3.5 Rational Functions

Transcription

1 0 Chapter Polnomial and Rational Functions Rational Functions For a rational function, find the domain and graph the function, identifing all of the asmptotes Solve applied problems involving rational functions Now we turn our attention to functions that represent the quotient of two polnomials Whereas the sum, difference, or product of two polnomials is a polnomial, in general the quotient of two polnomials is not itself a polnomial A rational number can be epressed as the quotient of two integers, pq, where q 0 A rational function is formed b the quotient of two polnomials, pq, where q 0 Here are some eamples of rational functions and their graphs f() f() f() f() 6 f() f() Rational Function A rational function is a function f that is a quotient of two polnomials, that is, f p, q where p and q are polnomials and where q is not the zero polnomial The domain of f consists of all inputs for which q 0

2 Section Rational Functions 0 domains of functions review section GCM The Domain of a Rational Function EXAMPLE Consider f Find the domain and graph f Solution When the denominator is 0, we have, so the onl input that results in a denominator of 0 is Thus the domain is,or,, The graph of this function is the graph of translated to the right units Two versions of the graph on a graphing calculator are shown below CONNECTED MODE DOT MODE Using CONNECTED mode can lead to an incorrect graph In CON- NECTED mode, a graphing calculator connects plotted points with line segments In DOT mode, it simpl plots unconnected points In the first graph, the graphing calculator has connected the points plotted on either side of the -value with a line that appears to be the vertical line (It is not actuall vertical since it connects the last point to the left of with the first point to the right of ) Since is not in the domain of the function, the vertical line cannot be part of the graph We will see later in this section that vertical lines like, although not part of the graph, are important in the construction of graphs If ou have a choice when graphing rational functions, use DOT mode EXAMPLE Determine the domain of each of the functions illustrated at the beginning of this section Solution The domain of each rational function will be the set of all real numbers ecept those values that make the denominator 0 To determine those eceptions, we set the denominator equal to 0 and solve for

3 06 Chapter Polnomial and Rational Functions FUNCTION f f DOMAIN 0,or,0 0, 0,or,0 0, f f 6 f f and,or,,,,or,, and,or,,,,or,, As a partial check of the domains, we can observe the discontinuities (breaks) in the graphs of these functions (See page 0) From left: f() 6 7 From right: Vertical asmptote: Asmptotes Look at the graph of f, shown at left (Also see Eample ) Let s eplore what happens as -values get closer and closer to from the left We then eplore what happens as -values get closer and closer to from the right From left: f 00 0,000,000,000 From right: f , ,999,000,000,000, , ,000,000,000 We see that as -values get closer and closer to from the left, the function values (-values) decrease without bound (that is, the approach negative infinit, ) Similarl, as the -values approach from the right, the function values increase without bound (that is, the approach positive infinit, ) We write this as f l as l and f l as l We read f l as l as f decreases without bound as approaches from the left We read f l as l as f increases without bound as approaches from the right The notation l means that gets as close to as possible without being equal to The vertical line is said to be a vertical asmptote for this curve

4 Section Rational Functions 07 In general, the line a is a vertical asmptote for the graph of f if an of the following is true: f l as l a or f l as l a, or f l as l a or f l as l a The following figures show the four was in which a vertical asmptote can occur a a a a f() f() f() f() f() as a f() as a f() as a f() as a The vertical asmptotes of a rational function f pq are found b determining the zeros of q that are not also zeros of p If p and q are polnomials with no common factors other than constants, we need determine onl the zeros of the denominator q Figure f() Figure g() Determining Vertical Asmptotes For a rational function f pq, where p and q are polnomials with no common factors other than constants, if a is a zero of the denominator, then the line a is a vertical asmptote for the graph of the function EXAMPLE Determine the vertical asmptotes for the graph of each of the following functions a) f b) g 8 Solution a) We factor to find the zeros of the denominator: 8 The zeros of the denominator are and Thus the vertical asmptotes are the lines and (See Fig ) b) We factor to find the zeros of the denominator: Solving 0 we get 0 or 0 0 or 0 or The zeros of the denominator are 0,, and Thus the vertical asmptotes are the lines 0,, and (See Fig )

5 08 Chapter Polnomial and Rational Functions f() Horizontal asmptote: 0 Looking again at the graph of f, shown at left (also see Eample ), let s eplore what happens to f as increases without bound (approaches positive infinit, ) and as decreases without bound (approaches negative infinit, ) increases without bound: ,000,000 f decreases without bound: ,000,000 f We see that as and as l l 0 l l 0 Since 0 is the equation of the -ais, we sa that the curve approaches the -ais asmptoticall and that the -ais is a horizontal asmptote for the curve In general, the line b is a horizontal asmptote for the graph of f if either or both of the following are true: f l b as l or f l b as l The following figures illustrate four was in which horizontal asmptotes can occur In each case, the curve gets close to the line b either as l or as l Keep in mind that the smbols and conve the idea of increasing without bound and decreasing without bound, respectivel b b f() b f() b f() f() f() b as f() b as f() b as f() b as How can we determine a horizontal asmptote? As gets ver large or ver small, the value of the polnomial function p is dominated b the function s leading term Because of this, if p and q have the same degree, the value of pq as l or as l is dominated b the ratio of the numerator s leading coefficient to the denominator s leading coefficient

6 Section Rational Functions f() For f, we see that the numerator,, is dominated b and the denominator,, is dominated b, so f approaches, or as gets ver large or ver small:,or, as l, and l,or, as l l We sa that the curve approaches the horizontal line asmptoticall and that is a horizontal asmptote for the curve It follows that when the numerator and the denominator of a rational function have the same degree, the line ab is the horizontal asmptote, where a and b are the leading coefficients of the numerator and the denominator, respectivel EXAMPLE Find the horizontal asmptote: f 7 0 Solution The numerator and the denominator have the same degree The ratio of the leading coefficients is 7, so the line 7, or 06, is the horizontal asmptote X X Y To check Eample, we could use a graphing calculator to evaluate the function for a ver large and a ver small value of (See the window at left) Another check, one that is useful in calculus, is to multipl b, using : f As becomes ver large, each epression with a power of in the denominator tends toward 0 Specificall, as l or as l, we have f l, or f l The horizontal asmptote is 7, or 06 We now investigate the occurrence of a horizontal asmptote when the degree of the numerator is less than the degree of the denominator

7 0 Chapter Polnomial and Rational Functions EXAMPLE Find the horizontal asmptote: f Solution We let p, q, and f pqnote that as l, the value of q grows much faster than the value of p Because of this, the ratio pq shrinks toward 0 As l, the ratio pq behaves in a similar manner The horizontal asmptote is 0, the -ais This is the case for all rational functions for which the degree of the numerator is less than the degree of the denominator Note in Eample that 0, the -ais, is the horizontal asmptote of f The following statements describe the two was in which a horizontal asmptote occurs Determining a Horizontal Asmptote When the numerator and the denominator of a rational function have the same degree, the line ab is the horizontal asmptote, where a and b are the leading coefficients of the numerator and the denominator, respectivel When the degree of the numerator of a rational function is less than the degree of the denominator, the -ais, or 0, is the horizontal asmptote When the degree of the numerator of a rational function is greater than the degree of the denominator, there is no horizontal asmptote The following statements are also true The graph of a rational function never crosses a vertical asmptote The graph of a rational function might cross a horizontal asmptote but does not necessaril do so EXAMPLE 6 Graph g Include and label all asmptotes Solution Since 0 is the zero of the denominator, the -ais, 0, is the vertical asmptote Note also that the degree of the numerator is the same as the degree of the denominator Thus,, or, is the horizontal asmptote To draw the graph, we first draw the asmptotes with dashed lines Then we compute and plot some ordered pairs and draw the two branches of the curve We can check the graph with a graphing calculator

8 Section Rational Functions g() Horizontal asmptote Vertical asmptote g 9 9 Sometimes a line that is neither horizontal nor vertical is an asmptote Such a line is called an oblique asmptote,or a slant asmptote EXAMPLE 7 Find all the asmptotes of f Solution The line is the vertical asmptote because is the zero of the denominator There is no horizontal asmptote because the degree of the numerator is greater than the degree of the denominator When the degree of the numerator is greater than the degree of the denominator, we divide to find an equivalent epression: Now we see that when l or l, l 0 and the value of f l This means that as becomes ver large, the graph of f gets ver close to the graph of Thus the line is the oblique asmptote f()

9 Chapter Polnomial and Rational Functions Occurrence of Lines as Asmptotes of Rational Functions For a rational function f pq, where p and q have no common factors other than constants: Vertical asmptotes occur at an -values that make the denominator 0 The -ais is the horizontal asmptote when the degree of the numerator is less than the degree of the denominator A horizontal asmptote other than the -ais occurs when the numerator and the denominator have the same degree An oblique asmptote occurs when the degree of the numerator is greater than the degree of the denominator There can be onl one horizontal asmptote or one oblique asmptote and never both An asmptote is not part of the graph of the function The following is an outline of a procedure that we can follow to create accurate graphs of rational functions To graph a rational function f pq, where p and q have no common factor other than constants: Find the real zeros of the denominator Determine the domain of the function and sketch an vertical asmptotes Find the horizontal or the oblique asmptote, if there is one, and sketch it Find the zeros of the function The zeros are found b determining the zeros of the numerator These are the first coordinates of the -intercepts of the graph Find f0 This gives the -intercept 0, f0, of the function Find other function values to determine the general shape Then draw the graph EXAMPLE 8 Graph: f 7 6 Solution We find the zeros of the denominator b solving Since 7 6, the zeros are and Thus the domain ecludes and and is,,,

10 Section Rational Functions The graph has vertical asmptotes and We sketch these as dashed lines Because the degree of the numerator is less than the degree of the denominator, the -ais, 0, is the horizontal asmptote To find the zeros of the numerator, we solve 0 and get Thus, is the zero of the function, and the pair,0 is the -intercept We find f0: f Thus, 0, is the -intercept We find other function values to determine the general shape and then draw the graph Note that the graph of this function crosses its horizontal asmptote at (w, 0) 0 f() (0, q) s 7 6 EXAMPLE 9 Graph: g 6 Solution We find the zeros of the denominator b solving 6 0 Since 6, the zeros are and Thus the domain ecludes the -values and and is,,, The graph has vertical asmptotes and We sketch these as dashed lines The numerator and the denominator have the same degree, so the horizontal asmptote is determined b the ratio of the leading coefficients:, or Thus is the horizontal asmptote We sketch it with a dashed line

11 Chapter Polnomial and Rational Functions To find the zeros of the numerator, we solve 0 The solutions are and Thus, and are the zeros of the function and the pairs, 0 and, 0 are the -intercepts We find g0: g Thus, 0, 6 is the -intercept We find other function values to determine the general shape and then draw the graph g() 6 (, 0) (, 0) (0, Z) 6 Curve crosses the line at (, ) ( )/( 6), Xscl, Yscl 0 The magnified portion of the graph in Eample 9 at left shows another situation in which a graph can cross its horizontal asmptote The point where g crosses can be found b setting g and solving for : Subtracting Adding 6 The point of intersection is, Let s observe the behavior of the curve after it crosses the horizontal asmptote at (See the graph at left) It continues to decrease for a short interval and then begins to increase, getting closer and closer to as l Graphs of rational functions can also cross an oblique asmptote The graph of f

12 Section Rational Functions shown below crosses its oblique asmptote Remember, graphs can cross horizontal or oblique asmptotes, but the cannot cross vertical asmptotes Oblique asmptote f() 0 X X Y ERROR ERROR GCM Let s now graph a rational function f pq, where p and q have a common factor EXAMPLE 0 Graph: g Solution We first epress the denominator in factored form: g The domain of the function is and, or,,,the zeros of the denominator are and, and the zero of the numerator is Since is the onl zero of the denominator that is not a zero of the numerator, the graph of the function has as its onl vertical asmptote The degree of the numerator is less than the degree of the denominator, so 0 is the horizontal asmptote There are no zeros of the function and thus no -intercepts, because is the onl zero of the numerator and is not in the domain of the function Since g0, 0, is the -intercept We draw the graph indicating the hole when with an open circle g() ( )( )

13 6 Chapter Polnomial and Rational Functions 7 7 The rational epression can be simplified Thus, g, where and The graph of g is the graph of with the point where missing To determine the coordinates of the hole, we substitute for in g : g Thus the hole is located at, With certain window dimensions, the hole is visible on a graphing calculator Applications T() Xscl, Yscl 0 Maimum 0 X Y Xscl, Yscl EXAMPLE Temperature During an Illness The temperature T, in degrees Fahrenheit, of a person during an illness is given b the function Tt t, t 986 where time t is given in hours since the onset of the illness a) Graph the function on the interval 0, 8 b) Find the temperature at t 0,,,,, and c) Find the horizontal asmptote of the graph of Tt Complete: Tt l as t l d) Give the meaning of the answer to part (b) in terms of the application e) Find the maimum temperature during the illness Solution a) The graph is shown at left b) We have T0 986, T 006, T 00, T 9969, T 989, and T c) Since Tt t t t t 986, t the horizontal asmptote is 986, or 986 Then it follows that Tt l 986 as t l d) As time goes on, the temperature returns to normal, which is 986 e) Using the MAXIMUM feature on a graphing calculator, we find the maimum temperature to be 006 at t hr

14 Section Rational Functions 7 A Visualizing the Graph Match the function with its graph F B f A f C G f D 6 C 6 6 f H f G 6 6 f F H 6 D 6 E 6 7 f B 8 f I 8 9 f J 6 0 f E Answers on page A- I J

15 8 Chapter Polnomial and Rational Functions Eercise Set In Eercises 6, use our knowledge of asmptotes and intercepts to match the equation with one of the graphs (a) (f), which follow List all asmptotes Check our work using a graphing calculator 0 80 a) b) c) 80 d) e) 0 f ) f 8 f 8 (f); 0 f 8 f 8 6 f 8 f 8 (b); 8 Determine the vertical asmptotes of the graph of each of the following functions 7 g h 7 g f, f h 0,, g, f, 8 Determine the horizontal asmptote of the graph of each of the following functions f 6 7 h g No horizontal asmptote 6 0 Determine the oblique asmptote of the graph of each of the following functions h 8 0 g f 6 h g 6 f g 6 f 6 h 0 Answers to Eercises and can be found on p IA-7

16 Section Rational Functions 9 Make a hand-drawn graph for each of the following Be sure to label all the asmptotes List the domain and the - and -intercepts Check our work using a graphing calculator 7 f 8 g 9 h 0 f 6 g h f f f 6 f 7 f 8 f 9 f 0 f f f f f f 6 f 9 7 f 8 f 9 f 0 f f 9 f f f g 6 f 7 f 8 f 9 f 60 f 6 f Answers to Eercises 7 6 can be found on pp IA-8 through IA-0

17 0 Chapter Polnomial and Rational Functions 6 f 6 f 6 f 6 f 66 f 6 67 f 68 f Find a rational function that satisfies the given conditions for each of the following Answers ma var, but tr to give the simplest answer possible 69 Vertical asmptotes, 70 Vertical asmptotes, ; -intercept, 0 7 Vertical asmptotes, ; horizontal asmptote ; -intercept, 0 7 Oblique asmptote 7 Medical Dosage The function Nt 08t 000 t gives the bod concentration Nt, in parts per million, of a certain dosage of medication after time t, in hours a) Graph the function on the interval, and complete the following: Nt l as t l b) Eplain the meaning of the answer to part (a) in terms of the application, t 7 Average Cost The average cost per DVD, in dollars, for a compan to produce DVDs on eercising is given b the function A a) Graph the function on the interval 0, and complete the following: A l as l b) Eplain the meaning of the answer to part (a) in terms of the application 7 Population Growth The population P,in thousands, of Lordsburg is given b Pt 00t, t 9 where t is the time, in months a) Graph the function on the interval 0, b) Find the population at t 0,,, and 8 months c) Find the horizontal asmptote of the graph and complete the following: Pt l 0 as t l Pt l as t l d) Eplain the meaning of the answer to part (c) in terms of the application e) Find the maimum population and the value of t that will ield it 8,96 at t months 76 Minimizing Surface Area The Hold-It Container Co is designing an open-top rectangular bo, with a square base, that will hold 08 cubic centimeters 00, 0 a) Epress the surface area S as a function of the length of a side of the base b) Use a graphing calculator to graph the function on the interval 0, c) Estimate the minimum surface area and the value of that will ield it Answers to Eercises 6 7, 7(a), 7(b), 7(d), and 76 can be found on pp IA-0 and IA-

18 Section Rational Functions 77 Graph and using the same viewing window Eplain how the parabola can be thought of as a nonlinear asmptote for Collaborative Discussion and Writing 78 Eplain wh the graph of a rational function cannot have both a horizontal and an oblique asmptote 79 Under what circumstances will a rational function have a domain consisting of all real numbers? Skill Maintenance In each of Eercises 80 88, fill in the blank with the correct term Some of the given choices will not be used Others will be used more than once -intercept -intercept odd function even function domain range slope distance formula midpoint formula horizontal lines vertical lines point slope equation slope intercept equation difference quotient f f f f 80 A function is a correspondence between a first set, called the the, and a second set, called, such that each member of the corresponds to eactl one member of the [] domain, range, domain, range 8 The of a line containing, and, is given b [] slope 8 The of the line with slope m and -intercept 0, b is m b [] slope intercept equation 8 The of the line with slope m passing through, is m [] point slope equation 8 A(n) is a point a,0 [] -intercept 8 For each in the domain of an odd function f, [7] f f 86 are given b equations of the tpe a [] Vertical lines 87 The is, [] midpoint formula 88 A(n) is a point 0, b [] -intercept Snthesis Find the nonlinear asmptote of the function 89 f 90 f Graph the function 9 f f 6 Find the domain of the function 9 f 7, 7, 9 f, 7, Answers to Eercises 77, 9, and 9 can be found on p IA-