The Rational Functions Unit

Transcription

1 The Rational Functions Unit (Level IV Pre-Calculus) NSSAL (Draft) C. David Pilmer 04 (Last Updated: March 04)

2 This resource is the intellectual property of the Adult Education Division of the Nova Scotia Department of Labour and Advanced Education. The following are permitted to use and reproduce this resource for classroom purposes. Nova Scotia instructors delivering the Nova Scotia Adult Learning Program Canadian public school teachers delivering public school curriculum Canadian non-profit tuition-free adult basic education programs Nova Scotia Community College instructors delivering the Academic Careers and Connections mathematics curriculum or core college programs The following are not permitted to use or reproduce this resource without the written authorization of the Adult Education Division of the Nova Scotia Department of Labour and Advanced Education. Upgrading programs at post-secondary institutions (eception: NSCC) Core programs at post-secondary institutions (eception: NSCC) Public or private schools outside of Canada Basic adult education programs outside of Canada Individuals, not including teachers or instructors, are permitted to use this resource for their own learning. They are not permitted to make multiple copies of the resource for distribution. Nor are they permitted to use this resource under the direction of a teacher or instructor at a learning institution. Acknowledgments The Adult Education Division would like to thank the following university professors for reviewing this resource to ensure all mathematical concepts were presented correctly and in a manner that supported our learners. Dr. Robert Dawson (Saint Mary s University) Dr. Genevieve Boulet (Mount Saint Vincent University) Dr. Jeff Hooper (Acadia University) The Adult Education Division would also like to thank the following NSCC instructors for piloting this resource and offering suggestions during its development. John Archibald (Truro Campus) Elliott Churchill (Waterfront Campus) Carissa Dulong (Truro Campus) Barbara Gillis (Burridge Campus) Alice Veenema (Kingstec Campus) The Adult Education Division would also like to thank Antoine Jarjoura, CSAP Math and Science Curriculum Consultant, for reviewing this resource and offering feedback.

3 Table of Contents Introduction.. Prerequisite Knowledge Negotiated Completion Date The Big Picture and Suggested Timelines. Pre-Calculus Relations Concept Map... Common Errors Seen in First Year University Math Classes.. ii iii iv v vi vii Introduction to Rational Functions: Part. Introduction to Rational Functions: Part Asymptotes, Points of Discontinuity and Zeros: Part 9 Asymptotes, Points of Discontinuity and Zeros: Part 6 Rational Equations 9 Rational Inequalities. 40 Partial Fractions. 48 Answers 57 NSSAL i Draft 0 C. D. Pilmer

4 Introduction Level IV Pre-Calculus Pre-Calculus is strictly designed for learners who are intending on enrolling in science or engineering programs at the university level. The prerequisite for this course is ALP Level IV Academic Math with a minimum mark of 85%, PSP Advanced Math with a minimum mark of 80%, or PSP Pre-Calculus Math with a minimum mark of 80%. We also recommend that the learner have a strong work ethic and the ability to work independently. The delivery of the Pre-Calculus curriculum will likely vary from campus to campus. Many campuses will not have the resources to offer this course on their schedule. In these cases, the learner will have to take it as a correspondence course or as an online course. Larger campuses may be able to offer the course, but it is likely that the learner will be placed in a class that is predominately populated with Level IV Academic Math learners. There is a possibility that Academic Careers and Connections (ACC) will use the ALP Pre-Calculus curriculum within their own program. If this is the case, ACC and ALP learners may be found in the same classroom. Ultimately, the delivery model used with Pre-Calculus will be at the discretion of the Nova Scotia Community College and the Academic Chairs at the various campuses. There is a strong likelihood that you may be the only Pre-Calculus learner in the class or at your campus. For this reason, it may be a challenge to obtain assistance for etended periods of time. This being said, we have made every effort to make the material found in this resource as accessible as possible. One of the major differences between the Pre-Calculus resources and the Academic Math resources can be seen in the answer key. In the Academic Math resources, the answer keys only contained the final answers. The answer key in this resource and other Pre- Calculus resources is far more etensive. For many of the questions we have also provided hints, and with the more challenging questions, we have provided complete solutions. Use this feature judiciously; do not refer to the answer key before making a concerted effort to answer the question on your own. Neither do we want learners to seek assistance from an instructor without first attempting to use the answer key to resolve their problem. In the near future, we hope to create online instructional videos to support our Pre-Calculus learners. Ultimately, learners with a solid understanding of Academic Math concepts, a great work ethic, and the ability to work independently, will flourish in this course. Although some of the topics can be challenging, remember that you can do it. Most of you will have taken ALP Level IV Academic Mathematics prior to taking this course. In that academic course, there was a heavy emphasis on real-world application questions, and the use of the graphing calculator. In Pre-Calculus, one will encounter application questions and have to use graphing calculators, but to a far lesser degree. There is a greater emphasis on the pure mathematics and the automaticity to recall relevant mathematical facts and concepts. This parallels the epectations that one will encounter in a first year university calculus course. Regardless of this, the material in this resource is presented in a manner to foster understanding (opposed to encouraging memorization). NSSAL ii Draft 0 C. D. Pilmer

5 The NSSAL Pre-Calculus Print Resources If you are not affiliated with the Nova Scotia School for Adult Learning (NSSAL) and choosing to use this resource, you may be wondering why some of the early topics that appear in our Pre- Calculus resources seem to be more introductory in nature compared to topics found in traditional Pre-Calculus programs and tetbooks. The reason stems from the fact that the prerequisite within the ALP system for our Pre-Calculus course is Level IV Academic Math. Traditional Pre-Calculus courses usually have an advanced level mathematics course as their prerequisite. NSSAL does not have the resources or the number of learners necessary to offer an advanced math prerequisite. For this reason our Pre-Calculus includes introductory material that is typically found in an advanced math prerequisite. Prerequisite Knowledge Learners who are starting the Level IV Pre-Calculus Rational Functions Unit should be familiar with the concepts eamined in the Level IV Academic Rational Epressions and Radicals Unit and the Level IV Pre-Calculus Polynomial Functions Unit. These include: Adding and subtracting rational epressions e.g ( )( ) ( )( ) 9 ( + ) ( + 7)( + 4)( + ) ( + 7) ( + 4)( + )( + 7) 9 ( + ) ( + 7) ( + 7)( + 4)( + ) ( + 7)( + 4)( + ) 5 5 ( + 7)( + 4)( + ) Multiplying and dividing rational epressions 6 p + 5 p p 4 p e.g. p + 4 p p 8 p+ 6 6 p + 5p p 8 p+ 6 p + 4 p p 4 p ( ) ( )( ) p( p+ 5) ( ) ( ) p p+ 5 p 4 p+ 8 p 4 p p 4 p + 8 NSSAL iii Draft 0 C. D. Pilmer

6 Identifying polynomial functions e.g. y polynomial, specifically cubic or third order polynomial e.g. e.g. y + not polynomial y polynomial, specifically quartic or fourth order polynomial Dividing two polynomial epressions using long division or synthetic division e.g. ( ) ( ) Long Division: Therefore: ( ) ( ) Synthetic Division: Negotiated Completion Date After working for a few days on this unit, sit down with your instructor and negotiate a completion date for this unit. Start Date: Completion Date: Instructor Signature: Student Signature: NSSAL iv Draft 0 C. D. Pilmer

7 The Big Picture and Suggested Timelines The following flow chart shows the si required units for the 00 hour Level IV Pre-Calculus course. These have been presented in a suggested order. Sequences and Series (5 hours) Functional Form, Recursive Form, Arithmetic Sequences and Series, Geometric Sequences and Series Trigonometric Equations and Identities Unit, Part (7 hours) Unit Circle, Special Rotations, Evaluating Trigonometric Epressions, Trigonometric Equations, Radian Measure, Trigonometric Identities Trigonometric Equations and Identities Unit, Part (4 hours) Trigonometric Equations Involving Tangents, Principal Solutions, More Trigonometric Identities The Euler Number Unit (7 hours) The Euler Number, Natural Logarithms, Eponential Functions Using Base e, Logistic Functions Interval Notation Unit (5 hours) Domain, Range, Increasing, Decreasing, Concave Upwards, Concave Downwards Polynomial Functions Unit ( hours) Graphs of Polynomial Functions, Determining the Equation of a Polynomial Function, Polynomial Equations and Inequalities Rational Functions Unit (9 hours) Graphs of Rational Functions, Rational Equations and Inequalities, Partial Fractions Irrational Functions Unit (0 hours) Comple Numbers Unit (0 hours) NSSAL v Draft 0 C. D. Pilmer

8 The Pre-Calculus Relations Concept Map Relations A relation shows the relationship between two sets of numbers Function A relation is a function when for every member of the first set (typically 's), there is only one corresponding member in the second set (typically y's). Conics Circles, ellipses, hyperbolas, and parabolas Algebraic Functions These functions can be described using algebraic operations (addition, subtraction, multiplication, division, eponentiation). Transcendental Functions These functions cannot be described using algebraic operations. Rational Functions The variable has an integral constant eponent. e.g. y or y Irrational Functions (or Radical Functions) The variable has a non-integral constant eponent. e.g. y or y Polynomial Functions The variable has a positive integral constant eponent. e.g. y Absolute Value Functions e.g. y or y Trigonometric Functions e.g. y sin, y cot Logarithmic Functions e.g. y log, y ln Eponential Functions e.g. y, y e NSSAL vi Draft 0 C. D. Pilmer

9 Common Errors Seen in First Year University Math Classes Do not walk into a first year university mathematics classroom without being aware of the following common errors. At this point in your Pre-Calculus course, you may not have encountered all of the concepts associated with the errors listed below. However, by the end of the course, you should be familiar with all of the concepts. ( ) + y is not equal to ( ) + y is not equal to + y ; rather, it is equal to + y ; rather, it is equal to + y + y. + y + y + y. + y is not equal to +. y + y is not equal to + y. The epression y + y cannot be simplified to + (i.e. we cannot cancel out the y's.) is not equal to 0; rather, it is undefined. 0 We know that cos( y) sin cos +, but that does not mean that sin + cos. + is not equal to cos + cos y; rather, it is equal to cos cos y sin sin y. sin is not equal to sin ; rather, it equal to sin cos is not equal to cos. : rather, it is another way to say arccos. cos 0 b is not equal to 0, it is equal to (as long as b is not equal to 0; when b 0, the epression is undefined). The solutions to the equations 9 and 9 are different. The solution to the equation 9 is. The solution to the equation 9 is or -. 5 and ( 5) log a ( b c) ( ) are different epressions; 5 5 while ( 5) 5. + is not equal to log b+ log c; rather, the appropriate law of logarithms is log bc log b + log c. a a a a a If by the end of your Pre-Calculus course, you are still unsure why some of these common errors are incorrect, sit down with your instructor and go over them. Understanding these errors is far more important than merely memorizing them. NSSAL vii Draft 0 C. D. Pilmer

10 Introduction to Rational Functions: Part First Application: Average Annual Cost of a Dishwasher Suppose you buy a dishwasher for $450, knowing that electrical cost per year for operating that dishwasher is $5. The average annual cost of that dishwasher over its first year of operation would be worked out in the following manner. We are assuming that there are no repair costs associated with the appliance () Average Annual Cost $50 The average annual cost of that dishwasher over its first two years of operation would be worked out in the following manner () Average Annual Cost $76 The average annual cost of that dishwasher over its first three years of operation would be worked out in the following manner () Average Annual Cost $0 Questions. Determine the average annual cost of that dishwasher over its first four years of operation. Show your work.. Complete the following table of values. Years of Operation Annual Average Cost of Dishwasher $50 $ Why does it make sense that the average annual cost is going down over time? NSSAL Draft 0 C. D. Pilmer

11 4. Determine the equation of the function that describes the average annual cost, C, of the dishwasher in terms of number, n, of years of operations. 5. (a) Using a graphing calculator, graph the function from question 4. Use the WINDOW settings shown on the right. Make a very rough sketch the curve in the space provided on the right. (b) Why did we use the WINDOW settings provided in part (a), as opposed to using settings that show all four quadrants of the Cartesian coordinate system? 6. (a) Below we have listed the types of functions you have encountered in this course and in Level IV Academic Math polynomial functions (constant, linear, quadratic, cubic, ) trigonometric functions ( y sin, y cos, y tan, y sec, ) eponential functions logarithmic functions logistic functions Based solely on the shape of the curve in question 5, which one of the previously encountered functions looks most like the graph in question 5? (b) How do you know that you are not dealing with the function you stated in part (a) of this question? NSSAL Draft 0 C. D. Pilmer

12 7. (a) If your friend, Nashi, bought a slightly more epensive dishwasher ($50 after taes), yet more efficient dishwasher (Average Annual Electrical Cost of $), what would be the equation of the function that describes the average annual cost, C, of the dishwasher in terms of number, n, of years of operations. (b) Which dishwasher, yours or Nashi's, has a lower average annual cost for ten years of operation? How much lower? Second Application: Loosening the Bolt The amount of force that needs to be eerted on the end of a wrench to loosen a rusty bolt is dependent upon the length of the wrench. Longer wrenches require less force. Shorter wrenches require more force. The following function describes the relationship between the force, f, and the length, l, for a particular situation (i.e. a specific rusted bolt). 600 f f is measured in Newtons, and l is measured in centimetres l Questions 8. How much force is needed to loosen the bolt using 8 cm wrench? 9. If you only wanted to eert 400 Newtons of force to loosen the bolt, how long would the wrench have to be? NSSAL Draft 0 C. D. Pilmer

13 0. (a) Complete the following table for this given situation. Length of Wrench (cm) Force Eerted (Newtons) (b) Is there a common ratio between the successive "force eerted" values? YES NO. Using a graphing calculator, graph the function that describes the relationship between the forces eerted and the length of the wrench. Use the WINDOW settings shown on the right. Make a very rough sketch the curve in the space provided on the right. NSSAL 4 Draft 0 C. D. Pilmer

14 Introduction to Rational Functions: Part n 600 In the previous section, you encountered the functions C and f. When n l we viewed the graphs of these functions with the restricted WINDOW settings, they initially appeared to be eponential decay curves. However, after thinking about it for a short period of time, we realized that these were not eponential; their equations were not of the appropriate k c forms y ab or y ae, and we were unable to see the appropriate common ratio in the table of values. If we viewed the graphs of these two functions such that we could see all four quadrants of Cartesian coordinate system, it would provide additional evidence that the functions are not eponential. We shown this below. (Note that we have changed to the variables and y.) y 600 y This function has: a horizontal asymptote at y 5 a vertical asymptote at 0 This function has: a horizontal asymptote at y 0 a vertical asymptote at 0 Both of these functions are called rational functions and they both form curves called hyperbolas. Not all rational functions form hyperbolas, but we will talk about that later. So what is a rational function? Rational Functions p( ) A rational function is a function that can be epressed as f ( ), where p( ) and q( ) q( ) are polynomial functions and q( ) 0. Translated: A rational function is the ratio of two polynomial functions where the denominator cannot be the constant function y 0. NSSAL 5 Draft 0 C. D. Pilmer

15 Consider: y. It's a rational function because it is the ratio of the linear function p q. ( ) to the linear function ( ) 600 y. It's a rational function because it is the ratio of the constant function q. p( ) 600 to the linear function ( ) It's a rational function because it is the ratio of the linear function + + p + to the quadratic function q( ) + +. y ( ) y. It's a rational function because it is the ratio of the quadratic function 5+ p 4 to the quadratic function q( ) 5+. ( ) y 7. It's a rational function because it is the ratio of the cubic function ( ) to the linear function q( ) 7. Using this definition, we can see that all polynomial functions can also be classified as rational functions. Consider the following. 9 The linear function y 9 can also be written as y, making it the ratio of the linear function p( ) 9 to the constant function q( ). The quadratic function y 7 + can also be written as p + 7 y, making it q. the ratio of the quadratic function p( ) + 7 to the constant function ( ) However, not all rational functions are polynomial functions. You may have noticed that at the beginning of every Pre-Calculus unit, you were supplied with the "Pre-Calculus Relations Concept Map." This map shows how we classify the variety of mathematical functions and relations that we have and will be encountering in this math course and other courses. We have provided the same concept map on the net page. NSSAL 6 Draft 0 C. D. Pilmer

16 Pre-Calculus Relations Concept Map Relations A relation shows the relationship between two sets of numbers Function A relation is a function when for every member of the first set (typically 's), there is only one corresponding member in the second set (typically y's). Conics Circles, ellipses, hyperbolas, and parabolas Algebraic Functions These functions can be described using algebraic operations (addition, subtraction, multiplication, division, eponentiation). Transcendental Functions These functions cannot be described using algebraic operations. Rational Functions The variable has an integral constant eponent. e.g. y or y Irrational Functions (or Radical Functions) The variable has a non-integral constant eponent. e.g. y or y Polynomial Functions The variable has a positive integral constant eponent. Absolute Value Functions e.g. y or y Trigonometric Functions e.g. y sin, y cot Logarithmic Functions e.g. y log, y ln Eponential Functions e.g. y, y e At this point in time, you have encountered everything on this concept map with the eception of conics, irrational functions, and absolute value functions. However, you will be working with these in upcoming units. Please note, there are other transcendental functions beyond the three families of functions listed in our concept map. NSSAL 7 Draft 0 C. D. Pilmer

17 Eample Fully classify each of the following functions. 5 (a) y + (b) g( ) tan (c) y (d) y e (e) f ( ) (f) y 4 5 (g) y log h + (h) ( ) ( ) Answers: (a) algebraic and rational (b) transcendental and trigonometric (c) algebraic, rational and polynomial (specifically cubic) (d) transcendental and eponential (e) algebraic, rational and polynomial (specifically quadratic) (f) algebraic and rational (g) transcendental and logarithmic (h) can be epressed as y, therefore algebraic and rational + Questions. Fully classify each of the following functions. y sin (b) (a) ( ) y (c) 6 y (d) g( ) ln ( + ) (e) y (f) h( ) + 5 ( 7)( ) (g) f ( ) 5( 0.7) (h) + 8 y 5 NSSAL 8 Draft 0 C. D. Pilmer

18 Asymptotes, Points of Discontinuity and Zeros: Part The graphs of rational functions come in a variety of shapes with a variety of features. some have horizontal asymptotes some have vertical asymptotes some have oblique asymptotes (also called slant asymptotes) some have points of discontinuity (i.e. holes in the graph) some have zeros (i.e. -intercepts) A rational function may have most of these features or none of these features. In this section, we eamine the equations of a variety of rational functions and their corresponding graphs to see if the equations actually reveal some of those special features listed above (asymptotes, points of discontinuity, zeros).. Consider the factors in the numerator and denominator of each rational epression in relation to the various features found on the graph.. Consider the degree of the numerator in relation to the degree of the denominator. More specifically, is the numerator of the same degree, a higher degree, or a lower degree relative to the denominator? For eample, in the function y, the numerator is of lower + + degree (first degree; linear) when compared to the degree of the denominator (second degree; quadratic). y 5 5 y or y + ( + ) horizontal asymptote at y 0 vertical asymptote at no oblique asymptotes no points of discontinuity no zeros horizontal asymptote at y 0 vertical asymptotes at 0 and - no oblique asymptotes no points of discontinuity no zeros NSSAL 9 Draft 0 C. D. Pilmer

19 y or y ( + )( ) ( ) 6 y horizontal asymptote at y 0 vertical asymptotes at -.5 and no oblique asymptotes no points of discontinuity zero at horizontal asymptote at y 0 vertical asymptote at no oblique asymptotes no points of discontinuity no zeros y + y + horizontal asymptote at y 0 no vertical asymptotes no oblique asymptotes no points of discontinuity no zeros horizontal asymptote at y vertical asymptote at - no oblique asymptotes no points of discontinuity zero at 0.5 NSSAL 0 Draft 0 C. D. Pilmer

20 y ( )( + ) y or y 4 ( + )( ) horizontal asymptote at y - vertical asymptote at no oblique asymptotes no points of discontinuity zero at 0 horizontal asymptote at y vertical asymptotes at - and no oblique asymptotes no points of discontinuity zeros at - and ( + )( ) y or y + + ( + )( ) + y or y horizontal asymptote at y no vertical asymptotes no oblique asymptotes no points of discontinuity zeros at - and no horizontal asymptotes vertical asymptote at 0 oblique asymptote at y + no points of discontinuity zeros at - and NSSAL Draft 0 C. D. Pilmer

21 + ( + ) or y y or y ( ) ( )( + ) y no horizontal asymptotes vertical asymptote at oblique asymptote at y 0.5 no points of discontinuity zeros at - and 0 no horizontal asymptotes vertical asymptotes at - and oblique asymptote at y no points of discontinuity zero at 0 Questions. Eight of the twelve rational functions had zeros (i.e. -intercepts). For eample, y + had a zero at 0.5. Eplain how you would be able to determine the zeros by merely working with the equation, as opposed to viewing the graph of the function.. Ten of the twelve rational functions had vertical asymptotes. For eample, y ( ) had a vertical asymptote at. Eplain how you would be able to determine the vertical asymptotes by merely working with the equation, as opposed to viewing the graph of the function. NSSAL Draft 0 C. D. Pilmer

22 . Five of the twelve rational functions had horizontal asymptotes at y 0 (i.e. along the -ais). For eample, y had a horizontal asymptote at y 0. Eplain how you would be able to determine if a rational function had a horizontal asymptote at y 0 by merely working with the equation, as opposed to viewing the graph of the function. 4. Four of the twelve rational functions had horizontal asymptotes that had equations other than y 0. For eample, y had a horizontal asymptote at y. Eplain how you + would be able to determine if a rational function had a horizontal asymptote at y 0 by merely working with the equation, as opposed to viewing the graph of the function Three of the twelve rational functions had oblique asymptotes. For eample, y had an oblique asymptote at y +. Eplain how you would be able to determine if a rational function has an oblique asymptote. You are not epected to know how to generate the equation of the oblique asymptote at this time; we will cover that topic in the net section. 6. Given what you have learned by answering questions through 5, determine the zeros and the equations of the asymptotes for each of these rational functions. In the case of oblique asymptotes that indicate whether one is present; presently you are unable to determine the equation of an oblique asymptote. Do not use graphing technology ( )( + 5) (a) y (b) y (c) y ( )( ) NSSAL Draft 0 C. D. Pilmer

23 7. We still need to consider points of discontinuities (i.e. holes in the graph). Eamine the following three rational functions and answer the corresponding question. ( + )( ) 6 y or y + + ( )( ) ( ) + y or y no horizontal asymptotes no vertical asymptotes no oblique asymptote at y 0.5 points of discontinuity at (-,-5) zero at ( ) ( )( ) 9 y or y 5+ 6 horizontal asymptote at y vertical asymptote at 0 no oblique asymptotes point of discontinuity at (,-) zero at Question: Eplain how you would be able to determine if a rational function has a point of discontinuity, and how to determine the coordinates of that point. horizontal asymptote at y 0 vertical asymptote at no oblique asymptotes point of discontinuity at (,) no zeros NSSAL 4 Draft 0 C. D. Pilmer

24 8. Given what you have learned by answering questions through 7, determine the zeros, points of discontinuity, and the equations of the asymptotes for each of these rational functions. In the case of oblique asymptotes, just indicate whether one is present; presently you are unable to determine the equation of an oblique asymptote. Do not use graphing technology ( + 4) (a) y (b) y or y ( )( ) (c) 4( ) ( )( ) 4 4 y or y (d) ( )( ) 8+ y 6 or y 4 4 (e) ( 6) ( )( ) + y or y (f) 5 9 or ( 5+ )( ) ( )( ) y y NSSAL 5 Draft 0 C. D. Pilmer

25 Asymptotes, Points of Discontinuity and Zeros; Part In the previous section, we eamined a variety of rational functions, eamined some of their special features (asymptotes, points of discontinuity, and zeros) and made inferences regarding how anyone can determine these special features by just eamining the equation of the function. Although all of the inferences we made in the last section were correct, we have not considered the mathematics behind each of these inferences; we now rectify this matter. Zeros ( ) ( ) ( ) equal to 0 and solving for. Why is this so? Consider each of the We determined that the zeros (or -intercepts) of a rational function of the form p q found by setting p following. What is y equal to in each case? y 0 y 0 5 y 0 6 y 0 4 y 7 0 can be y Hopefully, you concluded that all of the above y-values are equal to 0. If the numerator is equal to zero, it does not matter what the denominator is equal to because the value of y will be 0. (The only eception to this rule is 0 divided by 0; this is referred to as indeterminate.) Now consider the rational function y 7 +. When 7, then y ( 7) And as we know, zeros or -intercepts occur 5 when y 0. In the case of this rational function, the zero is at 7. Vertical Asymptotes ( ) ( ) ( ) equal to 0 and solving for. Why is this so? Consider each of the following. We determined that the vertical asymptotes of a rational function of the form p q by setting q What is y equal to in each case? y 4 0 y 8 0 y y y 9 0 can be found y 6 0 Hopefully you concluded that all of the above y-values are undefined. This stems from the fact that you cannot divide by 0. If you are unsure why, consider 4 divided by 0. Translated it means how many times does 0 go into 4? The answer is an infinite number of times. The answer is not a finite number; rather, it is undefined. Now consider the rational function y + 5. When 5, then y (undefined). Vertical asymptotes when occur when y is 0 undefined. Therefore, in the case of this rational function, the vertical asymptote is at 5. NSSAL 6 Draft 0 C. D. Pilmer

26 Horizontal Asymptotes In the previous section, we determined that there were two rules regarding horizontal asymptotes.. If the numerator of rational function is of lower degree than the denominator, the horizontal asymptote occurs at y 0.. If the numerator and denominator of the rational function are of the same degree, then the horizontal asymptote can be found by taking the leading numerical of the numerator and dividing it by the leading numerical coefficient of the denominator. The eplanations for these two rules rely totally on the concept of limits. We briefly talked about limits in the Sequence and Series Unit when we looked at infinite geometric series where the common ratio was less than one. We let the number of terms (n) in the series approach infinity( ), and considered the effect on the sum of the terms in the series. S S S n n ( r ) a r a lim n r a r n ( r ) If r is less than one and n is approaching infinity, then r n will approach zero. Let s consider other limit questions that have been completed for us below. lim 0 As approaches infinity, approaches 0. This can be seen in the following table of values lim 0 As approaches infinity, 5 approaches 0. This can be seen in the following table of values lim (no limit) As approaches infinity, also approaches infinity. Since it is not converging on a finite number, we say that no limit eists. This can be seen in the following table of values NSSAL 7 Draft 0 C. D. Pilmer

27 ( ) and let approach In the contet of rational functions, we are going to let approach infinity negative infinity ( ). Meaning we are trying to understand what is happening to the graph, specifically the y-values on the graph, when we go to the etreme right and the etreme left. With many rational functions, the function starts to approach a horizontal straight line (i.e. horizontal asymptote) as we move to the etreme to the right and left. A. Let s start with the rational function y. lim 0 lim Based on the work in the table above, we can see that the graph of the function y approach the function y 0 (i.e. horizontal asymptote) as we move the etreme right (where approaches infinity) and as we move to the etreme left (where approaches negative infinity). This conclusion is supported by the graph of the function, which we have supplied on the right. starts to We can use the notation to represent both and. It is read as the absolute value of is approaching infinity. Absolute values change negative numbers to positive numbers, and leave positive numbers as positive numbers. For eample,, , and. This isn t the formal definition of an absolute value but it will suffice until we can tackle this topic in detail in the net unit. Consider the -values in the two tables of values above. Hopefully you can see that the absolute values of either of the sets of -values are approaching infinity. Instead of writing lim 0 and will write lim 0. ± lim 0, we can simply write lim 0. Other tetbooks + B. Let s consider the rational function y + 4. We know that its horizontal asymptote should be at y 0 because the numerator is of lower degree when compared to the denominator. However, we are going to use limits to understand NSSAL 8 Draft 0 C. D. Pilmer

28 why this is so. Again we want to know what is happening to the etreme right and left on the graph of this function. To do so, we will let the absolute value of approach infinity. lim lim lim Divide everything in the numerator and denominator by. Simplify. Evaluate the limit. + Since lim 0, then we know y as an asymptote at y 0. C. Let s consider the rational function y We know that its horizontal asymptote should be at y because when the numerator and denominator are of the same degree, we merely divide the leading numerical coefficients. However, we are going to use limits to understand why this is so. Again we want to know what is happening to the etreme right and left on the graph of this function. To do so, we will let the absolute value of approach infinity. 6 4 lim lim lim Divide everything in the numerator and denominator by. Simplify. Evaluate the limit. NSSAL 9 Draft 0 C. D. Pilmer

29 6 4 Since lim, then we know y as an asymptote at y. Please note that with both eamples involving horizontal asymptotes, at one point in the question we divided everything in the numerator and denominator of the rational epression by the highest power of found amongst the different terms. Oblique Asymptotes In the last section we learned how to identify whether an oblique asymptote eisted (numerator one degree higher than denominator), but were unaware how to determine the equation of that asymptote. To determine the equation, one must know how to divide one polynomial epression by another polynomial epression (done using long division or synthetic division) and how to work with limits Consider the rational function y. + 4 We use long division to epress the rational function in another form remainder Therefore, y can also be epressed as y What happens to the when approaches infinity? The approaches zero. Therefore the rational function y starts to + 4 approach the linear function y 5 (i.e. the oblique asymptote) as we move to the etreme right and left on the graph. We attended to show this in the accompanying graph where the rational function and its oblique asymptote (dotted line) have been plotted simultaneously. NSSAL 0 Draft 0 C. D. Pilmer

30 Questions. Determine the limit, if one eists. Complete the corresponding tables to support your conclusion. (a) 0 lim (b) lim Determine a limit, if one eists. Please note that we are dealing with approaching infinity, rather than approaching infinity in each of these cases. (a) lim (b) lim (c) lim (d) lim ( + ) (e) 7 lim 5 (f) + 8 lim (g) lim ( 9) + (h) lim 8 NSSAL Draft 0 C. D. Pilmer

31 . Determine the oblique asymptote for each of these rational functions. Show all your work and eplain how limits were used to arrive at your final answer. + 9 (a) y + 5 (b) y A rational function and its horizontal asymptote are supplied. Use limits to confirm that this is the appropriate horizontal asymptote (a) y, horizontal asymptote at y 5 8 NSSAL Draft 0 C. D. Pilmer

32 8 (b) y + 7, horizontal asymptote at y 0 (c) y, horizontal asymptote at y (d) y , horizontal asymptote at y 0 NSSAL Draft 0 C. D. Pilmer

33 5. Si graphs and si functions are provided. Match the graph to the appropriate rational function. Graph I Graph II Graph III Graph IV Graph V Graph VI Graph VII Graph VIII Graph IX (a) y (b) y (c) y (d) y + (e) y + 6 (f) y (g) y (h) y + (i) y + 4 NSSAL 4 Draft 0 C. D. Pilmer

34 6. Is it possible to have a rational function that has both a horizontal asymptote and oblique asymptote? Eplain. 7. Write the equation of a possible rational function with each set of characteristics. (a) vertical asymptotes at and -4, horizontal asymptote at y, zeros at and -, no oblique asymptotes, and no points of discontinuity. (b) vertical asymptote at, horizontal asymptote at y 0, zero at -, no oblique asymptotes, and a point of discontinuity whose -coordinate is 5. (c) vertical asymptote at -, no horizontal asymptote, zeros at -5 and, an oblique asymptote, and no points of discontinuity. (d) vertical asymptotes at and -5, horizontal asymptote at y, zero at, no oblique asymptote, and no points of discontinuity. (e) vertical asymptotes at - and -6, no horizontal asymptote, zeros at -5, 0, and, an oblique asymptote, and a point of discontinuity whose -coordinate is -7. (f) vertical asymptote at 7, horizontal asymptote at y, zero at -4, no oblique asymptotes, and a point of discontinuity whose -coordinate is 0. NSSAL 5 Draft 0 C. D. Pilmer

35 8. Consider the following functions. a f ( ) g( ) b ( a) ( b)( c) What characteristic(s) do all three functions share? ( ) h ( c)( a) ( b)( c) 9. Write the equation of a possible rational function shown in each graph. (a) Points of Discontinuity at (,5) and (-,-4) (b) Passes through the points (-, -6), (-,0), (-,), and (0,0) Point of Discontinuity at (,-6) 0. Using interval notation, state the domain and range of each function (a) y (b) ( )( + 4) y + 4 NSSAL 6 Draft 0 C. D. Pilmer

36 . In a previous section of this unit, you encountered the following two rational functions. y or y ( + )( ) ( )( + ) 6 y or y The first function, y, has a horizontal asymptote at y 0, yet the function 6 intersects the asymptote at (,0). The second function, y, has an oblique asymptote at y, yet the function intersects that asymptote at (0,0). We have previously defined an asymptote as a line a function approaches but never touches; that's obviously not the case here. What's going on? NSSAL 7 Draft 0 C. D. Pilmer

37 . Si tables of values are supplied below. Each of these tables of values comes from a rational function of one of these three forms. b y y a y a ( a ) where a and b are any real number Determine the equation of each function. Do not use technology or graph paper to do this question. (a) (b) (c) y y y undefined undefined undefined (d) (e) (f) y y y undefined undefined undefined NSSAL 8 Draft 0 C. D. Pilmer

38 Rational Equations Eample Solve the equation (a) Graphically (b) Algebraically Answers: (a) We would graph the rational function y using technology, and then + find the -values that correspond to the y-value of. By looking at the graph on the right, we can estimate that the -values of -4 and are the desired answers. We cannot be sure that these are the correct answers until we substitute these -values into the original equation and see if both sides of the equation are equal to each other. Check: Check: ( 4) + 6( 4) + 4 ( ) + 6 ( ) + 4 ( 4) + ( 4) ( ) + ( ) (b) ( + )( ) ( + ) Multiply both sides by ( )( ) or Obviously solving rational equations algebraically is the preferred technique. NSSAL 9 Draft 0 C. D. Pilmer

39 Eample Solve the equation + + (a) Graphically (b) Algebraically Answers: (a) Graph the rational function y and the + linear function y + simultaneously on the same coordinate system and look for the point(s) of intersections. We only need the - coordinates of these points. We cannot be sure that these are the correct answers until we substitute these -values into the original equation and see if both sides of the equation are equal to each other. Check: Check: ( ) ( ) + ( ) ( ) + ( ) + ( ) (b) + + ( + ) ( + )( + ) Multiply both sides by ( + )( ) 0 or NSSAL 0 Draft 0 C. D. Pilmer

40 Eample A group of physics students conducted an eperiment where they eamined light intensity, l, with respect to the distance, d, from a light source. The intensity was measured in lu, and the distance is measured in centimetres. They obtained the following equation l d (a) According to the mathematical model provided, what should the light intensity reading be when you are 5 centimetres from the source? (b) According to the mathematical model provided, what distance should one be from the source to obtain a light intensity reading of 850 lu? Answers: (a) l d l l 65 l 544 lu (b) l d d d ( 850 ) d d 850d d 00 d ± cm from the light source. Eample 4 Given f ( ), find the value(s) of for which f ( ). + Answers: f ( ) + + ( + )( ) ( + ) Multiply both sides by Set the quadratic equation equal to zero. ( )( + ) 0 Factor 0 or + 0 or NSSAL Draft 0 C. D. Pilmer

41 Eample 5 Solve Answer: Method In a previous math course, you learned how to solve linear equations involving fractions by multiplying everything on both sides of the equations by a number that eliminated all the fractions. By doing so, the linear equation was much easier to solve. 4 e.g ( 0) 6 4 6( 5) We use the same strategy when dealing with the equation. We eliminate the denominators by +. multiplying everything on both sides of the equation by ( )( ) ( )( + ) ( )( + ) ( )( + ) + ( )( + )( 6) ( + )( 4) ( )( 8) or 5 ( )( ) Method Make a common denominator on the right hand side of the equation to complete the subtraction. Following that, eliminate the denominators in two separate steps; first multiply +. everything by, and later multiply everything by ( )( ) NSSAL Draft 0 C. D. Pilmer

42 ( + ) 8( ) ( )( + ) ( + )( ) 4( + ) 8( ) ( )( + ) 6 Making a common denominator ( )( + ) ( )( + ) Multiply both sides by. ( )( + ) ( )( + ) ( )( + )( 6) ( )( + ) Multiply both sides by ( )( + ). ( )( + ) ( )( ) or 5 Eample 6 If f ( ) 7 and g( ), find when f ( ) g( ) Answer: 7 7 ( ) ( )( ) ( ) 7 ( )( ) ( ) ( )( ) NSSAL Draft 0 C. D. Pilmer

43 b ± b 4ac a ± () 4()( ) ± 8 ± + or 0.44 or.44 Eample 7 4 Solve Answer: The big mistake learners make with this question is they attempt to eliminate the denominators by multiplying everything by ( + )( + ) without first factoring the quadratic epression. Once factored, you should notice that the denominators can be eliminated by merely multiplying by ( )( + ) ( )( ) ( )( ) ( )( ) 4 ( ) + ( + )( ) or 5 ( )( ) ( )( ) ( )( )( ) NSSAL 4 Draft 0 C. D. Pilmer

44 Questions. A variety of graphs are provided below. Use these graphs to solve each of the following equations. No work needs to be shown. y and other functions + 4 y and other functions + (a) 4 + (b) 4 + (c) + (e) 4 + (d) (f) 4 + ( + ) Praveen is running in the municipal elections. He decides to print up pamphlets so that he can clearly articulate his stances on a variety of municipal issues. The cost per pamphlet (or unit cost) can be described by the following function, where C is the cost per pamphlet in dollars and n is the number of pamphlets printed. 0.5n + 00 C n (a) What type of function are we dealing with? Eplain. (b) If Praveen wants to get the unit cost of a pamphlet down to $0.65, how many will he have to print? (b) If Praveen has 50 pamphlets printed up, what will be the unit cost? NSSAL 5 Draft 0 C. D. Pilmer

45 . Solve each of the following rational equations. (a) 4 + (b) (c) (d) (e) 9 (f) NSSAL 6 Draft 0 C. D. Pilmer

46 4. Given f ( ) + and g( ) 4, g. (a) Find the value(s) of for which ( ) (b) Find f ( 6). (c) Find the value(s) of for which f ( ). (d) Find the value(s) of for which f ( ) g( ). 5. The following chart provides the purchase price and the average annual electrical costs associated with specific household appliances. Appliance Purchase Price of Appliance (after ta) Average Annual Electrical Cost Microwave $97 $ Clothes Dryer $58 $75 Washing Machine $608 $79 Refrigerator $98 $9 (a) Determine the equation of the function that describes the average annual cost, C, of the washing machine in terms of number, n, of years of operation. We are assuming that there are no additional costs (e.g. repair costs) associated with the appliance. (b) Determine the average annual cost of that washing machine over its first eight years of operation. (c) After how many years of operation will the average annual cost of the washing machine be $95? Does this answer make sense given the contet of the problem? NSSAL 7 Draft 0 C. D. Pilmer

47 6. If, then find y 7. Given z a + by, solve for the indicated variable: (a) (b) y (c) a NSSAL 8 Draft 0 C. D. Pilmer

48 8. Mr. McKillop, a math instructor, has provided you with the following proof that. He challenges you to find the error. Please do so. Let A B A AB A B AB B ( A B)( A+ B) B( A B) ( A B)( A+ B) B( A B) A B A B A+ B B B+ B B B B B B B B????? NSSAL 9 Draft 0 C. D. Pilmer

49 Rational Inequalities Eample Solving Rational Inequalities Graphically In the previous section, we used the graph on the right to solve the rational equation. We + found that was equal to -4 or. So how can we use this graph to solve the following inequalities? (a) > (b) (c) < (d) + Answers: (a) We are looking for the -values that generate y-values that are above the line y. Recognize that vertical asymptotes should never be part of the solution as the function is undefined at those values of. 4, 0, ( ) ( ) (b) This inequality is very similar to the last inequality but we now also include the -values that on the line y. [ 4, ) ( 0,] (c) We are looking for the -values that generate y-values that are below the line y. (, 4) (, 0) (, ) (d) This inequality is very similar to the last inequality but we now also include the -values that on the line y. (, 4] (, 0) [, ) Solving Rational Inequalities Algebraically e.g. > > 0 + ( ) 6 4 ( + ) ( ) + + > 0 Making a common denominator ( + ) + NSSAL 40 Draft 0 C. D. Pilmer

50 ( + ) ( + ) ( )( ) ( + ) ( ) >0 4 >0 > 0 Instead of looking at the inequality inequality ( )( ) ( + ) ( 4)( ) function y ( + ) + + >, we are now working with the >0. In this new contet, we are thinking of the rational and trying to find the values of that generate y-values that are greater than zero. As we did with polynomial functions, we will use the roots of the equation (i.e. zeros) and test the regions between them using a number line. However, with rational inequalities, we must also consider the vertical asymptotes. In the case of this new rational function, it has zeros at -4 and, and vertical asymptotes at 0 and -. Place these values on the number line and test the regions between them. We test the values in the epression ( 4 )( ). + ( ) Test 5 {-0.6} Test.5 {.9} -4-0 Test {-} Test 0.5 {.9} Test {-0.6} (Note: Use the TABLE feature on the TI-84 to test values.) In this case, we are looking for the positive regions. 4, 0, Final Answer: ( ) ( ) If the question had been + +, which would ultimately change to ( 4)( ) ( + ) vertical asymptotes. The answer in that case would be [ 4, ) ( 0,]. 0, then the answer would include the zeros, but obviously not the NSSAL 4 Draft 0 C. D. Pilmer

51 Eample Solving Rational Inequalities Graphically In the previous section, we used the graph on the right to solve the rational equation +. We + found that was equal to - or. So how can we use this graph to solve the following inequalities? (a) + + (b) < + + (c) + + (d) > + + Answers: (a) In this case, we want to find the values of for which y-values for the rational function y are equal to or less than the y-values for the linear function y +. + (Translated: Where is the rational function below or intersecting the linear function.) Remember that the vertical asymptote cannot be included in the solution. (, ] (, ] (b) This inequality is very similar to the last inequality but we do not want include the points of intersection. (, ) (, ) (c) In this case, we want to find the values of for which y-values for the rational function y are equal to or greater than the y-values for the linear function y +. + (Translated: Where is the rational function above or intersecting the linear function.) [, ) [, ) (d) This inequality is very similar to the last inequality but we do not want include the points of intersection.,, ( ) ( ) Solving Rational Inequalities Algebraically e.g Making a common denominator + + NSSAL 4 Draft 0 C. D. Pilmer

52 ( + )( ) ( )( ) Instead of looking at the inequality +, we are now working with the + inequality ( + )( ) 0. In this new contet, we are thinking of the rational + ( + )( ) function y and trying to find the values of that generate y-values that + are less than zero. This rational function has zeros at and -, and has a vertical asymptote at -; all three of these values must be incorporated into the number line so that the regions between them can be tested. Test {-.5} Test.5 {.5} - - Test 0 {-4} Test {.5} (Note: Use the TABLE feature on the TI-84 to test values.) In this case, we are looking for the negative regions and the zeros.,, Final Answer: ( ] ( ] Eample If f ( ) and g( ) 5 Answer: f ( ) < g( ) 5 < 5 0< + 5 ( )( ) 0 < + ( )( ), find the values of for which f ( ) < g( ). NSSAL 4 Draft 0 C. D. Pilmer