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College Algebra

by: Ms. Ally Koelpin

College Algebra MATH 1320

Ms. Ally Koelpin
GPA 3.62


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This 17 page Class Notes was uploaded by Ms. Ally Koelpin on Thursday October 22, 2015. The Class Notes belongs to MATH 1320 at Texas Tech University taught by Staff in Fall. Since its upload, it has received 27 views. For similar materials see /class/226471/math-1320-texas-tech-university in Mathematics (M) at Texas Tech University.


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Date Created: 10/22/15
MATH 1 320 College Algebra Course Author Dr Hal Bennett Your grader may be d irmt am the author MATH 1320 features 3 hours credit 1 proctored diagnostic examination which must be taken and graded before doing any of the lesson assignments 12 lessons containing Introductions Lesson Objectives How to Proceeds Discussions and Lesson Assignments 1 self administered midcourse examination to be graded by your instructor 1 proctored final examination 1 textbook prerequisites reasonable training in high school algebra and geometry SATM of 470 or B or better in MATH 0302 All lesson assignments must be submitted via surface mail only MATH 1320 v10 Published by Division of Outreach and Distance Education Texas Tech University Box 42191 Lubbock TX 794092191 Outreach 81 Distance Education Course Development Instructional Designer Dr Hal Bennett Copyright 1995 by the Board of Regents for the College of Arts and Sciences acting for and on behalf of Texas Tech University Lubbock Texas 79409 All rights reserved TABLE OF CONTENTS Introduction to MATH 1320 College Algebra Course Lessons Lesson One Lesson Two Lesson Three Lesson Four Lesson Five Lesson Six Lesson Seven Lesson Eight Lesson Nine Lesson Ten Lesson Eleven Lesson Twelve V D 39 o u r 39 139 1 Basic Concepts anal Operations of Algebra Part One 5 Basic Concepts anal Operations of Algebra Part Tum 1 1 Equations Inequalities anal Problem Solving Part 711 17 Equations Inequalities anal Problem Solving Part Tum 23 Functions Graphs anal Transformations Part One 29 Functions Graphs anal Transformations Part Tum 3 5 7 I 39 J T 39 L39 4 1 Polynomial anal Rational Functions Part One 45 Polynomial anal Rational Functions Part Two 49 Exponential anal Logarithmic Functions Part One 55 Exponential anal Logarithmic Functions Part Tum 61 Lesson Thirteen Systems of 39quotl quot 65 Lesson Fourteen Sequences and Sem39es 71 Final Examination Dirertinns 77 introduction College Algebra Welcome to MATH 1320 Before proceeding with this course you Are you ready for should be sure that you are in the course that will best suit your this course needs and for which you have the sufficient prerequisites Prerequisites for MATH 1320 are as follows 0 High School Algebra I and 11 high school geometry and an SAT math score of at least 470 or an ACT math score of at least 19 01 o a grade of A or B in the Texas Tech University MATH 0302 Intermediate Algebra course or its equivalent If you do not meet either of these prerequisites you are at high risk of failure in this course because you probably do not have sufficient training in basic algebra skills As a remedy we advise you to earn an A or B in MATH 0302 Particularly if it has been several years since you took high school algebra you might need to brush up on your skills If you do meet either or both of these prerequisites you should be ready to attempt MATH 1320 MATH 1320 is designed for students who have at least an intermediate Overview proficiency in algebra and geometry This course covers basic algebra inequalities functions and graphs polynomials and rational functions exponential and logarithmic functions systems of equations and sequences and series MATH 1320 v10 Introduction v Diagnostic Exam Texas Tech University If you have met the prerequisites listed above you should have the background to be able to begin this course If you have not met those prerequisites your progress may be difficult The course consists of 14 lessons and a final exam The sequence is as follows 1 The first lesson is a diagnostic exam that will assess your ability to perform the fundamental operations and understand the basic concepts that you ll need to proceed in the course 2 The next four lessons Two through Five will help you review and sharpen your basic algebra skills You will also develop more advanced algebraic skills 3 Lessons Six and Seven introduce more advanced material followed by a self administered midcourse exam counted as Lesson Eight 4 Lesson Eight is a self administered midcourse exam 5 Lessons Nine through Fourteen consist of new more advanced topics to be mastered 6 The course ends with a proctored final examination Lesson One is a proctored supervised diagnostic examination that tests your ability to perform basic algebraic operations These basic skills are absolutely essential for your success in this course Our experience indicates that if you score less than 60 on the diagnostic exam it is very likely that you will not succeed in the course If you do not pass the diagnostic exam we strongly recommend that you enroll immediately in MATH 0302 Intermediate Algebra Success in MATH 0302 will greatly improve your basic algebra skills and make it much more likely that you will pass MATH 1320 If you do not pass the diagnostic exam and you also choose not to take MATH 0302 you will need to sign the form included in Lesson One that indicates your intention to continue in MATH 1320 against the recommendation of the Division of Outreach 8 Distance Education Without the basic algebra skills you need you will be at great risk of not passing the course vi Introduction MATH 1320 V10 Outreach amp Distance Education IMPORTANT You need to take the diagnostic exam with 60 days of your enrollment in order to take advantage of the exchange options listed below If you do not pass the diagnostic exam and you want to enroll in MATH 0302 through ODE follow this procedure within 60 days of your enrollment date 1 Apply for a full transfer of tuition and fees to 0302 If you re enroll later in 1320 you will have to pay the fees for that course again N If you bought your textbook at the ODE Bookstore you may exchange it for credit toward the 0302 textbook if it is within the time specified in the Policies 8 Forms Guide See the policy guide for more information If you do not pass the MATH 1320 diagnostic exam and you do not want to or cannot continue either with MATH 1320 or MATH 0302 then follow the procedure below again within 60 days of your enrollment date 1 Go through the official withdrawal procedures described in your Policies 8 Forms Guide and apply for a full refund on tuition and fees 2 Bring your textbook back to the ODE Bookstore if that is where you bought it and if it is within the time specified in the Policies 8 Forms Guide and the book is still in new condition you will receive a refund See the policy guide for more information If you do pass the diagnostic exam but you wish to withdraw from MATH 1320 or transfer to another course our regular refund policy applies See Requesting Refundsquot in your Policies 8 Forms Guide Lesson Eight is the midcourse examination This exam is not proctored Midcourse Exam supervised but it is graded by the course instructor As nearly as you can you should simulate the conditions of the final exam that is take it in one sitting as a closed book test in a place where you won t be distracted and time yourself according to the instructions included in Lesson Eight MATH 1320 v10 Introduction vii Final Exam Course Objectives Texas Tech University Being honest with yourself on this exam will allow you to gauge with a fair degree of accuracy how well you are doing in the course If you look in the book for answers or get help with the problems or give yourself more than the allotted time or allow yourself to be distracted too much by outside factors you will defeat the purpose of including this checkpoint exam that is you will not know how YOU are doing in this course The whole point of including this lesson is to give you an idea of how you re doing so far and if the idea you get is inaccurate then it hurts only you When you have completed Lessons One through Fourteen and they have been graded and returned to you you will be approved to take the final examination The final examination is a proctored supervised exam You will need to complete an Examination Application from the Policies 8 Forms Guide and send it to Outreach 8 Distance Education in one of the postage paid envelopes provided You must take the exam on or before your expiration date No matter what your other lesson grades are you must pass the final exam in order to pass the course After completing this course you should be able to work basic algebraic problems including word problems solve equations and inequalities and write the solution sets use the quadratic formula to solve equations understand and graph certain functions solve polynomial and rational functions and inequalities graph exponential and logarithmic functions and solve exponential and logarithmic equations solve systems of equations by various methods recognize and solve sequences and series viii Introduction MATH 1320 V10 Outreach amp Distance Education The required textbook for this course is Textbook Bittinger M L 8 Beecher J A 1993 College Algebra 2nd ed New York Addison Wesley Note It s a good idea not to purchase this textbook until you have received a grade on the diagnostic examination Lesson One If you do decide to buy it before you have received your grade on the diagnostic exam be sure to keep the book in new condition in case you need to get a full credit or refund That is don t write anything your name etc in the book or do anything else that would downgrade the book from its new condition If you have purchased the textbook already and you do not pass the diagnostic examination refer to the refund information in the Diagnostic Exam section on pages vi and vii of this course guide Any course you take by correspondence requires you to be very self How to use the textbook reliant often more so than you have had to be for your on campus courses MATH 1320 is no exception to that rule This course depends on you to use the resources available in the textbook to acquire the skills you will need to do the assignments and the exams well enough to pass the course Your textbook is constructed in a way that will facilitate self study It was chosen because of its readability and because it offers easy access to the information you need When you are studying for a lesson you can take advantage of the following features of the textbook Objectives Objectives are listed at the beginning of each chapter section By the time you get to the end of the lesson you should have accomplished these objectives if you have not you need to go back and do more work Please note that the objectives from the textbook are reproduced in this course guide as lesson objectives Seeing all the objectives in a long list rather than split up as they are in the textbook may seem intimidating but don t let that bother you If you master the material in each section you will be able to accomplish each objective one by one Text and Boxes Make sure you read the body of the text the main column and understand the concepts described therein Theorems definitions and explanations of key concepts often are included in MATH 1320 v10 Introduction ix Cumulative Knowledge Don t skip anything Texas Tech University shaded boxes in the main column of text Cautions about common mistakes are included in unshaded boxes Examples Many detailed examples of the theories you need to know are included in the main column of the text Before you attempt to do the homework exercises be absolutely sure you understand the examples Margin Practice Exercises To check your understanding of the theory do the problems in the margins These problems correspond to the theories and examples in the main column of the text Answers to the margin exercises are in the back of the book Exercise Sets and Answers The assignments that you will send in to be graded by your instructor require you to work selected problems from the exercise sets at the end of each section If a particular problem gives you trouble try working the problems before and after for which answers are provided at the back of the book at least one of those problems should be similar to the one that is giving you trouble PostChapter Reviews and Tests Chapter review exercises and a chapter test are provided for your benefit at the end of each chapter Take the chapter tests as closed book tests giving yourself a reasonable amount of time to complete them Check your answers with those given in the back of the book Successfully completing the chapter test indicates that you are ready to go on to the next chapter This one fact will determine your success in this course more than any other assuming you already have the basic skills You need to be able to understand all the concepts and do all the problems at each step along the way before you go on to the next step Even when you understand most of this chapter most of the next one most of the one after that and so on that means you have not understood all of the information and what s really happening is that you re accumulating gaps in knowledge If you do this eventually things will start to break down somewhere toward the end of the course and certainly by the final exam You ll find yourself frustrated when you take the final trying to work a problem that seems straightforward but that you can t remember how to work because you never figured out how to perform a simple operation that you need to be able to solve the problem Or you might be finishing up a fairly complex problem on the final completely unaware that things went wrong with a simple operation way back when x Introduction MATH 1320 V10 Outreach amp Distance Education you started doing the problem oblivious to the fact that everything you do after that is leading you further from the answer On the positive side if you really do the reading the practice exercises the chapter reviews and tests and all the lesson assignments you should do very well in this course MATH 1320 is neither easy nor particularly difficult and people just like you have taken it successfully by making sure not to omit any of the steps that it takes to learn the material There is simply no substitute for knowing the material and knowing how to perform algebraic operations there is no way of sliding through You must work for it and if you do work for it you should be all right Remember mathematics is not a spectator sport Work lots of problems You will not master this material without working problems Students in this course often use tutors While theoretically the use of a Tl ors tutor might be justifiable in practice using a tutor can be very detrimental Despite their best intentions students often find themselves depending too much on the tutor and thus get a false impression of how well they are doing in the course In short a tutor often becomes a crutch that the student never throws away Typically the student who relies too much on a tutor will make A s and B s on all the lessons with the tutor helping the student finish all the exercises but will fail the final sometimes scoring 30 points or more below a passing grade For these reasons we recommend that if you do decide to use a tutor you should do so correctly That is you should use a tutor only minimally and always make sure that you can work the problems from each lesson by yourself with no assistance before you go on to the next lesson When you come to a proctored exam the diagnostic or the final you must be prepared to do all the work yourself The lesson assignments as a group will count 40 and the final Grading Procedures examination will count 60 of your final grade You will be given a numeric grade on the lessons which will be based on the percentage of correct responses You must pass the final exam to pass the course MATH 1320 v10 Introduction xi About the Author Texas Tech University Harold R Bennett PhD is a professor of mathematics at Texas Tech University He received his master s and doctoral degrees from Arizona State University He joined the faculty of Texas Tech University in 1968 and has been an instructor with Outreach 8 Distance Education for over a decade xii Introduction MATH 1320 V10 two Basic Concepts and Operations of Algebra Part One With this lesson you will begin your review of basic concepts of Illtl39OdUCtion algebra Lessons Two and Three cover Chapter 1 in the text At the end of Lesson Three you should be able to complete the self check chapter test at the end of Chapter 1 successfully If you passed the diagnostic exam especially if you did very well on it you might wonder why you re being required to review these basic skills again There are two answers First the review you will complete in Lessons Two through Five is more in depth and is designed to sharpen your skills in preparation for more advanced material Second there are some important concepts that you may not have seen before And finally it s never a bad idea to get more practice on these skills anyway After completing this lesson you should be able to Lesson Objectives 0 identify various types of real numbers add subtract multiply and divide positive and negative real numbers simplify expressions with integer exponents convert between decimal and scientific notation and solve problems in scientific notation carry out calculations using the rules for order of operations MATH 1320 v10 Lesson Two 5 How to Proceed 3 4 Texas Tech University simplify expressions involving absolute value leaving as little as possible inside the absolute value signs determine the degree of each term of a polynomial and the degree of the polynomial itself add polynomials and other algebraic expressions find the opposite of an algebraic expression subtract polynomials and other algebraic expressions multiply any two polynomials striving for speed and accuracy Read the Introduction and Objectives for this lesson These will alert you to terms and concepts you need to learn Read the Discussion in this lesson that corresponds to Section 11 in the text This discussion will tell you what is important in your study of the textbook section Carefully read Section 11 in the text working the problems in the margin as indicated If you want to learn the ideas in the text you must work the problems as you go Answers to the margin problems and to odd numbered questions from exercise sets are in the back of the book Pay particular attention to theorems and definitions presented in the blue boxes because they represent the main ideas of the section Once you feel confident that you have mastered the concepts in Section 11 complete the selected problems from Exercise Set 11 that are assigned at the end of this lesson If a particular problem gives you trouble try working the problems before and after for which answers are provided at the back of the book at least one of those problems should be similar to the one that is giving you trouble Repeat steps 2 4 for each of the sections assigned in this lesson Sections 12 13 and 14 After you finish all the exercises in the Lesson Assignment place a Lesson Cover Sheet on top of them staple the whole stack together and mail them to Outreach 8 Distance Education according to the 6 Lesson Two MATH 1320 V10 Outreach amp Distance Education instructions given in the Policies 8 Forms Guide Be sure that you complete and mail all of the assignments from this lesson When you feel that you have mastered all of the concepts for this lesson you may proceed to Lesson Three Remember do the reading margin problems and assignments one section at time rather than trying to do them all at once Because the information presented in the course is cumulative it is difficult to do well in one section before you have mastered the material in the previous one This section covers the real number system Real numbers include DiSCUSSion natural numbers whole numbers integers rational numbers and irrational numbers Notice the textbook s explanation of how these various categories of numbers overlap and interrelate Section 11 The book reviews how to add positive and negative real numbers You ll see an explanation of properties of addition that govern algebraic operations like commutative law associative law additive identity and additive inverses You re probably familiar with these ideas although you may not have known what they were called and may not have wanted to know 1 One important item here is the idea of additive inverses the book warns u I you not to fall into the trap of reading every sign as meaning negative less than zero Understanding the quot sign as an opposite or an additive inverse will save your life on exams For instance if x is a negative number then x is a positive number Do you see why The opposite of a negative value is a positive For example if x 4 then x which you should think of as the additive inverse of 4 equals 4 And if x is zero then x is still zero neither negative nor positive So not every expression preceded by a quot will be negative Multiplying numbers also involves laws that are similar to the laws of addition One of the more initially complicated ones but one that is essential to algebraic manipulation is the distributive law You must understand this law to be able to do problems Subtraction and division are the opposites of addition and multiplication respectively and you can see the similarities between the laws that govern their operation and those that govern their counterparts MATH 1320 v10 Lesson Two 7 Texas Tech University Section 12 This section covers exponential scientific and absolute value notation You undoubtedly know what exponents are but you may not be as clear about how negative exponents work Look at the caution in the boxed text for an explanation Notice what zero as an exponent does it makes any nonzero real number value become 1 Now there s something you might not have remembered from high school and something that you wouldn t just be able to figure out by common sense More rules and theorems are explained in this section Don t skip over these just because you think they might be old hat Your success on the exercises and the exams in this course will depend on how well you have mastered these basic ideas When a student is having chronic problems with algebra that are keeping him or her from passing exams you d be surprised how rarely they occur because of an inability to master a complex operation usually they happen because the student missed some really basic idea way back at the beginning of the course Always keep this in mind You just can t do algebra by instinct and common sense You have to leam how to do it Once you ve practiced enough it might or even should begin to eel instinctive but don t let yourself believe that it s all just common sense because it s not You have to learn it Especially note that 4 i and also that 4 17 i 4 4 17 The book mentions scienti c notation a concept you probably remember and tells you how to manipulate it Order of operations is another concept that is essential if you want to work problems correctly Absolute value is a relatively simple concept but some of its properties Theorem 9 are less obvious Be sure you know them 8 Lesson Two MATH 1320 v10 Outreach amp Distance Education This section discusses addition and subtraction of algebraic expressions Note that not all letters that represent numbers are variables therefore a polynomial can be in one variable even if you see more than one letter there since any other letter can be a constant You may wonder how the degree of the term 10x3y can be 4 but remember that the implied exponent of y is 1 which added to the explicit exponent 3 for x makes 4 Note how similar terms are combined and simplified exactly how an expression within parentheses is affected by an opposite sign outside of the parentheses and how a subtraction problem can be converted into an addition problem and then simplified This section is on the multiplication of algebraic expressions Once again the distributive law becomes essential here for any polynomials being multiplied The book gives you a device the acronym FOIL first outside inside last for remembering how to multiply or square two binomials and how to find the product of a sum and a difference of the same two expressions Knowing how and when to use this device will help you on problems and exams For all problems show all your work Be sure to circle your answer and circle only your answer Before attempting the lesson assignment carefully study all examples and do all the practice exercises in the textbook s margins If you have trouble with a problem attempt the problems immediately before and after it adjacent problems are often similar to an assigned problem Most of your problems are even numbered answers for odd numbered problems are at the back of the book Exercise Set 11 pages 9 10 Exercises 3 4 6 13 14 24 42 50 Exercise Set 12 pages 18 19 Exercises 2 10 18 36 42 44 54 88 94 98 Exercise Set 13 page 23 Exercises 2 8 12 18 20 22 Exercise Set 14 pages 26 27 Exercises 2 6 8 16 20 28 30 34 42 46 Section 13 Section 14 Lesson Two Assignment MATH 1320 V10 Lesson Two 9


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