Polynomial and Rational Functions

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1 Polnomial and Rational Functions Figure -mm film, once the standard for capturing photographic images, has been made largel obsolete b digital photograph. (credit film : modification of work b Horia Varlan; credit memor cards : modification of work b Paul Hudson) Chapter Outline. Comple Numbers. Quadratic Functions. Power Functions and Polnomial Functions. Graphs of Polnomial Functions. Dividing Polnomials. Zeros of Polnomial Functions.7 Rational Functions.8 Inverses and Radical Functions.9 Modeling Using Variation Introduction Digital photograph has dramaticall changed the nature of photograph. No longer is an image etched in the emulsion on a roll of film. Instead, nearl ever aspect of recording and manipulating images is now governed b mathematics. An image becomes a series of numbers, representing the characteristics of light striking an image sensor. When we open an image file, software on a camera or computer interprets the numbers and converts them to a visual image. Photo editing software uses comple polnomials to transform images, allowing us to manipulate the image in order to crop details, change the color palette, and add special effects. Inverse functions make it possible to convert from one file format to another. In this chapter, we will learn about these concepts and discover how mathematics can be used in such applications. This OpenSta book is available for free at 97

2 78 CHAPTER Polnomial and Rational Functions Learning Objectives In this section, ou will: Use arrow notation. Solve applied problems involving rational functions. Find the domains of rational functions. Identif vertical asmptotes. Identif horizontal asmptotes. Graph rational functions..7 Rational Functions Suppose we know that the cost of making a product is dependent on the number of items,, produced. This is given b the equation C() =, If we want to know the average cost for producing items, we would divide the cost function b the number of items,. The average cost function, which ields the average cost per item for items produced, is, f () = Man other application problems require finding an average value in a similar wa, giving us variables in the denominator. Written without a variable in the denominator, this function will contain a negative integer power. In the last few sections, we have worked with polnomial functions, which are functions with non-negative integers for eponents. In this section, we eplore rational functions, which have variables in the denominator. Using Arrow Notation We have seen the graphs of the basic reciprocal function and the squared reciprocal function from our stud of toolkit functions. Eamine these graphs, as shown in Figure, and notice some of their features. Graphs of Toolkit Functions f () = f () = Figure Several things are apparent if we eamine the graph of f () = _.. On the left branch of the graph, the curve approaches the -ais ( = 0) as.. As the graph approaches = 0 from the left, the curve drops, but as we approach zero from the right, the curve rises.. Finall, on the right branch of the graph, the curves approaches the -ais ( = 0) as. To summarize, we use arrow notation to show that or f () is approaching a particular value. See Table.

3 SECTION.7 Rational Functions 79 Smbol Meaning a approaches a from the left ( < a but close to a) a + approaches a from the right ( > a but close to a) f () f () f () a approaches infinit ( increases without bound) approaches negative infinit ( decreases without bound) The output approaches infinit (the output increases without bound) The output approaches negative infinit (the output decreases without bound) The output approaches a Table Arrow Notation Local Behavior of f ( ) = Let s begin b looking at the reciprocal function, f () = _. We cannot divide b zero, which means the function is undefined at = 0; so zero is not in the domain. As the input values approach zero from the left side (becoming ver small, negative values), the function values decrease without bound (in other words, the approach negative infinit). We can see this behavior in Table. We write in arrow notation f () = ,000 Table as 0, f () As the input values approach zero from the right side (becoming ver small, positive values), the function values increase without bound (approaching infinit). We can see this behavior in Table f () = ,000 We write in arrow notation See Figure. Table As 0 +, f (). As 0 + f () As f () 0 As f () 0 As 0 f () Figure This OpenSta book is available for free at

4 80 CHAPTER Polnomial and Rational Functions This behavior creates a vertical asmptote, which is a vertical line that the graph approaches but never crosses. In this case, the graph is approaching the vertical line = 0 as the input becomes close to zero. See Figure. = 0 Figure vertical asmptote A vertical asmptote of a graph is a vertical line = a where the graph tends toward positive or negative infinit as the inputs approach a. We write As a, f (), or as a, f (). End Behavior of f ( ) = _ As the values of approach infinit, the function values approach 0. As the values of approach negative infinit, the function values approach 0. See Figure. Smbolicall, using arrow notation As, f () 0, and as, f () 0. As f () 0 As 0 + f () As f () 0 As 0 f () Figure Based on this overall behavior and the graph, we can see that the function approaches 0 but never actuall reaches 0; it seems to level off as the inputs become large. This behavior creates a horizontal asmptote, a horizontal line that the graph approaches as the input increases or decreases without bound. In this case, the graph is approaching the horizontal line = 0. See Figure. = 0 = 0 Figure

5 SECTION.7 Rational Functions 8 horizontal asmptote A horizontal asmptote of a graph is a horizontal line = b where the graph approaches the line as the inputs increase or decrease without bound. We write As or, f () b. Eample Using Arrow Notation Use arrow notation to describe the end behavior and local behavior of the function graphed in Figure. Solution at = Figure Notice that the graph is showing a vertical asmptote at =, which tells us that the function is undefined As, f (), and as +, f (). And as the inputs decrease without bound, the graph appears to be leveling off at output values of, indicating a horizontal asmptote at =. As the inputs increase without bound, the graph levels off at. Tr It # As, f () and as, f (). Use arrow notation to describe the end behavior and local behavior for the reciprocal squared function. Eample Using Transformations to Graph a Rational Function Sketch a graph of the reciprocal function shifted two units to the left and up three units. Identif the horizontal and vertical asmptotes of the graph, if an. Solution Shifting the graph left and up would result in the function f () = + + or equivalentl, b giving the terms a common denominator, The graph of the shifted function is displaed in Figure f () = + = 7 7 Figure 7 This OpenSta book is available for free at 7 =

6 8 CHAPTER Polnomial and Rational Functions Notice that this function is undefined at =, and the graph also is showing a vertical asmptote at =. As, f (), and as +, f (). As the inputs increase and decrease without bound, the graph appears to be leveling off at output values of, indicating a horizontal asmptote at =. As ±, f (). Analsis Notice that horizontal and vertical asmptotes are shifted left and up along with the function. Tr It # Sketch the graph, and find the horizontal and vertical asmptotes of the reciprocal squared function that has been shifted right units and down units. Solving Applied Problems Involving Rational Functions + 7 In Eample, we shifted a toolkit function in a wa that resulted in the function f () =. This is an eample of + a rational function. A rational function is a function that can be written as the quotient of two polnomial functions. Man real-world problems require us to find the ratio of two polnomial functions. Problems involving rates and concentrations often involve rational functions. rational function A rational function is a function that can be written as the quotient of two polnomial functions P() and Q(). P() f () = Q() = a p p + a p p a + a 0, Q() 0 b q q + b q q b + b 0 Eample Solving an Applied Problem Involving a Rational Function A large miing tank currentl contains 00 gallons of water into which pounds of sugar have been mied. A tap will open pouring 0 gallons per minute of water into the tank at the same time sugar is poured into the tank at a rate of pound per minute. Find the concentration (pounds per gallon) of sugar in the tank after minutes. Is that a greater concentration than at the beginning? Solution Let t be the number of minutes since the tap opened. Since the water increases at 0 gallons per minute, and the sugar increases at pound per minute, these are constant rates of change. This tells us the amount of water in the tank is changing linearl, as is the amount of sugar in the tank. We can write an equation independentl for each: water: W(t) = t in gallons sugar: S(t) = + t in pounds The concentration, C, will be the ratio of pounds of sugar to gallons of water + t C(t) = t The concentration after minutes is given b evaluating C(t) at t =. + C() = () 7 = 0 This means the concentration is 7 pounds of sugar to 0 gallons of water. At the beginning, the concentration is + 0 C(0) = (0) = 0 7 Since > = 0.0, the concentration is greater after minutes than at the beginning. 0

7 SECTION.7 Rational Functions 8 Analsis To find the horizontal asmptote, divide the leading coefficient in the numerator b the leading coefficient in the denominator: 0 = 0. Notice the horizontal asmptote is = 0.. This means the concentration, C, the ratio of pounds of sugar to gallons of water, will approach 0. in the long term. Tr It # There are,00 freshmen and,00 sophomores at a prep rall at noon. After p.m., 0 freshmen arrive at the rall ever five minutes while sophomores leave the rall. Find the ratio of freshmen to sophomores at p.m. Finding the Domains of Rational Functions A vertical asmptote represents a value at which a rational function is undefined, so that value is not in the domain of the function. A reciprocal function cannot have values in its domain that cause the denominator to equal zero. In general, to find the domain of a rational function, we need to determine which inputs would cause division b zero. domain of a rational function The domain of a rational function includes all real numbers ecept those that cause the denominator to equal zero. How To Given a rational function, find the domain.. Set the denominator equal to zero.. Solve to find the -values that cause the denominator to equal zero.. The domain is all real numbers ecept those found in Step. Eample Finding the Domain of a Rational Function + Find the domain of f () = _ 9. Solution Begin b setting the denominator equal to zero and solving. 9 = 0 = 9 = ± The denominator is equal to zero when = ±. The domain of the function is all real numbers ecept = ±. Analsis A graph of this function, as shown in Figure 8, confirms that the function is not defined when = ±. = 0 = Figure 8 There is a vertical asmptote at = and a hole in the graph at =. We will discuss these tpes of holes in greater detail later in this section. This OpenSta book is available for free at

8 8 CHAPTER Polnomial and Rational Functions Tr It # Find the domain of f () = ( )( ). Identifing Vertical Asmptotes of Rational Functions B looking at the graph of a rational function, we can investigate its local behavior and easil see whether there are asmptotes. We ma even be able to approimate their location. Even without the graph, however, we can still determine whether a given rational function has an asmptotes, and calculate their location. Vertical Asmptotes The vertical asmptotes of a rational function ma be found b eamining the factors of the denominator that are not common to the factors in the numerator. Vertical asmptotes occur at the zeros of such factors. How To Given a rational function, identif an vertical asmptotes of its graph.. Factor the numerator and denominator.. Note an restrictions in the domain of the function.. Reduce the epression b canceling common factors in the numerator and the denominator.. Note an values that cause the denominator to be zero in this simplified version. These are where the vertical asmptotes occur.. Note an restrictions in the domain where asmptotes do not occur. These are removable discontinuities. Eample Identifing Vertical Asmptotes + Find the vertical asmptotes of the graph of k() =. Solution First, factor the numerator and denominator. + k() = + = ( + )( ) To find the vertical asmptotes, we determine where this function will be undefined b setting the denominator equal to zero: ( + )( ) = 0 =, Neither = nor = are zeros of the numerator, so the two values indicate two vertical asmptotes. The graph in Figure 9 confirms the location of the two vertical asmptotes. = = = 7 Figure 9

9 SECTION.7 Rational Functions 8 Removable Discontinuities Occasionall, a graph will contain a hole: a single point where the graph is not defined, indicated b an open circle. We call such a hole a removable discontinuit. For eample, the function f () = ma be re-written b factoring the numerator and the denominator. f () = ( + )( ) ( + )( ) Notice that + is a common factor to the numerator and the denominator. The zero of this factor, =, is the location of the removable discontinuit. Notice also that is not a factor in both the numerator and denominator. The zero of this factor, =, is the vertical asmptote. See Figure 0. Removable discontinuit at = 7 9 Vertical asmptote at = Figure 0 removable discontinuities of rational functions A removable discontinuit occurs in the graph of a rational function at = a if a is a zero for a factor in the denominator that is common with a factor in the numerator. We factor the numerator and denominator and check for common factors. If we find an, we set the common factor equal to 0 and solve. This is the location of the removable discontinuit. This is true if the multiplicit of this factor is greater than or equal to that in the denominator. If the multiplicit of this factor is greater in the denominator, then there is still an asmptote at that value. Eample Identifing Vertical Asmptotes and Removable Discontinuities for a Graph Find the vertical asmptotes and removable discontinuities of the graph of k() =. Solution Factor the numerator and the denominator. k() = ( )( + ) Notice that there is a common factor in the numerator and the denominator,. The zero for this factor is =. This is the location of the removable discontinuit. Notice that there is a factor in the denominator that is not in the numerator, +. The zero for this factor is =. The vertical asmptote is =. See Figure. This OpenSta book is available for free at

10 8 CHAPTER Polnomial and Rational Functions = 8 Figure The graph of this function will have the vertical asmptote at =, but at = the graph will have a hole. Tr It # Find the vertical asmptotes and removable discontinuities of the graph of f () = +. Identifing Horizontal Asmptotes of Rational Functions While vertical asmptotes describe the behavior of a graph as the output gets ver large or ver small, horizontal asmptotes help describe the behavior of a graph as the input gets ver large or ver small. Recall that a polnomial s end behavior will mirror that of the leading term. Likewise, a rational function s end behavior will mirror that of the ratio of the leading terms of the numerator and denominator functions. There are three distinct outcomes when checking for horizontal asmptotes: Case : If the degree of the denominator > degree of the numerator, there is a horizontal asmptote at = 0. + Eample: f () = + In this case, the end behavior is f () _ = _. This tells us that, as the inputs increase or decrease without bound, this function will behave similarl to the function g() = _, and the outputs will approach zero, resulting in a horizontal asmptote at = 0. See Figure. Note that this graph crosses the horizontal asmptote. = 0 8 = = p( ) Figure Horizontal Asmptote = 0 when f( ) =, q( ) 0 where degree of p < degree of q. q( ) Case : If the degree of the denominator < degree of the numerator b one, we get a slant asmptote. In this case, the end behavior is f () Eample: f () = + _ =. This tells us that as the inputs increase or decrease without bound, this function will behave similarl to the function g() =. As the inputs grow large, the outputs will grow and not level off, so this graph has no horizontal asmptote. However, the graph of g() = looks like a diagonal line, and since f will behave similarl to g, it will approach a line close to =. This line is a slant asmptote.

11 SECTION.7 Rational Functions 87 To find the equation of the slant asmptote, divide +. The quotient is +, and the remainder is. The slant asmptote is the graph of the line g() = +. See Figure = + = p( ) Figure Slant Asmptote when f( ) =, q( ) 0 where degree of p > degree of q b. q( ) Case : If the degree of the denominator = degree of the numerator, there is a horizontal asmptote at = _, where a b n and b n are the leading coefficients of p() and q() for f () = p(), q() 0. n q() Eample: f () = + + In this case, the end behavior is f () _ =. This tells us that as the inputs grow large, this function will behave like the function g() =, which is a horizontal line. As ±, f (), resulting in a horizontal asmptote at =. See Figure. Note that this graph crosses the horizontal asmptote. 9 9 = = Figure Horizontal Asmptote when f ( ) = p( _ ), q( ) 0 where degree of p = degree of q. q( ) Notice that, while the graph of a rational function will never cross a vertical asmptote, the graph ma or ma not cross a horizontal or slant asmptote. Also, although the graph of a rational function ma have man vertical asmptotes, the graph will have at most one horizontal (or slant) asmptote. It should be noted that, if the degree of the numerator is larger than the degree of the denominator b more than one, the end behavior of the graph will mimic the behavior of the reduced end behavior fraction. For instance, if we had the function f () = + with end behavior f () =, the end behavior of the graph would look similar to that of an even polnomial with a positive leading coefficient. ±, f () 9 = a n horizontal asmptotes of rational functions The horizontal asmptote of a rational function can be determined b looking at the degrees of the numerator and denominator. Degree of numerator is less than degree of denominator: horizontal asmptote at = 0. Degree of numerator is greater than degree of denominator b one: no horizontal asmptote; slant asmptote. Degree of numerator is equal to degree of denominator: horizontal asmptote at ratio of leading coefficients. This OpenSta book is available for free at

12 88 CHAPTER Polnomial and Rational Functions Eample 7 Identifing Horizontal and Slant Asmptotes For the functions below, identif the horizontal or slant asmptote. a. g() = 0 + b. h() = + c. k() = p() Solution For these solutions, we will use f () =, q() 0. q() a. g() = _ 0 : The degree of p = degree of q =, so we can find the hori zontal asmptote b taking the ratio + of the leading terms. There is a horizontal asmptote at = _ or =. b. h() = _ + : The degree of p = and degree of q =. Since p > q b, there is a slant asmptote found + at _ +. + The quotient is and the remainder is. There is a slant asmptote at =. c. k() = _ + : The degree of p = < degree of q =, so there is a horizontal asmptote = 0. 8 Eample 8 Identifing Horizontal Asmptotes + t In the sugar concentration problem earlier, we created the equation C(t) = t. Find the horizontal asmptote and interpret it in contet of the problem. Solution Both the numerator and denominator are linear (degree ). Because the degrees are equal, there will be a horizontal asmptote at the ratio of the leading coefficients. In the numerator, the leading term is t, with coefficient. In the denominator, the leading term is 0t, with coefficient 0. The horizontal asmptote will be at the ratio of these values: t, C(t) 0 This function will have a horizontal asmptote at = 0. This tells us that as the values of t increase, the values of C will approach. In contet, this means that, as more time 0 goes b, the concentration of sugar in the tank will approach one-tenth of a pound of sugar per gallon of water or pounds per gallon. 0 Eample 9 Identifing Horizontal and Vertical Asmptotes Find the horizontal and vertical asmptotes of the function ( )( + ) f () = ( )( + )( ) Solution First, note that this function has no common factors, so there are no potential removable discontinuities. The function will have vertical asmptotes when the denominator is zero, causing the function to be undefined. The denominator will be zero at =,, and, indicating vertical asmptotes at these values. The numerator has degree, while the denominator has degree. Since the degree of the denominator is greater than the degree of the numerator, the denominator will grow faster than the numerator, causing the outputs to tend towards zero as the inputs get large, and so as ±, f () 0. This function will have a horizontal asmptote at = 0. See Figure. = 0 8 = = Figure =

13 SECTION.7 Rational Functions 89 Tr It # Find the vertical and horizontal asmptotes of the function: ( )( + ) f () = ( )( + ) intercepts of rational functions A rational function will have a -intercept when the input is zero, if the function is defined at zero. A rational function will not have a -intercept if the function is not defined at zero. Likewise, a rational function will have -intercepts at the inputs that cause the output to be zero. Since a fraction is onl equal to zero when the numerator is zero, -intercepts can onl occur when the numerator of the rational function is equal to zero. Eample 0 Finding the Intercepts of a Rational Function Find the intercepts of f () = ( )( + ) ( )( + )( ). Solution We can find the -intercept b evaluating the function at zero (0 )(0 + ) f (0) = (0 )(0 + )(0 ) = 0 = = 0. The -intercepts will occur when the function is equal to zero: ( )( + ) 0 = This is zero when the numerator is zero. ( )( + )( ) 0 = ( )( + ) =, The -intercept is (0, 0.), the -intercepts are (, 0) and (, 0). See Figure. (, 0) (, 0) 0 (0, 0.) = = = Figure 7 8 = 0 Tr It #7 Given the reciprocal squared function that is shifted right units and down units, write this as a rational function. Then, find the - and -intercepts and the horizontal and vertical asmptotes. This OpenSta book is available for free at

14 90 CHAPTER Polnomial and Rational Functions Graphing Rational Functions In Eample 9, we see that the numerator of a rational function reveals the -intercepts of the graph, whereas the denominator reveals the vertical asmptotes of the graph. As with polnomials, factors of the numerator ma have integer powers greater than one. Fortunatel, the effect on the shape of the graph at those intercepts is the same as we saw with polnomials. The vertical asmptotes associated with the factors of the denominator will mirror one of the two toolkit reciprocal functions. When the degree of the factor in the denominator is odd, the distinguishing characteristic is that on one side of the vertical asmptote the graph heads towards positive infinit, and on the other side the graph heads towards negative infinit. See Figure 7. = = 0 Figure 7 When the degree of the factor in the denominator is even, the distinguishing characteristic is that the graph either heads toward positive infinit on both sides of the vertical asmptote or heads toward negative infinit on both sides. See Figure 8. = = 0 Figure 8 For eample, the graph of f () = ( + ) ( ) is shown in Figure 9. ( + ) ( ) f () = ( + ) ( ) ( + ) ( ) 8 8 (, 0) (, 0) = = = Figure 9

15 SECTION.7 Rational Functions 9 At the -intercept = corresponding to the ( + ) factor of the numerator, the graph bounces, consistent with the quadratic nature of the factor. At the -intercept = corresponding to the ( ) factor of the numerator, the graph passes through the ais as we would epect from a linear factor. At the vertical asmptote = corresponding to the ( + ) factor of the denominator, the graph heads towards positive infinit on both sides of the asmptote, consistent with the behavior of the function f () = _. At the vertical asmptote =, corresponding to the ( ) factor of the denominator, the graph heads towards positive infinit on the left side of the asmptote and towards negative infinit on the right side, consistent with the behavior of the function f () =. How To Given a rational function, sketch a graph.. Evaluate the function at 0 to find the -intercept.. Factor the numerator and denominator.. For factors in the numerator not common to the denominator, determine where each factor of the numerator is zero to find the -intercepts.. Find the multiplicities of the -intercepts to determine the behavior of the graph at those points.. For factors in the denominator, note the multiplicities of the zeros to determine the local behavior. For those factors not common to the numerator, find the vertical asmptotes b setting those factors equal to zero and then solve.. For factors in the denominator common to factors in the numerator, find the removable discontinuities b setting those factors equal to 0 and then solve. 7. Compare the degrees of the numerator and the denominator to determine the horizontal or slant asmptotes. 8. Sketch the graph. Eample Graphing a Rational Function Sketch a graph of f () ( + )( ) = ( + ) ( ). Solution We can start b noting that the function is alread factored, saving us a step. Net, we will find the intercepts. Evaluating the function at zero gives the -intercept: (0 + )(0 ) f (0) = (0 + ) (0 ) = To find the -intercepts, we determine when the numerator of the function is zero. Setting each factor equal to zero, we find -intercepts at = and =. At each, the behavior will be linear (multiplicit ), with the graph passing through the intercept. We have a -intercept at (0, ) and -intercepts at (, 0) and (, 0). To find the vertical asmptotes, we determine when the denominator is equal to zero. This occurs when + = 0 and when = 0, giving us vertical asmptotes at = and =. There are no common factors in the numerator and denominator. This means there are no removable discontinuities. Finall, the degree of denominator is larger than the degree of the numerator, telling us this graph has a horizontal asmptote at = 0. To sketch the graph, we might start b plotting the three intercepts. Since the graph has no -intercepts between the vertical asmptotes, and the -intercept is positive, we know the function must remain positive between the asmptotes, letting us fill in the middle portion of the graph as shown in Figure 0. This OpenSta book is available for free at

16 9 CHAPTER Polnomial and Rational Functions Figure 0 The factor associated with the vertical asmptote at = was squared, so we know the behavior will be the same on both sides of the asmptote. The graph heads toward positive infinit as the inputs approach the asmptote on the right, so the graph will head toward positive infinit on the left as well. For the vertical asmptote at =, the factor was not squared, so the graph will have opposite behavior on either side of the asmptote. See Figure. After passing through the -intercepts, the graph will then level off toward an output of zero, as indicated b the horizontal asmptote. = 0 = = Figure Tr It #8 ( + ) Given the function f () = ( ), use the characteristics of polnomials and rational functions to describe ( ) ( ) its behavior and sketch the function. Writing Rational Functions Now that we have analzed the equations for rational functions and how the relate to a graph of the function, we can use information given b a graph to write the function. A rational function written in factored form will have an -intercept where each factor of the numerator is equal to zero. (An eception occurs in the case of a removable discontinuit.) As a result, we can form a numerator of a function whose graph will pass through a set of -intercepts b introducing a corresponding set of factors. Likewise, because the function will have a vertical asmptote where each factor of the denominator is equal to zero, we can form a denominator that will produce the vertical asmptotes b introducing a corresponding set of factors. writing rational functions from intercepts and asmptotes If a rational function has -intercepts at =,,..., n, vertical asmptotes at = v, v,, v m, and no i = an v j, then the function can be written in the form: ( ) p ( ) p ( n ) p n f () = a ( v ) q ( v ) q ( v m ) q n where the powers p i or q i on each factor can be determined b the behavior of the graph at the corresponding intercept or asmptote, and the stretch factor a can be determined given a value of the function other than the -intercept or b the horizontal asmptote if it is nonzero.

17 SECTION.7 Rational Functions 9 How To Given a graph of a rational function, write the function.. Determine the factors of the numerator. Eamine the behavior of the graph at the -intercepts to determine the zeroes and their multiplicities. (This is eas to do when finding the simplest function with small multiplicities such as or but ma be difficult for larger multiplicities such as or 7, for eample.). Determine the factors of the denominator. Eamine the behavior on both sides of each vertical asmptote to determine the factors and their powers.. Use an clear point on the graph to find the stretch factor. Eample Writing a Rational Function from Intercepts and Asmptotes Write an equation for the rational function shown in Figure. 7 Figure Solution The graph appears to have -intercepts at = and =. At both, the graph passes through the intercept, suggesting linear factors. The graph has two vertical asmptotes. The one at = seems to ehibit the basic behavior similar to, with the graph heading toward positive infinit on one side and heading toward negative infinit on the other. The asmptote at = is ehibiting a behavior similar to _, with the graph heading toward negative infinit on both sides of the asmptote. See Figure. Vertical asmptotes -intercepts Figure We can use this information to write a function of the form ( + )( ) f () = a ( + )( ) This OpenSta book is available for free at

18 9 CHAPTER Polnomial and Rational Functions To find the stretch factor, we can use another clear point on the graph, such as the -intercept (0, ). (0 + )(0 ) = a (0 + )(0 ) = a _ 8 a = _ = _ This gives us a final function of f () ( + )( ) = ( + )( ). Access these online resources for additional instruction and practice with rational functions. Graphing Rational Functions ( Find the Equation of a Rational Function ( Determining Vertical and Horizontal Asmptotes ( Find the Intercepts, Asmptotes, and Hole of a Rational Function (

19 SECTION.7 Section Eercises 9.7 section EXERCISES Verbal. What is the fundamental difference in the algebraic representation of a polnomial function and a rational function?. If the graph of a rational function has a removable discontinuit, what must be true of the functional rule?. Can a graph of a rational function have no -intercepts? If so, how?. What is the fundamental difference in the graphs of polnomial functions and rational functions?. Can a graph of a rational function have no vertical asmptote? If so, how? Algebraic For the following eercises, find the domain of the rational functions.. f () = + 7. f () = + 8. f () = f () = + + For the following eercises, find the domain, vertical asmptotes, and horizontal asmptotes of the functions. 0. f () =. f () = +. f () = 9. f () = +. f () = + 7. f () =. f () = f () = + 8. f () = 9. f () = For the following eercises, find the - and -intercepts for the functions. 0. f () + = +. f () =. f () = f () = f () = 9 For the following eercises, describe the local and end behavior of the functions.. f () = +. f () = 7. f () = 8. f () = + 9. f () = + For the following eercises, find the slant asmptote of the functions. 0. f () = + +. f () = 0. f () = 8 8. f () = +. f () = + + This OpenSta book is available for free at

20 9 CHAPTER Polnomial and Rational Functions GRAPHICAl For the following eercises, use the given transformation to graph the function. Note the vertical and horizontal asmptotes.. The reciprocal function shifted up two units.. The reciprocal function shifted down one unit and left three units. 7. The reciprocal squared function shifted to the right units. 8. The reciprocal squared function shifted down units and right unit. For the following eercises, find the horizontal intercepts, the vertical intercept, the vertical asmptotes, and the horizontal or slant asmptote of the functions. Use that information to sketch a graph. 9. p() = + 0. q() =. s() = ( ). r() = ( + ). f () = + 8. g() = a() = _ +. b() = _ 7. h() = _ + 8. k() = 0 9. w() = ( )( + )( ) 0. z() = ( + ) ( ) ( + ) ( ) ( )( + )( + ) For the following eercises, write an equation for a rational function with the given characteristics.. Vertical asmptotes at = and =, -intercepts at (, 0) and (, 0), -intercept at (0, ). Vertical asmptotes at = and =, -intercepts at (, 0) and (, 0), -intercept at (0, 7). Vertical asmptotes at = and =, -intercepts at (, 0) and (, 0), horizontal asmptote at = 7. Vertical asmptotes at = and =, -intercepts at (, 0) and (, 0), horizontal asmptote at =. Vertical asmptote at =, double zero at =, -intercept at (0, ). Vertical asmptote at =, double zero at =, -intercept at (0, ) For the following eercises, use the graphs to write an equation for the function

21 SECTION.7 Section Eercises Numeric For the following eercises, make tables to show the behavior of the function near the vertical asmptote and reflecting the horizontal asmptote.. f () =. f () = 7. f () = + 8. f () = ( ) 9. f () = + + Technolog For the following eercises, use a calculator to graph f (). Use the graph to solve f () > f () = + 7. f () = 7. f () = ( )( + ) 7. f () = + ( )( ) 7. f () = ( + ) ( ) ( + ) Etensions For the following eercises, identif the removable discontinuit. 7. f () = 7. f () = f () = f () = f () = This OpenSta book is available for free at

22 98 CHAPTER Polnomial and Rational Functions Real-World ApplICATIOns For the following eercises, epress a rational function that describes the situation. 80. A large miing tank currentl contains 00 gallons of water, into which 0 pounds of sugar have been mied. A tap will open, pouring 0 gallons of water per minute into the tank at the same time sugar is poured into the tank at a rate of pounds per minute. Find the concentration (pounds per gallon) of sugar in the tank after t minutes. For the following eercises, use the given rational function to answer the question. 8. A large miing tank currentl contains 00 gallons of water, into which 8 pounds of sugar have been mied. A tap will open, pouring 0 gallons of water per minute into the tank at the same time sugar is poured into the tank at a rate of pounds per minute. Find the concentration (pounds per gallon) of sugar in the tank after t minutes. 8. The concentration C of a drug in a patient s bloodstream t hours after injection in given b t C(t) = _. What happens to the concentration of + t the drug as t increases? 8. The concentration C of a drug in a patient s bloodstream t hours after injection is given b 00t C(t) = _. Use a calculator to approimate the t + 7 time when the concentration is highest. For the following eercises, construct a rational function that will help solve the problem. Then, use a calculator to answer the question. 8. An open bo with a square base is to have a volume of 08 cubic inches. Find the dimensions of the bo that will have minimum surface area. Let = length of the side of the base. 8. A right circular clinder has volume of 00 cubic inches. Find the radius and height that will ield minimum surface area. Let = radius. 8. A rectangular bo with a square base is to have a volume of 0 cubic feet. The material for the base costs 0 cents/square foot. The material for the sides costs 0 cents/square foot. The material for the top costs 0 cents/square foot. Determine the dimensions that will ield minimum cost. Let = length of the side of the base. 87. A right circular clinder with no top has a volume of 0 cubic meters. Find the radius that will ield minimum surface area. Let = radius. 88. A right circular clinder is to have a volume of 0 cubic inches. It costs cents/square inch to construct the top and bottom and cent/square inch to construct the rest of the clinder. Find the radius to ield minimum cost. Let = radius.