Rational Functions, Equations, and Inequalities

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2 Chapter 5 Rational Functions, Equations, and Inequalities GOALS You will be able to Graph the reciprocal functions of linear and quadratic functions Identif the ke characteristics of rational functions from their equations and use these characteristics to sketch their graphs Solve rational equations and inequalities with and without graphing technolog Determine average and instantaneous rates of change in situations that are modelled b rational functions? When polluted water flows into a clean pond, how does the concentration of pollutant in the pond change over time? What tpe of function would model this change? 5

3 5 Getting Started Stud Aid For help, see the Review of Essential Skills found at the Nelson Advanced Functions website. Question Appendi R-3, 3,, 8 R- 7 R-8 SKILLS AND CONCEPTS You Need. Factor each epression. a) 3 d) b) 3 5 e) c) 6 9 f) 9 3a a Simplif each epression. State an restrictions on the variables, if necessar. 8s 5 a) d) 5(5 ) 6m n 3 8 b) e) 8m 3 n a ab 5b c) f ) 3 a 7ab 5b 3. Simplif each epression, and state an restrictions on the variable. a) 3 c) b) 3 d) Simplif each epression, and state an restrictions on the variable. a) 5 d) b) 3 e) c) 6 f ) a 9a 8 a a 5 5. Solve and check. a) c) b) d) Sketch the graph of the reciprocal function f () 5 and describe its characteristics. Include the domain and range, as well as the equations of the asmptotes. 6 Getting Started

4 Getting Started 7. List the transformations that need to be applied to 5 to graph each of the following reciprocal functions. Then sketch the graph. a) f () 5 c) f () 5 b) f () d) f () 5 3( ) 8. Describe the steps that are required to divide two rational epressions. 9 Use our description to simplif APPLYING What You Know Painting Houses Ton can paint the eterior of a house in si working das. Rebecca takes nine das to complete the same painting job.? How long will Rebecca and Ton take to paint a similar house, if the work together? A. What fraction of the job can Ton complete in one da? What fraction of the job can Rebecca complete? B. Write a numerical epression to represent the fraction of the job that Rebecca and Ton can complete in one da, if the work together. C. Let represent the number of das that Rebecca and Ton, working together, will take to complete the job. Eplain wh represents the fraction of the job Rebecca and Ton will complete in one da when the work together. D. Use our answers for parts B and C to write an equation. Determine the lowest common denominator for the rational epressions in our equation. Rewrite the equation using the lowest common denominator. E. Solve the equation ou wrote in part D b collecting like terms and comparing the numerators on the two sides of the equation. F. What is the amount of time Rebecca and Ton will take to paint a similar house, when the work together? Chapter 5 7

5 5. Graphs of Reciprocal Functions YOU WILL NEED graph paper coloured pencils or pens graphing calculator or graphing software 5 f() = 5 5 GOAL Sketch the graphs of reciprocals of linear and quadratic functions. INVESTIGATE the Math Owen has noted some connections between the graphs of f () 5 and its reciprocal function g() 5. Both graphs are in the same quadrants for the same -values. When f () 5, there is a vertical asmptote for g(). f () is alwas increasing, and g() is alwas decreasing. 5 g() =? How are the graphs of a function and its reciprocal function related? A. Eplain wh the graphs of f () 5 and g() 5 are in the same quadrants over the same intervals. Does this relationship hold for m() 5 and n () 5? Does this relationship hold for an function and its reciprocal function? Eplain. B. What graphical characteristic in the reciprocal function do the zeros of the original function correspond to? Eplain. C. Eplain wh the reciprocal function g() 5 is decreasing when f () 5 is increasing. Does this relationship hold for n () 5 and m() 5? Eplain how the increasing and decreasing intervals of a function and its reciprocal are related. D. What are the -coordinates of the points where f () and g() intersect? Will the points of intersection for an function and its reciprocal alwas have the same -coordinates? Eplain. E. Eplain wh the graph of g() has a horizontal asmptote. What is the equation of this asmptote? Will all reciprocal functions have the same horizontal asmptote? Eplain Graphs of Reciprocal Functions

6 F. On graph paper, draw the graph of p() 5. In a table like the one below, note the characteristics of the graph of p() and use this information to help ou determine the characteristics of the reciprocal function q() Characteristics p() 5 q() 5 zeros and/or vertical asmptotes interval(s) on which the graph is above the -ais (all values of the function are positive) interval(s) on which the graph is below the -ais (all values of the function are negative) interval(s) on which the function is increasing interval(s) on which the function is decreasing point(s) where the -value is point(s) where the -value is G. On the same graph, draw the vertical asmptotes for the reciprocal function. Then use the rest of the information determined in part F to draw the graph for q() 5. H. Verif our graphs b entering p() and q() in a graphing calculator using the friendl window setting shown. I. Repeat parts F to H for the following pairs of functions. a) p() 5 and q() 5 b) p() 5 3 and q() 5 3 c) p() 5 ( )( 3) and q() 5 d) p() 5 ( ) and q() 5 ( ) ( )( 3) J. Write a summar of the relationships between the characteristics of the graphs of a) a linear function and its reciprocal function b) a quadratic function and its reciprocal function Tech Support On a graphing calculator, the length of the displa screen contains 9 piels, and the width contains 6 piels. When the domain, Xma-Xmin, is cleanl divisible b 9, and the range, Yma-Ymin, is cleanl divisible b 6, the window is friendl. This means that ou can trace without using ugl decimals. A friendl window is useful when working with rational functions. Use brackets when entering reciprocal functions in the Y 5 editor of a graphing calculator. For eample, to graph the function f() 5, enter Y 5. ( ) Chapter 5 9

7 Reflecting K. How did knowing the positive/negative intervals and the increasing/decreasing intervals for p() 5 help ou draw the graph for p() 5? L. Wh are some numbers in the domain of a function ecluded from the domain of its reciprocal function? What graphical characteristic of the reciprocal function occurs at these values? M. What common characteristics are shared b all reciprocals of linear and quadratic functions? APPLY the Math EXAMPLE Connecting the characteristics of a linear function to its corresponding reciprocal function Given the function f () 5, a) determine the domain and range, intercepts, positive/negative intervals, and increasing/decreasing intervals b) use our answers for part a) to sketch the graph of the reciprocal function Solution a) f () 5 is a linear function. D 5 5PR6 R 5 5PR6 From the equation, the -intercept is. f () 5 when 5 5 The -intercept is. The domain and range of most linear functions are the set of real numbers. A linear function f() 5 m b has -intercept b. The -intercept occurs where f() = 5 f () is positive when P(`, ) and negative when P(, `). f () is decreasing when P(`, `). Sketch the graph of f() to determine the positive and negative intervals. The line 5 is above the -ais for all -values less than and below the -ais for all -values greater than. This is a linear function with a negative slope, so it is decreasing over its entire domain Graphs of Reciprocal Functions

8 b) The reciprocal function is g() 5. D 5 5PR 6 R 5 5 PR 6 The -intercept is.5. The vertical asmptote is 5 and the horizontal asmptote is 5. The reciprocal function is positive when P(`, ) and negative when P(, `). It is increasing when P(`, ) and when P(, `). The graph of g() 5 intersects the graph of g() 5 at (, ) and (3, ). All the -values of points on the reciprocal function are reciprocals of the -values on the original function. There is a vertical asmptote at the zero of the original function. The reciprocals of a linear function alwas have the -ais as a horizontal asmptote. The positive/negative intervals are alwas the same for both functions. Because the original function is alwas decreasing, the reciprocal function is alwas increasing. The reciprocal of is, and the reciprocal of is. Thus, the two graphs intersect at an points with these -values. 5. = = 5 = Use all this information to sketch the graph of the reciprocal function. EXAMPLE Connecting the characteristics of a quadratic function to its corresponding reciprocal function Given the function f () 5 9 a) determine the domain and range, intercepts, positive/negative intervals, and increasing/decreasing intervals b) use our answers for part a) to sketch the graph of the reciprocal function Solution a) f () 5 9 is a quadratic function. D 5 5PR6 The domain of a quadratic function is the set of real numbers. f () 5 9, so the -intercept is 9. R 5 5PR # 96 The graph of f() is a parabola that opens down. The verte is at (, 9), so # 9. Chapter 5 5

9 f () (3 )(3 ) The -intercepts are 3 and 3. = 9 Factor and determine the -intercepts. 5 The parabola 5 9 is above the -ais for -values between 3 and 3. The graph is below the -ais for -values less than 3 and for -values greater than 3. 5 f () is positive when P(3, 3) and negative when P(`, 3) and when P(3, `). f () is increasing when P(`, ) and decreasing when P(, `). b) The reciprocal function is g() 5. 9 The vertical asmptotes are 53and 5 3. D 5 5PR 636 The horizontal asmptote is 5. The -intercept is. 9 There is a local minimum value at a,. 9 b R 5 UPR, or $ 9 V The -values increase as increases from ` to. The -values decrease as increases from to `. Vertical asmptotes occur at each zero of the original function, so these numbers must be ecluded from the domain. The reciprocals of all quadratic functions have the -ais as a horizontal asmptote. The -intercept of the original function is 9, so the -intercept of the reciprocal function is. 9 When the original function has a local maimum point, the reciprocal function has a corresponding local minimum point Graphs of Reciprocal Functions

10 The reciprocal function is positive when P(3, 3) and negative when P(`, 3) and when P(3, `). It is decreasing when P(`, 3) and when P(3, ), and increasing when P(, 3) and when P(3, `). f () 5 when 9 5 and f () 5when " 56" The positive/negative intervals are alwas the same for both functions. Where the original function is decreasing, ecluding the zeros, the reciprocal function is increasing (and vice versa). A function and its reciprocal intersect at points where 56. Solve the corresponding equations to determine the -coordinates of the points of intersection. 5. The graph of g() 5 intersects the graph of f () at (", ), (", ) and at (", ), (", ). = 9 5 = 9 Use all this information to sketch the graph of the reciprocal function. 5 = 3 = 3 In Summar Ke Idea You can use ke characteristics of the graph of a linear or quadratic function to graph the related reciprocal function. Need to Know All the -coordinates of a reciprocal function are the reciprocals of the -coordinates of the original function. The graph of a reciprocal function has a vertical asmptote at each zero of the original function. A reciprocal function will alwas have 5 as a horizontal asmptote if the original function is linear or quadratic. A reciprocal function has the same positive/negative intervals as the original function. (continued) Chapter 5 53

11 Intervals of increase on the original function are intervals of decrease on the reciprocal function. Intervals of decrease on the original function are intervals of increase on the reciprocal function. If the range of the original function includes and/or, the reciprocal function will intersect the original function at a point (or points) where the -coordinate is or. If the original function has a local minimum point, the reciprocal function will have a local maimum point at the same -value (and vice versa). A linear function and its reciprocal = = A quadratic function and its reciprocal = (+) = (+) = = 3 = Both functions are negative when P(`, ) and positive when P(, `). The original function is increasing when P(`, `). The reciprocal function is decreasing when P(`, ) or (, `). Both functions are negative when P(3, ) and positive when P(`, 3) or (, `). The original function is decreasing when P(`, ) and increasing when P(, `). The reciprocal function is increasing when P(`, 3) or (3, ) and decreasing when P(, ) or (, `). CHECK Your Understanding. Match each function with its equation on the net page. Then identif which function pairs are reciprocals. a) c) e) b) d) f) 5 5. Graphs of Reciprocal Functions

12 A B C. For each pair of functions, determine where the zeros of the original function occur and state the equations of the vertical asmptotes of the reciprocal function, if possible. a) b) c) d) e) f) ( ) 5 f () 5 6, g() 5 6 f () 5 3, g() 5 3 f () 5 5, g() 5 5 f () 5 5, g() 5 5 f () 5, g() 5 f () 5 5 3, g() D E F 5 5 ( ) Sketch the graph of each function. Use our graph to help ou sketch the graph of the reciprocal function. a) f () 5 5 b) f () 5 6 PRACTISING. a) Cop and complete the following table f() f() b) Sketch the graphs of 5 f () and 5. f () c) Find equations for 5 f () and 5. f () 5. State the equation of the reciprocal of each function, and determine the equations of the vertical asmptotes of the reciprocal. Verif our results using graphing technolog. a) f () 5 e) f () 53 6 b) f () 5 5 f) f () 5 ( 3) c) f () 5 g) f () 5 3 d) f () 5 5 h) f () 5 3 Chapter 5 55

13 6. Sketch the graph of the reciprocal of each function. a) c) 6 = f () 6 = f () 6 6 b) d) 6 6 = f () 6 = f () 6 7. Sketch each pair of graphs on the same aes. State the domain and range of each reciprocal function. a) 5 5, 5 b) 5 3, Draw the graph of 5 f () and 5 on the same aes. f () a) f () 5 d) f () 5 ( 3) b) f () 5 ( ) 3 e) f () 5 c) f () 5 3 f) f () 5( ) 9. For each function, determine the domain and range, intercepts, K positive/negative intervals, and increasing/decreasing intervals. State the equation of the reciprocal function. Then sketch the graphs of the original and reciprocal functions on the same aes. a) f () 5 8 c) f () 5 b) f () 5 3 d) f () 5. Wh do the graphs of reciprocals of linear functions alwas have vertical asmptotes, but the graphs of reciprocals of quadratic functions sometimes do not? Provide sketches of three different reciprocal functions to illustrate our answer Graphs of Reciprocal Functions

14 5. k. An equation of the form 5 has a graph that closel matches b c the graph shown. Find the equation. Check our answer using graphing technolog.. A chemical compan is testing the effectiveness of a new cleaning A solution for killing bacteria. The test involves introducing the solution into a sample that contains approimatel bacteria. The number of bacteria remaining, b(t), over time, t, in seconds is given b the equation b(t) 5. t a) How man bacteria will be left after s? b) After how man seconds will onl 5 bacteria be left? c) After how man seconds will onl one bacterium be left? d) This model is not alwas accurate. Determine what sort of inaccuracies this model might have. Assume that the solution was introduced at t 5. e) Based on these inaccuracies, what should the domain and range of the equation be? 6 = = 6 3. Use our graphing calculator to eplore and then describe the ke T characteristics of the famil of reciprocal functions of the form g() 5. Make sure that ou include graphs to support our n descriptions. a) State the domain and range of g(). b) For the famil of functions f () 5 n, the -intercept changes as the value of n changes. Describe how the -intercept changes and how this affects g(). c) If graphed, at what point would the two graphs f () and g() intersect?. Due to a basketball tournament, our friend has missed this class. C Write a concise eplanation of the steps needed to graph a reciprocal function using the graph of the original function (without using graphing technolog). Use an eample, and eplain the reason for each step. Etending 5. Sketch the graphs of the following reciprocal functions. a) 5 c) " b) 5 d) sin 6. Determine the equation of the function in the graph shown. 6 6 Chapter 5 57

15 YOU WILL NEED 5. graph paper coloured pencils or pens graphing calculator or graphing software Eploring Quotients of Polnomial Functions EXPLORE the Math Each row shows the graphs of two polnomial functions. A. GOAL Eplore graphs that are created b dividing polnomial functions. f() = g() = rational function a function that can be epressed as f() 5 p(), q() where p() and q() are polnomial functions, q() (e.g., f() 5 3,, and f() 5,, are rational functions, but f() 5 is #,, not because its denominator is not a polnomial) B. C. m() = p() = 5 n() = + 5 q() =? What are the characteristics of the graphs that are created b dividing two polnomial functions? A. Using the given functions, write the equation of the rational function 5 f (). Enter this equation into Y of the equation editor of a g() graphing calculator. Graph this equation using the window settings shown, and draw a sketch Eploring Quotients of Polnomial Functions

16 B. Describe the characteristics of the graph ou created in part A b answering the following questions: i) Where are the zeros? ii) Are there an asmptotes? If so, where are the? iii) What are the domain and range of this function? iv) Is it a continuous function? Eplain. v) Are there an values of 5 f () that are undefined? What g() feature(s) of the graph is (are) related to these values? vi) Describe the end behaviours of this function. vii) Is the resulting graph a function? Eplain. C. Write the equation defined b 5 g(). Predict how the graph of f () this function will differ from the graph of 5 f (). Graph this g() function using our graphing calculator, and draw a sketch. D. Describe the characteristics of the graph ou created in part C b answering the questions in part B. Tech Support 5. When entering a rational function into a graphing calculator, use brackets around the epression in the numerator and the epression in the denominator. E. Repeat parts A through D for the functions in the other two rows. F. Using graphing technolog, and the same window settings ou used in part A, eplore the graphs of the following rational functions. Sketch each graph on separate aes, and note an holes or asmptotes. i) v) f () 5.5 f () 5 ii) vi) f () 5 f () 5 3 iii) iv) f () 5 3 f () 5 G. Eamine the graphs of the functions in parts i) and v) of part F at the point where. Eplain wh f () 5 5 has a hole where 5, but f () 5.5 has a vertical asmptote. Identif the other functions in part F that have holes and the other functions that have vertical asmptotes. vii) viii) f () 5 9 f () 5 3 Chapter 5 59

17 oblique asmptote an asmptote that is neither vertical nor horizontal, but slanted H. Redraw the graph of the rational function f () 5.5. Then enter the equation into Y of the equation editor. What do ou notice? Eamine all our other sketches in this eploration to see if an of the other functions have an oblique asmptote. I. Eamine the equations with graphs that have horizontal asmptotes in part F. Compare the degree of the epression in the numerator with the degree of the epression in the denominator. Is there a connection between the degrees in the numerator and denominator and the eistence of horizontal asmptotes? Eplain. Repeat for functions with oblique asmptotes. J. Investigate several functions of the form f () 5 a b. Note c d similarities and differences. Without graphing, how can ou predict where a horizontal asmptote will occur? K. Investigate graphs of quotients of quadratic functions. How are the different from graphs of quotients of linear functions? L. Summarize the different characteristics of the graphs of rational functions. Reflecting M. How do the zeros of the function in the numerator help ou graph the rational function? How do the zeros of the function in the denominator help ou graph the rational function? N. Eplain how ou can use the epressions in the numerator and the denominator of a rational function to decide if the graph has i) a hole ii) a vertical asmptote iii) a horizontal asmptote iv) an oblique asmptote 6 5. Eploring Quotients of Polnomial Functions

18 In Summar Ke Ideas The quotient of two polnomial functions results in a rational function which often has one or more discontinuities. The breaks or discontinuities in a rational function occur where the function is undefined. The function is undefined at values where the denominator is equal to zero. As a result, these values must be restricted from the domain of the function. The values that must be restricted from the domain of a rational function result in ke characteristics that define the shape of the graph. These characteristics include a combination of vertical asmptotes (also called infinite discontinuities) and holes (also called point discontinuities). The end behaviours of man rational functions are determined b either horizontal asmptotes or oblique asmptotes. 5. Need to Know p(a) A rational function, f() 5 p(), has a hole at 5 a if. This q(a) 5 q() occurs when p() and q() contain a common factor of ( a). For eample, f() 5 has the common factor of ( ) in the numerator and the denominator. This results in a hole in the graph of f() at 5. 6 f() = 6 p(a) A rational function, f() 5 p(), has a vertical asmptote at 5 a if. q(a) 5 p(a) q() For eample, f() 5 has a vertical asmptote at 5. f() = = A rational function, f() 5 p(), has a horizontal asmptote onl when the q() degree of p() is less than or equal to the degree of q(). For eample, f() 5 has a horizontal asmptote at 5. 6 f() = + = 6 = A rational function, f() 5 p(), has an oblique (slant) asmptote onl when q() the degree of p() is greater than the degree of q() b eactl. For eample, f() 5 has an oblique asmptote = 8 f() = + + = Chapter 5 6

19 FURTHER Your Understanding. Without using graphing technolog, match each equation with its corresponding graph. Eplain our reasoning. a) 5 d) 5 3 ( )( 3) b) 5 9 e) c) f) 5 5 ( 3) 3 A C E B D F. For each function, determine the equations of an vertical asmptotes, the locations of an holes, and the eistence of an horizontal or oblique asmptotes. a) 5 e) 5 i) 5 8 ( 3)( 5) b) 5 f ) 5 j) c) 5 5 g) k) d) 5 9 h) 5 l) Write an equation for a rational function with the properties as given. a) a hole at 5 b) a vertical asmptote anwhere and a horizontal asmptote along the -ais c) a hole at 5and a vertical asmptote at 5 d) a vertical asmptote at 5and a horizontal asmptote at 5 e) an oblique asmptote, but no vertical asmptote 6 5. Eploring Quotients of Polnomial Functions

20 5.3 Graphs of Rational Functions of the Form f() 5 a b c d GOAL Sketch the graphs of rational functions, given equations of the form f() 5 a b c d. YOU WILL NEED graph paper graphing calculator or graphing software INVESTIGATE the Math The radius, in centimetres, of a circular juice blot on a piece of paper towel is modelled b r(t) 5 t, where t is measured in seconds. According to t this model, the maimum size of the blot is determined b the location of the horizontal asmptote.? How can ou find the equation of the horizontal asmptote of a rational function of the form f() 5 a b c d? A. Without graphing, determine the domain, intercepts, vertical asmptote, and positive/negative intervals of the simple rational function f () 5. B. Cop the following tables, and complete them b evaluating f () for each value of. Eamine the end behaviour of f () b observing the trend in f () as grows positivel large and negativel large. What value does f () seem to approach? S ` f() 5 S ` f() 5 Chapter 5 63

21 C. Write an equation for the horizontal asmptote of the function in part B. D. Repeat parts A, B, and C for the functions g() 5 h() 5, and m() 5 3,. 3 5 E. Verif our results b graphing all the functions in part D on a graphing calculator. Note similarities and differences among the graphs. F. Make a list of the equations of the functions and the equations of their horizontal asmptotes. Discuss how the degree of the numerator compares with the degree of the denominator. Eplain how the leading coefficients of in the numerator and the denominator determine the equation of the horizontal asmptote. G. Determine the equation of the horizontal asmptote of the juice blot function r(t) 5 t What does this equation tell ou about the t. eventual size of the juice blot? Reflecting H. How do the graphs of rational functions with linear epressions in the numerator and denominator compare with the graphs of reciprocal functions? I. Eplain how ou determined the equation of a horizontal asmptote from i) end behaviour tables ii) the equation of the function APPLY the Math EXAMPLE Determine how the graph of Solution Selecting a strateg to determine how a graph approaches a vertical asmptote f () approaches its vertical asmptote. f () has a vertical asmptote with the equation 5. Near this asmptote, the values of the function will grow ver large in a positive direction or ver large in a negative direction. f() is undefined when 5. There is no common factor in the numerator and denominator Graphs of Rational Functions of the Form f() 5 a b c d

22 Choose a value of to the left and ver close to. This value is less than. f (.) 5 3(.) 5 (.) 5 3 The graph of a rational function never crosses a vertical asmptote, so choose -values that are ver close to the vertical asmptote, on both sides, to determine the behaviour of the function. 5.3 On the left side of the vertical asmptote, the values of the function are positive. As S, f () S `. The function increases to large positive values as approaches from the left. Choose a value of to the right and ver close to. This value is greater than. f (.9) 5 3(.9) 5 (.9) 57 On the right side of the vertical asmptote, the values of the function are negative. As S, f () S `. The function decreases to small negative values as approaches from the right. Make a sketch to show how the graph approaches the vertical asmptote. = EXAMPLE Using ke characteristics to sketch the graph of a rational function For each function, a) f () 5 b) f () 5 c) f () i) determine the domain, intercepts, asmptotes, and positive/negative intervals ii) use these characteristics to sketch the graph of the function iii) describe where the function is increasing or decreasing Chapter 5 65

23 Solution a) f () 5 3 i) D 5 5PR 36 f () 5, so the -intercept is 3. 3 f (), so there is no -intercept. The line 5 3 is a vertical asmptote. The line 5 is a horizontal asmptote. f () is negative when P(`, 3) and positive when P(3, `). ii) Confirm the behaviour of f () near the vertical asmptote. f (3.) 5, so as S 3, f () S ` on the right. f (.9) 5, so as S 3, f () S ` on the left. 6 f() = 3 6 The function f() 5 is 3 undefined when 5 3. An rational function equals zero when its numerator equals zero. The numerator is alwas, so f() can never equal zero. Since the numerator and denominator do not contain the common factor ( 3), f() has a vertical asmptote at 5 3. An rational function that is formed b a constant numerator and a linear function denominator has a horizontal asmptote at 5. The numerator is alwas positive, so the denominator determines the sign of f(). 3, when, 3 3. when. 3 Use all the information in part i) to sketch the graph. = 3 iii) From the graph, the function is decreasing on its entire domain: when P(`, 3) and when P(3, `) Graphs of Rational Functions of the Form f() 5 a b c d

24 b) f () 5 3 i) 3 3 f() is undefined when the denominator is zero D 5 5PR 3 f f () 5 or 3() 5 The -intercept is. f () 5 when The -intercept is. To determine the -intercept, let 5. To determine the -intercept, let 5. An rational function equals zero when its numerator equals zero. The line 5 is a vertical 3 asmptote. This is the value that makes f() undefined. The line 5 is a horizontal 3 asmptote. Eamine the signs of the numerator and denominator, and their quotient, to determine the positive>negative intervals. The ratio of the leading coefficients of the numerator and denominator is. 3, 3 3,, The vertical asmptote and the -intercept divide the set of real numbers into three intervals: Q`,, and (, `). 3 R, Q 3, R Choose numbers in each interval to evaluate the sign of each epression. f () is positive when P Q`, 3 R and when P(, `). f () is negative when P Q 3, R. Chapter 5 67

25 ii) 6 = 3 = 3 iii) From the graph, f () is increasing on its entire domain; that is, when P Q`, and when 3 R P Q 3, `R. f() = 3+ 6 When sketching the graph, it helps to shade the regions where there is no graph. Use the positive and negative intervals as indicators for these regions. For eample, since f() is positive on Q`,, there is no graph 3 R under the -ais on this interval. Draw the asmptotes, and mark the intercepts. Then draw the graph to approach the asmptotes. c) f () i) D 5 5PR 36 f (), so there is no -intercept. or 6 f () 5 3 () The -intercept is. f () 5 3 ( 3) f () has a hole, not a vertical asmptote, where 5 3. f () 5 3 ( 3) 5 f() is undefined when the denominator is zero. To determine the -intercept, let 5. To determine the -intercept, let 5. Onl consider when the numerator is zero; that is, when 3 5. Therefore, the numerator is zero at 5 3, but this has alread been ecluded from the domain. Factoring reveals a common factor ( 3) in the numerator and denominator. The graph has a hole at the point where 5 3. The value of the function is alwas for all values of, ecept when Graphs of Rational Functions of the Form f() 5 a b c d

26 5.3 f () is positive at all points in its domain. f() has no vertical asmptote or -intercept. There is onl one interval to consider: (`, `). For an value of, f() 5. ii) The graph is a horizontal line with the equation 5. There is a hole at f() = Use the information in part i) to sketch the graph. 5 iii) The function is neither increasing nor decreasing. It is constant on its entire domain. EXAMPLE 3 Solving a problem b graphing a rational function 3(7t 9) The function P(t) 5 models the population, in thousands, of a 3t town t ears since 99. Describe the population of the town over the net ears. Solution 3(7t 9) P(t) 5 3t Determine the initial population in 99, when t 5. P() 5 3(7() 9) 3() 5 35 Use the equation to help ou decide on the window settings. For the given contet, t $ and P(t).. Chapter 5 69

27 Graph P(t) to show the population for the ears after 99. The value that makes 3t 5 lies outside the domain of P(t). There is no vertical asmptote in the domain. TRACE along the curve to get an idea of how the population changed. In the first two ears, the population dropped b about 5 people. Then it began to level off and approach a stead value. There is a horizontal asmptote at P 5 3(7) Use the function equation to determine the equation of the horizontal asmptote. For large values of t, P(t) 8 3(7t). 3t Therefore, the leading coefficients in the numerator and denominator define the equation of the horizontal asmptote. The population of the town has been decreasing since 99. It was 35 in 99, but dropped b about 5 in the net two ears. Since then, the population has begun to level off and, according to the model, will approach a stead value of 7 people b. Multipl the values of the function b, since the population is given in thousands Graphs of Rational Functions of the Form f() 5 a b c d

28 In Summar Ke Ideas The graphs of most rational functions of the form f() 5 b and f() 5 a b have both a vertical asmptote and c d c d a horizontal asmptote. You can determine the equation of the vertical asmptote directl from the equation of the function b finding the zero of the denominator. You can determine the equation of the horizontal asmptote directl from the equation of the function b eamining the ratio of the leading coefficients in the numerator and the denominator. This gives ou the end behaviours of the function. To sketch the graph of a rational function, ou can use the domain, intercepts, equations of asmptotes, and positive/negative intervals. 5.3 Need to Know Rational functions of the form f() 5 b c d have a vertical asmptote defined b 5 d and a horizontal c asmptote defined b 5. For eample, see the graph of f() 5. 3 Most rational functions of the form f() 5 a b have a vertical c d asmptote defined b 5 d and a horizontal asmptote defined c b 5 a. For eample, see the graph of f () 5. c The eception occurs when the numerator and the denominator both contain a common linear factor. This results in a graph of a horizontal line that has a hole where the zero of the common factor occurs. As a result, the graph has no asmptotes. For eample, see the graph of f() 5 8 ( ) 5. ( ) 6 = 6 = 3 f() = 3 6 f() = = 6 = 6 8 f() = 6 Chapter 5 7

29 CHECK Your Understanding. Match each function with its graph. a) h() 5 5 c) b) m() 5 d) A 6 C 6 f () 5 3 g() B 5 D Consider the function f () 5 3. a) State the equation of the vertical asmptote. b) Use a table of values to determine the behaviour(s) of the function near its vertical asmptote. c) State the equation of the horizontal asmptote. d) Use a table of values to determine the end behaviours of the function near its horizontal asmptote. e) Determine the domain and range. f) Determine the positive and negative intervals. g) Sketch the graph. 3. Repeat question for the rational function f () Graphs of Rational Functions of the Form f() 5 a b c d

30 PRACTISING. State the equation of the vertical asmptote of each function. Then choose a strateg to determine how the graph of the function approaches its vertical asmptote. a) 5 3 c) b) 5 5 d) For each function, determine the domain, intercepts, asmptotes, and K positive/negative intervals. Use these characteristics to sketch the graph of the function. Then describe where the function is increasing or decreasing. a) f () 5 3 c) f () b) f () 5 5 d) f () 5 5( ) 6. Read each set of conditions. State the equation of a rational function of the form f () 5 a b that meets these conditions, and sketch the c d graph. a) vertical asmptote at 5, horizontal asmptote at 5 ; negative when P (`, ), positive when P(, `) ; alwas decreasing b) vertical asmptote at 5, horizontal asmptote at 5 ; -intercept 5, -intercept 5 ; positive when P(`, ) or (, `), negative when P(, ) c) hole at 5 3; no vertical asmptotes; -intercept 5 (,.5) d) vertical asmptotes at 5and 5 6, horizontal asmptote at 5 ; positive when P(`, ) or (6, `), negative when P(, 6) ; increasing when P(`, ), decreasing when P(, `) 7. a) Use a graphing calculator to investigate the similarities and T differences in the graphs of rational functions of the form f () 5 8, for n 5,,, and 8. n b) Use our answer for part a) to make a conjecture about how the function changes as the values of n approach infinit. c) If n is negative, how does the function change as the value of n approaches negative infinit? Choose our own values, and use them as eamples to support our conclusions. Chapter 5 73

31 8. Without using a graphing calculator, compare the graphs of the rational functions f () 5 3 and g() t 5 9. The function I(t) 5 gives the value of an investment, in t thousands of dollars, over t ears. a) What is the value of the investment after ears? b) What is the value of the investment after ear? c) What is the value of the investment after 6 months? d) There is an asmptote on the graph of the function at t 5. Does this make sense? Eplain wh or wh not. e) Choose a ver small value of t (a value near zero). Calculate the value of the investment at this time. Do ou think that the function is accurate at this time? Wh or wh not? f) As time passes, what will the value of the investment approach?. An amount of chlorine is added to a swimming pool that contains A pure water. The concentration of chlorine, c, in the pool at t hours is given b c(t) 5 t, where c is measured in milligrams per litre. t What happens to the concentration of chlorine in the pool during the h period after the chlorine is added?. Describe the ke characteristics of the graphs of rational functions of C the form f () 5 a b. Eplain how ou can determine these c d characteristics using the equations of the functions. In what was are the graphs of all the functions in this famil alike? In what was are the different? Use eamples in our comparison. Etending. Not all asmptotes are horizontal or vertical. Find a rational function that has an asmptote that is neither horizontal nor vertical, but slanted or oblique. 3. Use long division to rewrite f () in the form f () 5 a b c k. What does this tell ou about the end behaviour of the function? Graph the function. Include all asmptotes in our graph. Write the equations of the asmptotes.. Let,, h() g() f () 5 3, and 3 9 m() 5. a) Which of these rational functions has a horizontal asmptote? b) Which has an oblique asmptote? c) Which has no vertical asmptote? d) Graph 5 m(), showing the asmptotes and intercepts Graphs of Rational Functions of the Form f() 5 a b c d

32 5 Mid-Chapter Review FREQUENTLY ASKED Questions Q: How can ou use the graph of a linear or quadratic function to graph its reciprocal function? A: If ou have the graph of a linear or quadratic function, ou can draw the graph of its reciprocal function as follows:. Draw a vertical asmptote for the reciprocal function at each zero of the original function. The -ais is a horizontal asmptote for the reciprocal function, unless the original function is a constant function. Stud Aid See Lesson 5., Eamples and. Tr Mid-Chapter Review Questions and.. The reciprocal function has the same positive/negative intervals that the original function has, so ou can shade the regions where there will be no graph. 3. Mark an points where the -value of the original function is or. The reciprocal function also goes through these points.. The -intercept of the reciprocal function is the reciprocal of the -intercept of the original function. Determine and mark the -intercept of the reciprocal function. 5. If the original function is quadratic, the reciprocal function has a local maimum/minimum at the same -value as the verte of the quadratic function. The -value of this local maimum/minimum is the reciprocal of the -value of the verte. Determine and mark the local maimum/minimum point. 6. Draw the pieces of the graph of the reciprocal function through the marked points, approaching the asmptotes. Q: What are rational functions, and what are the characteristics of their graphs? A: Rational functions are quotients of polnomial functions. Their equations have the form f () 5 p(), where p() and q() are q() polnomial functions and q(). Graphs of rational functions ma have vertical, horizontal, or oblique asmptotes. Some rational functions have holes in their graphs. Stud Aid See Lesson 5.. Tr Mid-Chapter Review Question 3. Chapter 5 75

33 Stud See Lesson 5.. Tr Mid-Chapter Review Question. = = Stud Aid 6 6 Aid = f() = + 6 See Lesson 5.3, Eample. Tr Mid-Chapter Review Questions 5 to 8. f() = Q: Most rational functions have one or more discontinuities. Where and wh do these discontinuities occur? When is a rational function continuous? A: If the polnomial function in the denominator of a rational function has one or more zeros, the rational function will be discontinuous at these points. If a value of can be zero in both the numerator and the denominator of a rational function (that is, if the numerator and denominator have a common linear factor), the result is a hole. This tpe of discontinuit is called a point discontinuit. If a zero in the denominator does not correspond to a zero in the numerator, there will be a vertical asmptote at the -value. This is called an infinite discontinuit. For eample, f () 5 has a point 5 ( )( ) discontinuit where 5because is a zero of both the denominator, q() 5, and the numerator, p() 5. The graph of f () has an infinite discontinuit where 5 because is a zero of q() but not of p(). The graph also has a hole at 5 and a vertical asmptote at 5 p(). Note that p() but q() 5 q() 5,. If the polnomial function in the denominator of a rational function does not have an zeros, the rational function is continuous. Its graph is a smooth curve, with no breaks. For eample, f () 5 3 is a continuous rational function with a horizontal asmptote at 5. Q: How do ou determine the equations of the vertical and horizontal asmptotes of a rational function of the form f() 5 b and f() 5 a b? c d c d A: You can determine the equations of the vertical and horizontal asmptotes directl from the equation of a rational function of the form f () 5 b or f () 5 a b. The vertical asmptote c d c d occurs at the zero of the function in the denominator. The equation of the vertical asmptote is 5 d. The horizontal asmptote c describes the end behaviour of the function when S 6`. All rational functions of the form f () 5 a b have 5 a as a c d c horizontal asmptote. All rational functions of the form f () 5 (the -ais) as a horizontal asmptote. b c d have 5 76 Mid-Chapter Review

34 PRACTICE Questions Lesson 5.. State the reciprocal of each function, and determine the locations of an vertical asmptotes. a) b) c) d). For each function, determine the domain and range, intercepts, positive/negative intervals, and intervals of increase/decrease. Use this information to sketch the graphs of the function and its reciprocal. a) b) c) d) Lesson Different characteristics of the graph of a rational function are created b different characteristics of the function. List at least four characteristics of a graph and the characteristic of the function that causes each one. Make sure that ou include a characteristic of a continuous rational function in our list.. For each function, determine the equations of an vertical asmptotes, the locations of an holes, and the eistence of an horizontal asmptotes (other than the -ais) or oblique asmptotes. a) 5 b) 5 c) d) e) f () 5 3 f (q) 5q 6 f (z) 5 z z 5 f (d) 5 6d 7d 3 f () 5 6 f () 5 f () 5 6 f () Lesson 5.3 Mid-Chapter Review 5. List the functions that had a horizontal asmptote in question, and give the equation of the horizontal asmptote. 6. For each function, determine the domain, intercepts, asmptotes, and positive/negative intervals. Use these characteristics to sketch the graph of the function. Then describe where the function is increasing or decreasing. a) f () b) f () c) f () 5 d) f () 5 7. Kevin is tring to develop a reciprocal function to model some data that he has. He wants the horizontal asmptote to be 5 7. He also wants the graph to decrease and approach 5 7 as approaches infinit, so he chooses the equation Then he decides that he needs the vertical asmptote to be 5, so he changes the equation to What happened to the graph of Kevin s function? Did it give him the result he wanted? Eplain wh or wh not. 8. For the function f () 5 7 m find the n, values of m and n such that the vertical asmptote is at 5 6 and the -intercept is Create a rational function that has a domain of 5PRZ 6 and no vertical asmptote. Describe the graph of this function. Chapter 5 77

35 5. Solving Rational Equations YOU WILL NEED graphing calculator or graphing software GOAL Connect the solution to a rational equation with the graph of a rational function. LEARN ABOUT the Math When the work together, Stuart and Luc can deliver flers to all the homes in their neighbourhood in min. When Luc works alone, she can finish the deliveries in 3 min less time than Stuart can when he works alone.? When Stuart works alone, how long does he take to deliver the flers? EXAMPLE Selecting a strateg to solve a rational equation Determine the time that Stuart takes to deliver the flers when he works alone. Solution A: Creating an equation and solving it using algebra Let s minutes be the time that Stuart takes to deliver the flers when working alone. Luc takes (s 3) minutes when working alone. Choose a variable to represent Stuart's time and use it to write an epression for Luc's time. Luc delivers the flers in 3 min less time than Stuart. The fraction of deliveries made in one minute b Stuart working alone is s b Luc working alone is s 3 b Stuart and Luc working together is s s 3 5 Compare the rates at which the work. For eample, if Stuart took 8 min to deliver all the flers, he would deliver of the flers per minute. 8 s. 3 because Stuart takes longer than Luc to deliver the flers, and it is not possible for the denominators to be zero Solving Rational Equations

36 Multipl b the LCD. s(s 3) a s b 5 s(s 3) a s 3 b There are no common factors in the denominators, s(s 3) s s (s 3) s s(s 3) s 3 s(s 3) s 3 5 s(s 3) 5 s(s 3) (s 3) s 5 s(s 3) so the LCD (lowest common denominator) is the product of the three denominators s(s 3). Multipl each term b the LCD, and then simplif the resulting rational epressions to remove all the denominators. 5. s 56 s 5 s 3s 5 s 97s 56 5 (s 9)(s 6) s 5 6 or 9 s. 3 so 6 is not an admissible solution. LS 5 s s RS 5 Solve the quadratic equation b factoring or b using the quadratic formula. Remember to look for inadmissible solutions b carefull considering both the contet and the information given in the problem. Check the solution, s 5 9, b substituting it into the original equation. Since LS 5 RS, s 5 9 is the solution. It will take Stuart 9 min to deliver the flers when working alone. Chapter 5 79

37 Solution B: Using the graph of a rational function to solve a rational equation The equation that models the problem is where s represents s s 3 5, the time, in minutes, that Stuart takes to deliver the flers when working alone. s s 3 5 Subtract from each side. To solve the equation, find the zeros of the function f(s) 5 s s 3. Tech Support For help determining the zeros on a graphing calculator, see Technical Appendi, T-8. Graph f (s) 5 s s 3. Use the zero operation to determine the zeros. From the equation, ou can epect the graph to have vertical asmptotes at s 5 and s 5 3. Since ou are onl interested in finding the zeros, ou can limit the -values to those close to zero. The first zero for f (s) is s 5 6. Reject this solution since s. 3. Determine the other zero. You know that Luc takes 3 min less time than Stuart takes, so Stuart must take longer than 3 min. The solution s 5 6 is inadmissible. The solution is s 5 9. Stuart takes 9 min to deliver the flers when working alone Solving Rational Equations

38 Reflecting A. In Solution A, eplain how a rational equation was created using the times given in the problem. 5. B. In Solution B, eplain how finding the zeros of a rational function provided the solution to the problem. C. Where did the inadmissible root obtained in Solution A show up in the graphical solution in Solution B? How was this root dealt with? APPLY the Math EXAMPLE Using an algebraic strateg to solve simple rational equations Solve each rational equation. 3 a) b) Solution a) 3 5, 3 Determine an restrictions on the value of. ( ) ( 3) 5 ( 3) ( 3) 5 5 To verif, graph f () 5 3 and use the zero operation to determine the zero. Multipl both sides of the equation b the LCD, ( 3). Add to each side. From the equation, the graph will have a vertical asmptote at 5 3 and a horizontal asmptote at 5. The solution is 5. Chapter 5 8

39 b) 3 5,, ( )( ) a 3 b 5 ( )( ) a b ( ) ( ) a 3 b 5 ( )( ) ( )( 3) 5 ( )( ) To verif, graph f () 5 3 and use the zero operation to determine the zero. a b Note the restrictions. Multipl each side of the equation b the LCD, ( ) ( ). Simplif. Epand. Subtract, and add 5 to both sides. Solve the resulting linear equation. Adjust the window settings so ou can view enough of the graph to see all the possible zeros. The solution is 5.. EXAMPLE 3 Connecting the solution to a problem with the zeros of a rational function Salt water is flowing into a large tank that contains pure water. The concentration of salt, c, in the tank at t minutes is given b c(t) 5 t where c is measured 5 t, in grams per litre. When does the salt concentration in the tank reach 3.75 g> L? Solution If the salt concentration is 3.75, c(t) t 5 t (5 t) a t b (5 t) 5 t t (5 t) (5 t) (5 t) t t Set the function epression equal to t, and because t measures the time since the salt water started flowing, t $. Multipl both sides of the equation b the LCD, (5 t), and solve the resulting linear equation Solving Rational Equations

40 t 3.75t t t t 5 5 It takes 5 min for the salt concentration to reach 3.75 g> L. To verif, graph f (t) 5 t 5 t and g(t) , and determine where the functions intersect. Use inverse operations to solve for t. Use an appropriate window setting, based on the domain, t $. 5. The salt concentration reaches 3.75 g> L after 5 min. Use the intersect operation. Tech Support For help determining the point of intersection between two functions, see Technical Appendi, T-. EXAMPLE Using a rational function to model and solve a problem Rima bought a case of concert T-shirts for $5. She kept two T-shirts for herself and sold the rest for $56, making a profit of $ on each T-shirt. How man T-shirts were in the case? Solution Let the number of T-shirts in the case be. Buing price per T-shirt 5 5 Selling price per T-shirt 5 56 Rima paid $5 for T-shirts, so each T-shirt cost her 5 $. She kept two for herself, which left T-shirts for her to sell. Rima sold T-shirts for $56, so she charged 56 $ for each one. Chapter 5 83