16 0undefined  It is an even function, and its graph is symmetric with respect to the y-axis.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 23 Example 3 GRAPHING A RATIONAL FUNCTION Solution The equation is equivalent to Graph Give the domain and range and the intervals of the domain for which the function is increasing or decreasing. where This indicates that the graph will be shifted 2 units to the left and 1 unit down.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 24 Example 3 GRAPHING A RATIONAL FUNCTION Solution The horizontal shift affects the domain, now (– , – 2)  (– 2,  ), while the vertical shift affects the range, now (– 1,  ). Graph Give the domain and range and the intervals of the domain for which the function is increasing or decreasing.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 25 Example 3 GRAPHING A RATIONAL FUNCTION Solution The vertical asymptote has equation x = – 2, and the horizontal asymptote has equation y = – 1. This function is increasing on (– , –2) and decreasing on (–2,  ). Graph Give the domain and range and the intervals of the domain for which the function is increasing or decreasing.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 26 Asymptotes Let p(x) and q(x) define polynomials. Consider the rational function written in lowest terms, and real numbers a and b. 1. If  (x)    as x  a, then the line x = a is a vertical asymptote. 2. If  (x)  b as  x   , then the line y = b is a horizontal asymptote.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 27 Determining Asymptotes To find the asymptotes of a rational function defined by a rational expression in lowest terms, use the following procedures. 1. Vertical Asymptotes Find any vertical asymptotes by setting the denominator equal to 0 and solving for x. If a is a zero of the denominator, then the line x = a is a vertical asymptote.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 28 Determining Asymptotes 2. Other Asymptotes Determine any other asymptotes by considering three possibilities: (a) If the numerator has lesser degree than the denominator, then there is a horizontal asymptote y = 0 (the x-axis).

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 29 Determining Asymptotes 2. Other Asymptotes Determine any other asymptotes. Consider three possibilities: (b) If the numerator and denominator have the same degree, and the function is of the form where a n, b n ≠ 0, then the horizontal asymptote has equation

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 30 Determining Asymptotes 2. Other Asymptotes Determine any other asymptotes. Consider three possibilities: (c) If the numerator is of degree exactly one more than the denominator, then there will be an oblique (slanted) asymptote. To find it, divide the numerator by the denominator and disregard the remainder. Set the rest of the quotient equal to y to obtain the equation of the asymptote.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 31 Note The graph of a rational function may have more than one vertical asymptote, or it may have none at all. The graph cannot intersect any vertical asymptote. There can be at most one other (nonvertical) asymptote, and the graph may intersect that asymptote, as we shall see in Example 7.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 32 Example 4 FINDING ASYMPTOTES OF RATIONAL FUNCTIONS Solution To find the vertical asymptotes, set the denominator equal to 0 and solve. (a) For each rational function , find all asymptotes. Zero-factor property Solve each equation.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 33 Example 4 FINDING ASYMPTOTES OF RATIONAL FUNCTIONS The equations of the vertical asymptotes are x = ½ and x = – 3. To find the equation of the horizontal asymptote, divide each term by the greatest power of x in the expression. First, multiply the factors in the denominator.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 34 Example 4 FINDING ASYMPTOTES OF RATIONAL FUNCTIONS Now divide each term in the numerator and denominator by x 2 since 2 is the greatest power of x. Stop here. Leave the expression in complex form.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 35 Example 4 FINDING ASYMPTOTES OF RATIONAL FUNCTIONS As  x  increases without bound, the quotients all approach 0, and the value of  (x) approaches The line y = 0 (that is, the x-axis) is therefore the horizontal asymptote.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 36 Example 4 FINDING ASYMPTOTES OF RATIONAL FUNCTIONS Solution Set the denominator x – 3 equal to 0 to find that the vertical asymptote has equation x = 3. To find the horizontal asymptote, divide each term in the rational expression by x since the greatest power of x in the expression is 1. (b) For each rational function , find all asymptotes.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 37 Example 4 FINDING ASYMPTOTES OF RATIONAL FUNCTIONS Solution For each rational function , find all asymptotes. (b)

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 38 Example 4 FINDING ASYMPTOTES OF RATIONAL FUNCTIONS As  x  increases without bound, both approach 0, and  (x) approaches so the line y = 2 is the horizontal asymptote.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 39 Example 4 FINDING ASYMPTOTES OF RATIONAL FUNCTIONS Solution Setting the denominator x – 2 equal to 0 shows that the vertical asymptote has equation x = 2. If we divide by the greatest power of x as before (x 2 in this case), we see that there is no horizontal asymptote because (c) For each rational function , find all asymptotes. does not approach any real number as  x   , since is undefined.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 40 Example 4 FINDING ASYMPTOTES OF RATIONAL FUNCTIONS This happens whenever the degree of the numerator is greater than the degree of the denominator. In such cases, divide the denominator into the numerator to write the expression in another form. We use synthetic division, as shown. The result enables us to write the function as follows.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 41 Example 4 FINDING ASYMPTOTES OF RATIONAL FUNCTIONS For very large values of  x , is close to 0, and the graph approaches the line y = x + 2. This line is an oblique asymptote (slanted, neither vertical nor horizontal) for the graph of the function.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 42 Steps for Graphing Rational Functions A comprehensive graph of a rational function will show the following characteristics. 1. all x- and y-intercepts 2. all asymptotes: vertical, horizontal, and/or oblique 3. the point at which the graph intersects its nonvertical asymptote (if there is any such point) 4. the behavior of the function on each domain interval determined by the vertical asymptotes and x-intercepts

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 43 Graphing a Rational Function Let define a function where p(x) and q(x) are polynomials and the rational expression is written in lowest terms. To sketch its graph, follow these steps. Step 1 Find any vertical asymptotes. Step 2 Find any horizontal or oblique asymptotes. Step 3 Find the y-intercept by evaluating  (0).

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 44 Graphing a Rational Function Step 4 Find the x-intercepts, if any, by solving  (x) = 0. (These will be the zeros of the numerator, p(x).) Step 5 Determine whether the graph will intersect its nonvertical asymptote y = b or y = mx + b by solving  (x) = b or  (x) = mx + b.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 45 Graphing a Rational Function Step 6 Plot selected points, as necessary. Choose an x-value in each domain interval determined by the vertical asymptotes and x-intercepts. Step 7 Complete the sketch.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 46 Example 5 GRAPHING A RATIONAL FUNCTION WITH THE x-AXIS AS HORIZONTAL ASYMPTOTE Solution Graph Step 1 Since 2x 2 + 5x – 3 = (2x – 1)(x + 3), from Example 4(a), the vertical asymptotes have equations x = ½ and x = – 3. Step 2 Again, as shown in Example 4(a), the horizontal asymptote is the x-axis.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 47 Example 5 GRAPHING A RATIONAL FUNCTION WITH THE x-AXIS AS HORIZONTAL ASYMPTOTE Solution Graph Step 3 The y-intercept is – ⅓, since The y-intercept is the ratio of the constant terms.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 48 Example 5 GRAPHING A RATIONAL FUNCTION WITH THE x-AXIS AS HORIZONTAL ASYMPTOTE Solution Graph Step 4 The x-intercept is found by solving  (x) = 0. If a rational expression is equal to 0, then its numerator must equal 0. The x-intercept is – 1.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 49 Example 5 GRAPHING A RATIONAL FUNCTION WITH THE x-AXIS AS HORIZONTAL ASYMPTOTE Solution Graph Step 5 To determine whether the graph intersects its horizontal asymptote, solve y-value of horizontal asymptote Since the horizontal asymptote is the x-axis, the solution of this equation was found in Step 4. The graph intersects its horizontal asymptote at (– 1, 0).

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 50 Example 5 GRAPHING A RATIONAL FUNCTION WITH THE x-AXIS AS HORIZONTAL ASYMPTOTE Solution Graph Step 6 Plot a point in each of the intervals determined by the x-intercepts and vertical asymptotes, to get an idea of how the graph behaves in each interval.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 51 Example 5 GRAPHING A RATIONAL FUNCTION WITH THE x-AXIS AS HORIZONTAL ASYMPTOTE IntervalTest Point Value of  (x) Sign of  (x) Graph Above or Below x-Axis (– , –3) – 4NegativeBelow (–3, –1)– 2PositiveAbove (–1, ½ )0NegativeBelow (½,  ) 2PositiveAbove

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 52 Example 5 GRAPHING A RATIONAL FUNCTION WITH THE x-AXIS AS HORIZONTAL ASYMPTOTE Solution Graph Step 7 Complete the sketch. This function is decreasing on each interval of its domain— that is, on

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 53 Example 6 GRAPHING A RATIONAL FUNCTION THAT DOES NOT INTERSECT ITS HORIZONTAL ASYMPTOTE Solution Graph Steps 1 and 2 As determined in Example 4(b), the equation of the vertical asymptote is x = 3. The horizontal asymptote has equation y = 2.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 54 Example 6 GRAPHING A RATIONAL FUNCTION THAT DOES NOT INTERSECT ITS HORIZONTAL ASYMPTOTE Solution Graph Step 3  (0) = – ⅓, so the y-intercept is – ⅓.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 55 Example 6 GRAPHING A RATIONAL FUNCTION THAT DOES NOT INTERSECT ITS HORIZONTAL ASYMPTOTE Solution Graph Step 4 Solve  (x) = 0 to find any x-intercepts. If a rational expression is equal to 0, then its numerator must equal 0. x-intercept

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 56 Example 6 GRAPHING A RATIONAL FUNCTION THAT DOES NOT INTERSECT ITS HORIZONTAL ASYMPTOTE Solution Graph Step 5 The graph does not intersect its horizontal asymptote since  (x) = 2 has no solution. Set f (x) = 2. Multiply each side by x – 3. Subtract 2x. A false statement results. The solution set is Ø.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 57 Example 6 GRAPHING A RATIONAL FUNCTION THAT DOES NOT INTERSECT ITS HORIZONTAL ASYMPTOTE Solution Graph Steps 6 and 7 The points (– 4, 1), (1, – 3/2), and (6, 13/3) are on the graph and can be used to complete the sketch of this function, which decreases on every interval of its domain.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 58 Example 7 GRAPHING A RATIONAL FUNCTION THAT INTERSECTS ITS HORIZONTAL ASYMPTOTE Solution Graph Step 1 To find the vertical asymptote(s), solve x 2 + 8x + 16 = 0. Set the denominator equal to 0. Factor. Zero-factor property.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 59 Example 7 GRAPHING A RATIONAL FUNCTION THAT INTERSECTS ITS HORIZONTAL ASYMPTOTE Solution Graph Since the numerator is not 0 when x = – 4, the vertical asymptote has the equation x = – 4. Step 1 To find the vertical asymptote(s), solve x 2 + 8x + 16 = 0.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 60 Example 7 GRAPHING A RATIONAL FUNCTION THAT INTERSECTS ITS HORIZONTAL ASYMPTOTE Solution Graph Step 2 We divide all terms by x 2 and consider the behavior of each term as │x│ increases without bound to get the equation of the horizontal asymptote. Leading coefficient of numerator Leading coefficient of denominator

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 61 Example 7 GRAPHING A RATIONAL FUNCTION THAT INTERSECTS ITS HORIZONTAL ASYMPTOTE Solution Graph Step 3 The y-intercept is

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 62 Example 7 GRAPHING A RATIONAL FUNCTION THAT INTERSECTS ITS HORIZONTAL ASYMPTOTE Solution Graph Step 4 To find the x-intercept(s), if any, we solve  (x) = 0. Set the numerator equal to 0.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 63 Example 7 GRAPHING A RATIONAL FUNCTION THAT INTERSECTS ITS HORIZONTAL ASYMPTOTE Solution Graph Step 4 Divide by 3. Factor. Zero-factor property The x-intercepts are –1 and 2.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 64 Example 7 GRAPHING A RATIONAL FUNCTION THAT INTERSECTS ITS HORIZONTAL ASYMPTOTE Solution Graph Step 5 We set  (x) = 3 and solve to locate the point where the graph intersects the horizontal asymptote. Multiply each side by x 2 + 8x + 16.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 65 Example 7 GRAPHING A RATIONAL FUNCTION THAT INTERSECTS ITS HORIZONTAL ASYMPTOTE Solution Graph Step 5 Subtract 3x 2. Subtract 24x and add 6. Divide by – 27. The graph intersects its horizontal asymptote at (– 2, 3).

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 66 Example 7 GRAPHING A RATIONAL FUNCTION THAT INTERSECTS ITS HORIZONTAL ASYMPTOTE Solution Graph Steps 6 and 7 Some of the other points that lie on the graph are These are used to complete the graph.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 67 Behavior of Graphs of Rational Functions near Vertical Asymptotes Suppose that  (x) is a rational expression in lowest terms. If n is the largest positive integer such that (x – a) n is a factor of the denominator of  (x), the graph will behave in the manner illustrated.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 68 Behavior of Graphs In Section 3.4 we 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  (x) is defined by a rational expression in lowest terms. If n is the greatest positive integer such that (x – c) n is a factor of the numerator of  (x), the graph will behave in the manner illustrated.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 69 Behavior of Graphs

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 70 Example 8 GRAPHING A RATIONAL FUNCTION WITH AN OBLIQUE ASYMPTOTE Solution In shown in Example 4, the vertical asymptote has equation x = 2, and the graph has an oblique asymptote with equation y = x + 2. The y-intercept is –½, and the graph has no x-intercepts since the numerator, x 2 + 1, has no real zeros. Graph

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 71 Example 8 Graph Using the y-intercept, asymptotes, the points and and the general behavior of the graph near its asymptotes leads to the following graph. GRAPHING A RATIONAL FUNCTION WITH AN OBLIQUE ASYMPTOTE Solution The graph does not intersect its oblique asymptote because the following has no solution.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 72 Example 8 Solution Graph GRAPHING A RATIONAL FUNCTION WITH AN OBLIQUE ASYMPTOTE

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 73 A rational function that is not in lowest terms often has a “hole,” or point of discontinuity, in its graph.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 74 Example 9 GRAPHING A RATIONAL FUNCTION DEFINED BY AN EXPRESSION THAT IS NOT IN LOWEST TERMS Solution The domain of this function cannot include 2. The expression should be written in lowest terms. Graph Factor.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 75 Example 9 GRAPHING A RATIONAL FUNCTION DEFINED BY AN EXPRESSION THAT IS NOT IN LOWEST TERMS Solution 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 (2, 4). Graph

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 76 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. (Source: Mannering, F. and W. Kilareski, Principles of Highway Engineering and Traffic Analysis,2 nd ed., John Wiley & Sons.)

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 77 Example 10 MODELING TRAFFIC INTENSITY WITH A RATIONAL FUNCTION (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. The average arrival rate is 2.6 vehicles and the average admittance rate is 3.2 vehicles, so Solution

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 78 Example 10 MODELING TRAFFIC INTENSITY WITH A RATIONAL FUNCTION (b) The average number of vehicles waiting in line to enter the ramp is given by where 0  x < 1 is the traffic intensity. Graph  (x) and compute  (0.8125) for this parking ramp. Solution

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 79 (c) What happens to the number of vehicles waiting as the traffic intensity approaches 1? Example 10 MODELING TRAFFIC INTENSITY WITH A RATIONAL FUNCTION Solution From the graph we see that as x approaches 1, y =  (x) gets very large. Thus, the average number of waiting vehicles gets very large. This is what we would expect.