GRAPHS OF RATIONAL FUNCTIONS
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1 0 (0-) Chapter 0 Polnomial and Rational Functions. f() ( 0) ( 0). f() ( 0) ( 0). f() ( 0) ( 0). f() ( 0) ( 0) 0. GRAPHS OF RATIONAL FUNCTIONS In this section Domain Horizontal and Vertical Asmptotes Oblique Asmptotes Sketching the Graphs We first studied rational epressions in Chapter 6. In this section we will stud functions that are defined b rational epressions. Domain A rational epression was defined in Chapter 6 as a ratio of two polnomials. If a ratio of two polnomials is used to define a function, then the function is called a rational function. Rational Function P( ) If P() and Q() are polnomials with no common factor and f () Q for ( ) Q() 0, then f () is called a rational function. The domain of a rational function is the set of all real numbers ecept those that cause the denominator to have a value of 0. E X A M P L E Domain of a rational function Find the domain of each rational function. a) f () b) g()
2 0. Graphs of Rational Functions (0-) If the viewing window is too large, a rational function will appear to touch its asmptotes. 8 calculator close-up 8 Because the asmptotes are an important feature of a rational function, we should draw it so that it approaches but does not touch its asmptotes. E X A M P L E 8 f() = FIGURE 0. 8 Solution a) Since 0 onl for, the domain of f is the set of all real numbers ecept,. b) Since 0 for, the domain of g is the set of all real numbers ecluding and, and. Horizontal and Vertical Asmptotes Consider the simplest rational function f (). Its domain does not include 0, but 0 is an important number for the graph of this function. The behavior of the graph of f when is ver close to 0 is what interests us. For this function the -coordinate is the reciprocal of the -coordinate. When the -coordinate is close to 0, the -coordinate is far from 0. Consider the following tables of ordered pairs that satisf f () : , ,000 As gets closer and closer to 0 from above 0, the value of gets larger and larger. We sa that goes to positive infinit. As gets closer and closer to 0 from below 0, the values of are negative but gets larger and larger. We sa that goes to negative infinit. The graph of f gets closer and closer to the vertical line 0, and so 0 is called a vertical asmptote. On the other hand, as gets larger and larger, gets closer and closer to 0. The graph approaches the -ais as goes to infinit, and so the -ais is a horizontal asmptote for the graph of f. See Fig. 0. for the graph of f (). In general, a rational function has a vertical asmptote for ever number ecluded from the domain of the function. The horizontal asmptotes are determined b the behavior of the function when is large. Horizontal and vertical asmptotes Find the horizontal and vertical asmptotes for each rational function. a) f () b) g() c) h() Solution a) The denominator has a value of 0 if. So the lines and are vertical asmptotes. If is ver large, the value of is approimatel 0. So the -ais is a horizontal asmptote. b) The denominator has a value of 0 if. So the lines and are vertical asmptotes. If is ver large, the value of is approimatel 0. So the -ais is a horizontal asmptote.
3 (0-) Chapter 0 Polnomial and Rational Functions The graph for Eample (b) should consist of three separate pieces, but in connected mode the calculator connects the separate pieces. Even though the calculator does not draw a ver good graph of this function, it does support the conclusion that the horizontal asmptote is the -ais and the vertical asmptotes are and. calculator close-up 0 0 c) The denominator has a value of 0 if. So the line is a vertical asmptote. If is ver large, the value of h() is not approimatel 0. To understand the value of h(), we change the form of the rational epression b using long division: 6 mainder Writing the rational epression as quotient re, we get divisor h(). If is ver large, is approimatel 0, and so the -coordinate is approimatel. The line is a horizontal asmptote. Eample illustrates two important facts about horizontal asmptotes. If the degree of the numerator is less than the degree of the denominator, then the -ais is the horizontal asmptote. For eample, 7 has the -ais as a horizontal asmptote. If the degree of the numerator is equal to the degree of the denominator, then the ratio of the leading coefficients determines the horizontal asmptote. For eample, 7 has as its horizontal asmptote. The remaining case is when the degree of the numerator is greater than the degree of the denominator. This case is discussed net. Oblique Asmptotes Each rational function of Eample had one horizontal asmptote and a vertical asmptote for each number that caused the denominator to be 0. The horizontal asmptote 0 occurs because as gets larger and larger, the -coordinate gets closer and closer to 0. Some rational functions have a nonhorizontal line for an asmptote. An asmptote that is neither horizontal nor vertical is called an oblique asmptote or slant asmptote. E X A M P L E stud tip The last couple of weeks of the semester is not the time to slack off. This is the time to double our efforts. Make a schedule and plan ever hour of our time. Don t schedule anthing that isn t necessar. Get an earl start on studing for our final eams. Finding an oblique asmptote Determine all of the asmptotes for g(). Solution If 0, then. So the line is a vertical asmptote. Use long division to rewrite the function as quotient re mainder : d ivisor g() If is large, the value of is approimatel 0. So when is large, the value of g() is approimatel. The line is an oblique asmptote for the graph of g. We can summarize this discussion of asmptotes with the following strateg for finding asmptotes for a rational function.
4 0. Graphs of Rational Functions (0-) Strateg for Finding Asmptotes for a Rational Function P( ) Suppose f() is a rational function with the degree of Q() at least. Q ( ). Solve the equation Q() 0. The graph of f has a vertical asmptote corresponding to each solution to the equation.. If the degree of P() is less than the degree of Q(), then the -ais is a horizontal asmptote.. If the degree of P() is equal to the degree of Q(), then find the ratio of the leading coefficients. The horizontal line through that ratio is the horizontal asmptote.. If the degree of P() is greater than the degree of Q(), then use division to rewrite the function as quotient re mainder. divisor The equation formed b setting equal to the quotient gives us an oblique asmptote. M A T H A T W O R K Machines that do computations have been around for thousands of ears. The abacus, the slide rule, and the calculator have simplified computations. However, recent calculators have gone wa beond numerical computations. Graphing calculators now draw two- and three-dimensional graphs and even do smbolic computations. Modern calculators are great, but to use one effectivel ou must still learn the underling principles of mathematics. Consider the fairl simple process of drawing a graph of a function. The graph of a function is a picture of all ordered pairs of the function. When we graph a function, we tpicall plot a few ordered pairs and then use our knowledge about the function to draw a graph that shows all of the important features of the function. A tpical graphing calculator plots 96 ordered pairs of a function on a screen that is not much bigger than a postage stamp. The calculator does not generalize and does not make conclusions. We are still responsible for looking at what the calculator shows and making conclusions. For eample, if we graphed ( 00) with the window set for 0 0, we would not see the vertical asmptote 00. Would we believe the calculator and conclude that the graph has no vertical asmptote? If we graph with the window set for and 00 00, the graph will be too close to its horizontal and vertical asmptotes for us to see it. Would we conclude that there is no graph for? If ou have a graphing calculator, use it to help ou graph the functions in this chapter and to reinforce our understanding of the properties of functions that we are learning. Do not attempt to use it as a substitute for learning. MATHEMATICS MACHINES
5 (0-6) Chapter 0 Polnomial and Rational Functions Sketching the Graphs We now use asmptotes and smmetr to help us sketch the graphs of some rational functions. E X A M P L E calculator close-up This calculator graph supports the graph drawn in Fig Remember that the calculator graph can be misleading. The vertical lines drawn b the calculator are not part of the graph of the function. 0 Graphing a rational function Sketch the graph of each rational function. a) f () b) g() Solution a) From Eample (a) we know that the lines and are vertical asmptotes and the -ais is a horizontal asmptote. Draw the vertical asmptotes on the graph with dashed lines. Since all of the powers of are even, f () f (), and the graph is smmetric about the -ais. The ordered pairs (0, ), (0.9,.789), (.,.86), (, ), and, 8 are also on the graph. Use the smmetr to sketch the graph shown in Fig b) Draw the vertical asmptotes and from Eample (b) as dashed lines. The -ais is a horizontal asmptote. Because f() f (), the graph is smmetric about the origin. The ordered pairs (0, 0),,, (.9,.87), (.,.),,, and, graph shown in Fig are on the graph. Use the smmetr to get the 0 f() = g() = FIGURE 0.6 FIGURE 0.7 E X A M P L E Graphing a rational function Sketch the graph of each rational function. a) h() b) g()
6 0. Graphs of Rational Functions (0-7) calculator close-up This calculator graph supports the graph drawn in Fig Note that if is, there is no -coordinate because is the vertical asmptote Solution a) Draw the vertical asmptote and the horizontal asmptote from Eample (c) as dashed lines. The points (, ), 0,,, 0, (7,.), (, 7), and (,.) are on the graph shown in Fig b) Draw the vertical asmptote and the oblique asmptote from Eample as dashed lines. The points (, 6), 0,, (, 0), (, 6.), and (., 0) are on the graph shown in Fig h() = g() = + + FIGURE 0.8 FIGURE 0.9 WARM-UPS True or false? Eplain our answer.. The domain of f () is 9. False 9. The domain of f () is and. False. The domain of f () is and. False. The line is the onl vertical asmptote for the graph of f (). False. The -ais is a horizontal asmptote for the graph of f (). 9 True 6. The -ais is a horizontal asmptote for the graph of f (). False 7. The line is an asmptote for the graph of f (). True 8. The line is an asmptote for the graph of f (). False 9. The graph of f () is smmetric about the -ais. True 9 0. The graph of f () is smmetric about the origin. True
7 6 (0-8) Chapter 0 Polnomial and Rational Functions 0. EXERCISES Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences.. What is a rational function? A rational function is of the form f() P()Q(), where P() and Q() are polnomials with no common factor and Q() 0.. What is the domain of a rational function? The domain of a rational function is all real numbers ecept those that cause the denominator to be 0.. What is a vertical asmptote? A vertical asmptote is a vertical line that is approached b the graph of a rational function.. What is a horizontal asmptote? A horizontal asmptote is a horizontal line that is approached b the graph of a rational function.. What is an oblique asmptote? An oblique asmptote is a nonhorizontal, nonvertical line that is approached b the graph of a rational function. 6. What is a slant asmptote? A slant asmptote is the same as an oblique asmptote. Find the domain of each rational function. See Eample. 7. f () 8. f () 9. f () 0 0. f () 0. f () and 6. f () and 6 Determine all asmptotes for the graph of each rational function. See Eamples and. 7. f () Vertical: ; horizontal: -ais 8. f () Vertical: 9; horizontal: -ais 9. f () 6 Vertical:, ; horizontal: -ais 6. f () 6 Vertical:, ; horizontal: -ais 7. f () Vertical: 7; horizontal: 7 8. f () 8 Vertical: ; horizontal: 9. f () Vertical: ; oblique: 6 0. f () Vertical: ; oblique: Match each rational function with its graph a h.. f () c. f () e. f () b. f () d. f () g 6. f () h 7. f () f 8. f () a (a) (b) (c)
8 0. Graphs of Rational Functions (0-9) 7 (d) (e) (f) (g) (h) Determine all asmptotes and sketch the graph of each function. See Eamples and. 9. f (), -ais. f () 9,, -ais 0. f (), -ais. f (),, -ais
9 8 (0-0) Chapter 0 Polnomial and Rational Functions. f (), 8. f (),. f (), Find all asmptotes, -intercepts, and -intercepts for the graph of each rational function and sketch the graph of the function. 9. f () 0, 0. f () -ais, 0. f (), 0, 0, 6. f () -ais, 7. f (),. f () 6,, 0, 0,,, 0
10 0. Graphs of Rational Functions (0-) 9. f (), 0, (0, 0) 7. f (), 0, (0, 0). f () 0, 0, (, 0) 8. f (),, 0, (0, 0). f () 0, 0, (, 0) 9. f () 0, (0, ). f () 9 0,, 0,, 0 0. f () 0, (0, 0) 6. f () 0,, 0. f (),, (0, 0)
11 60 (0-) Chapter 0 Polnomial and Rational Functions. f (),, (0, 0) Solve each problem.. Oscillating modulators. The number of oscillating modulators produced b a factor in t hours is given b the polnomial function n(t) t 6t for t. The cost in dollars of operating the factor for t hours is given b the function c(t) 6t 00 for t. The average cost per modulator is given b the rational function f (t) 6 t 00 t 6t for t. Graph the function f. What is the average cost per modulator at time t 0 and time t 0? What can ou conclude about the average cost per modulator after a long period of time? f (0) $., f (0) $.6, average approaches 0 If it costs $,000 to manufacture each SUV, then the average cost per vehicle in dollars when vehicles are manufactured is given b the rational function, ,000,000 A(). a) What is the horizontal asmptote for the graph of this function? b) What is the average cost per vehicle when 0,000 vehicles are made? c) For what number of vehicles is the average cost $0,000? d) Graph this function for ranging from 0 to 00,000. a),000 b) $9,000 c) 0, Average cost of a pill. Assuming Pfizer spent a tpical $0 million to develop its latest miracle drug and $0.0 each to make the pills, then the average cost per pill in dollars when pills are made is given b the rational function 0.0 0,000,000 A().. Nonoscillating modulators. The number of nonoscillating modulators produced b a factor in t hours is given b the polnomial function n(t) 6t for t. The cost in dollars of operating the factor for t hours is given b the function c(t) 6t 00 for t. The average cost per modulator is given b the rational function f(t) 6t 00 for 6t t. Graph the function f. What is the average cost per modulator at time t 0 and t 0? What can ou conclude about the average cost per modulator after a long period of time? f (0) $7., f (0) $.6, average cost approaches $.00. Average cost of an SUV. Mercedes-Benz spent $700 million to design its new 998 SUV (Motor Trend, FIGURE FOR EXERCISE 6
12 0. Transformations of Graphs (0-) 6 a) What is the horizontal asmptote for the graph of this function? b) What is the average cost per pill when 00 million pills are made? c) For what number of pills is the average cost per pill $? d) Graph this function for ranging from 0 to 00 million. a) 0.0 b) $.60 c) 8,0,6 GRAPHING CALCULATOR EXERCISES Sketch the graph of each pair of functions in the same coordinate sstem. What do ou observe in each case? 7. f (), g() The graph of f () is an asmptote for the graph of g(). 8. f (), g() The graph of f () is an asmptote for the graph of g(). 9. f (), g() The graph of f () is an asmptote for the graph of g(). 60. f (), g() The graph of f () is an asmptote for the graph of g(). 6. f (), g() The graph of f () is an asmptote for the graph of g(). 6. f (), g() The graph of f () is an asmptote for the graph of g(). In this section Reflecting Translating Stretching and Shrinking calculator close-up With a graphing calculator, ou can quickl see the result of modifing the formula for a function. If ou have a graphing calculator, use it to graph the functions in the eamples. Eperimenting with it will help ou to understand the ideas in this section TRANSFORMATIONS OF GRAPHS We can discover what the graph of almost an function looks like if we plot enough points. However, it is helpful to know something about a graph so that we do not have to plot ver man points. For eample, it is important to know that the graph of a function is smmetric about the -ais before we begin calculating ordered pairs. In this section we will learn how one graph can be transformed into another b modifing the formula that defines the function. Reflecting Consider the graphs of f() and g() shown in Fig Notice that the graph of g is a mirror image of the graph of f. For an value of we compute the -coordinate of an ordered pair of f b squaring. For an ordered pair of g we square first and then find the opposite because of the order of operations. This gives a correspondence between the ordered pairs of f and the ordered pairs of g. For ever ordered pair on the graph of f there is a corresponding ordered pair directl below it on the graph of g, and these ordered pairs are the same distance from the -ais. We sa that the graph of g is obtained b reflecting the graph of f in the -ais or that g is a reflection of the graph of f. f() = g() = FIGURE Reflection The graph of f () is a reflection in the -ais of the graph of f ().
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