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TEACHING GUIDE FOR SENIOR HIGH SCHOOL
General Mathematics CORE SUBJECT
This Teaching Guide was collaboratively developed and reviewed by educators from public and private schools, colleges, and universities. We encourage teachers and other education
stakeholders to email their feedback, comments, and recommendations to the Commission on Higher Education, K to 12 Transition Program Management Unit - Senior High School
Support Team at [email protected]. We value your feedback and recommendations.
Commission on Higher Education in collaboration with the Philippine Normal University
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Printed in the Philippines by EC-TEC Commercial, No. 32 St. Louis Compound 7, Baesa, Quezon City, [email protected]
Published by the Commission on Higher Education, 2016 Chairperson: Patricia B. Licuanan, Ph.D.
Commission on Higher Education K to 12 Transition Program Management Unit Office Address: 4th Floor, Commission on Higher Education, C.P. Garcia Ave., Diliman, Quezon City Telefax: (02) 441-1143 / E-mail Address: [email protected]
DEVELOPMENT TEAM
Team Leader: Debbie Marie B. Verzosa, Ph.D.
Writers: Leo Andrei A. Crisologo, Lester C. Hao, Eden Delight P. Miro, Ph.D., Shirlee R. Ocampo, Ph.D., Emellie G. Palomo, Ph.D., Regina M. Tresvalles, Ph.D.
Technical Editors: Mark L. Loyola, Ph.D., Christian Chan O. Shio, Ph.D.
Copy Reader: Sheena I. Fe
Typesetters: Juan Carlo F. Mallari, Regina Paz S. Onglao
Illustrator: Ma. Daniella Louise F. Borrero
Cover Artists: Paolo Kurtis N. Tan, Renan U. Ortiz
CONSULTANTS THIS PROJECT WAS DEVELOPED WITH THE PHILIPPINE NORMAL UNIVERSITY.University President: Ester B. Ogena, Ph.D. VP for Academics: Ma. Antoinette C. Montealegre, Ph.D. VP for University Relations & Advancement: Rosemarievic V. Diaz, Ph.D.
Ma. Cynthia Rose B. Bautista, Ph.D., CHEDBienvenido F. Nebres, S.J., Ph.D., Ateneo de Manila University Carmela C. Oracion, Ph.D., Ateneo de Manila University Minella C. Alarcon, Ph.D., CHEDGareth Price, Sheffield Hallam University Stuart Bevins, Ph.D., Sheffield Hallam University
SENIOR HIGH SCHOOL SUPPORT TEAM CHED K TO 12 TRANSITION PROGRAM MANAGEMENT UNIT Program Director: Karol Mark R. Yee
Lead for Senior High School Support: Gerson M. Abesamis
Lead for Policy Advocacy and Communications: Averill M. Pizarro
Course Development Officers: Danie Son D. Gonzalvo, John Carlo P. Fernando
Teacher Training Officers: Ma. Theresa C. Carlos, Mylene E. Dones
Monitoring and Evaluation Officer: Robert Adrian N. Daulat
Administrative Officers: Ma. Leana Paula B. Bato, Kevin Ross D. Nera, Allison A. Danao, Ayhen Loisse B. Dalena
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Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
DepEd General Mathematics Curriculum Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Chapter 1 Functions
Lesson 1: Functions as Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Lesson 2: Evaluating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Lesson 3: Operations on Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Chapter 2 Rational Functions
Lesson 4: Representing Real-Life Situations Using Rational Functions . . . . . . . . . . . . . . 23
Lesson 5: Rational Functions, Equations, and Inequalities . . . . . . . . . . . . . . . . . . . . . . . . 28
Lesson 6: Solving Rational Equations and Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Lesson 7: Representations of Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Lesson 8: Graphing Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Chapter 3 One-to-One and Inverse Functions
Lesson 9: One-to-One Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Lesson 10: Inverse of One-to-One Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Lesson 11: Graphs of Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Chapter 4 Exponential Functions
Lesson 12: Representing Real-Life Situations Using Exponential Functions . . . . . . . . . . 88
Lesson 13: Exponential Functions, Equations, and Inequalities . . . . . . . . . . . . . . . . . . . . 94
Lesson 14: Solving Exponential Equations and Inequalities . . . . . . . . . . . . . . . . . . . . . . . 96
Lesson 15: Graphing Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Lesson 16: Graphing Transformations of Exponential Functions . . . . . . . . . . . . . . . . . . . 107
Chapter 5 Logarithmic Functions
Lesson 17: Introduction to Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
Lesson 18: Logarithmic Functions, Equations, and Inequalities . . . . . . . . . . . . . . . . . . . . 125
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Lesson 19: Basic Properties of Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Lesson 20: Laws of Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Lesson 21: Solving Logarithmic Equations and Inequalities . . . . . . . . . . . . . . . . . . . . . . . 136
Lesson 22: The Logarithmic Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
Chapter 6 Simple and Compound Interest
Lesson 23: Illustrating Simple and Compound Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
Lesson 24: Simple Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
Lesson 25: Compound Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
Lesson 26: Compounding More than Once a Year . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
Lesson 27: Finding Interest Rate and Time in Compound Interest . . . . . . . . . . . . . . . . . 185
Chapter 7 Annuities
Lesson 28: Simple Annuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
Lesson 29: General Annuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
Lesson 30: Deferred Annuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
Chapter 8 Basic Concepts of Stocks and Bonds
Lesson 31: Stocks and Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
Lesson 32: Market Indices for Stocks and Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
Lesson 33: Theory of Efficient Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
Chapter 9 Basic Concepts of Loans
Lesson 34: Business Loans and Consumer Loans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
Lesson 35: Solving Problems on Business and Consumer Loans (Amortization and Mortgage) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
Chapter 10 Logic
Lesson 36: Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
Lesson 37: Logical Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
Lesson 38: Constructing Truth Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
Lesson 39: Logical Equivalence and Forms of Conditional Propositions . . . . . . . . . . . . . . 285
Lesson 40: Valid Arguments and Fallacies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
Lesson 41: Methods of Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
Biographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
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IntroductionAs the Commission supports DepEds implementation of Senior High School (SHS), it upholds the vision and mission of the K to 12 program, stated in Section 2 of Republic Act 10533, or the Enhanced Basic Education Act of 2013, that every graduate of basic education be an empowered individual, through a program rooted on...the competence to engage in work and be productive, the ability to coexist in fruitful harmony with local and global communities, the capability to engage in creative and critical thinking, and the capacity and willingness to transform others and oneself.
To accomplish this, the Commission partnered with the Philippine Normal University (PNU), the National Center for Teacher Education, to develop Teaching Guides for Courses of SHS. Together with PNU, this Teaching Guide was studied and reviewed by education and pedagogy experts, and was enhanced with appropriate methodologies and strategies.
Furthermore, the Commission believes that teachers are the most important partners in attaining this goal. Incorporated in this Teaching Guide is a framework that will guide them in creating lessons and assessment tools, support them in facilitating activities and questions, and assist them towards deeper content areas and competencies. Thus, the introduction of the SHS for SHS Framework.
The SHS for SHS Framework The SHS for SHS Framework, which stands for Saysay-Husay-Sarili for Senior High School, is at the core of this book. The lessons, which combine high-quality content with flexible elements to accommodate diversity of teachers and environments, promote these three fundamental concepts:
SAYSAY: MEANING Why is this important?
Through this Teaching Guide, teachers will be able to facilitate an understanding of the value of the lessons, for each learner to fully engage in the content on both the cognitive and affective levels.
HUSAY: MASTERY How will I deeply understand this?
Given that developing mastery goes beyond memorization, teachers should also aim for deep understanding of the subject matter where they lead learners to analyze and synthesize knowledge.
SARILI: OWNERSHIP What can I do with this?
When teachers empower learners to take ownership of their learning, they develop independence and self-direction, learning about both the subject matter and themselves.
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The Parts of the Teaching Guide This Teaching Guide is mapped and aligned to the DepEd SHS Curriculum, designed to be highly usable for teachers. It contains classroom activities and pedagogical notes, and integrated with innovative pedagogies. All of these elements are presented in the following parts:
1. INTRODUCTION Highlight key concepts and identify the essential questions
Show the big picture
Connect and/or review prerequisite knowledge
Clearly communicate learning competencies and objectives
Motivate through applications and connections to real-life
2. MOTIVATION Give local examples and applications
Engage in a game or movement activity
Provide a hands-on/laboratory activity
Connect to a real-life problem
3. INSTRUCTION/DELIVERY Give a demonstration/lecture/simulation/hands-on activity
Show step-by-step solutions to sample problems
Give applications of the theory
Connect to a real-life problem if applicable
4. PRACTICE Provide easy-medium-hard questions
Give time for hands-on unguided classroom work and discovery
Use formative assessment to give feedback
5. ENRICHMENT Provide additional examples and applications
Introduce extensions or generalisations of concepts
Engage in reflection questions
Encourage analysis through higher order thinking prompts
Allow pair/small group discussions
Summarize and synthesize the learnings
6. EVALUATION Supply a diverse question bank for written work and exercises
Provide alternative formats for student work: written homework, journal, portfolio, group/individual projects, student-directed research project
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On DepEd Functional Skills and CHEDs College Readiness Standards As Higher Education Institutions (HEIs) welcome the graduates of the Senior High School program, it is of paramount importance to align Functional Skills set by DepEd with the College Readiness Standards stated by CHED.
The DepEd articulated a set of 21st century skills that should be embedded in the SHS curriculum across various subjects and tracks. These skills are desired outcomes that K to 12 graduates should possess in order to proceed to either higher education, employment, entrepreneurship, or middle-level skills development.
On the other hand, the Commission declared the College Readiness Standards that consist of the combination of knowledge, skills, and reflective thinking necessary to participate and succeed - without remediation - in entry-level undergraduate courses in college.
The alignment of both standards, shown below, is also presented in this Teaching Guide - prepares Senior High School graduates to the revised college curriculum which will initially be implemented by AY 2018-2019.
College Readiness Standards Foundational Skills DepEd Functional Skills
Produce all forms of texts (written, oral, visual, digital) based on: 1. Solid grounding on Philippine experience and culture; 2. An understanding of the self, community, and nation; 3. Application of critical and creative thinking and doing processes; 4. Competency in formulating ideas/arguments logically, scientifically,
and creatively; and 5. Clear appreciation of ones responsibility as a citizen of a multicultural
Philippines and a diverse world;
Visual and information literacies Media literacy Critical thinking and problem solving skills Creativity Initiative and self-direction
Systematically apply knowledge, understanding, theory, and skills for the development of the self, local, and global communities using prior learning, inquiry, and experimentation
Global awareness Scientific and economic literacy Curiosity Critical thinking and problem solving skills Risk taking Flexibility and adaptability Initiative and self-direction
Work comfortably with relevant technologies and develop adaptations and innovations for significant use in local and global communities;
Global awareness Media literacy Technological literacy Creativity Flexibility and adaptability Productivity and accountability
Communicate with local and global communities with proficiency, orally, in writing, and through new technologies of communication;
Global awareness Multicultural literacy Collaboration and interpersonal skills Social and cross-cultural skills Leadership and responsibility
Interact meaningfully in a social setting and contribute to the fulfilment of individual and shared goals, respecting the fundamental humanity of all persons and the diversity of groups and communities
Media literacy Multicultural literacy Global awareness Collaboration and interpersonal skills Social and cross-cultural skills Leadership and responsibility Ethical, moral, and spiritual values
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The General Mathematics Teaching Guide
Implementing a new curriculum is always subject to a new set of challenges.
References are not always available, and training may be too short to cover all the
required topics. Under these circumstances, providing teachers with quality resource
materials aligned with the curricular competencies may be the best strategy for
delivering the expected learning outcomes. Such is the rationale for creating a series of
teaching guides for several Grade 11 and 12 subjects. The intention is to provide
teachers a complete resource that addresses all expected learning competencies, as
stated in the Department of Educations offi cial curriculum guide.
This resource is a teaching guide for General Mathematics. The structure is quite
unique, re flective of the wide scope of General Mathematics: functions, business
mathematics, and logic. Each lesson begins with an introductory or motivational
activity. The main part of the lesson presents important ideas and provides several
solved examples. Explanations to basic properties, the rationale for mathematical
procedures, and the derivation of important formulas are also provided. The goal is to
enable teachers to move learners away from regurgitating information and towards an
authentic understanding of, and appreciation for, the subject matter.
The chapters on functions are an extension of the functions learned in Junior High
School, where the focus was primarily on linear, quadratic, and polynomial functions.
In Grade 11, learners will be exposed to other types of functions such as piecewise,
rational, exponential, and logarithmic functions. Related topics such as solving
equations and inequalities, as well as identifying the domain, range, intercepts, and
asymptotes are also included.
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The chapters on business mathematics in Grade 11 may be learners' first opportunity
to be exposed to topics related to financial literacy. Here, they learn about simple and
compound interest, annuities, loans, stocks, and bonds. These lessons can hopefully
prepare learners to analyze business-related problems and make sound financial
decisions.
The final chapter on logic exposes learners to symbolic forms of propositions (or
statements) and arguments. Through the use of symbolic logic, learners should be able
to recognize equivalent propositions, identify fallacies, and judge the validity of
arguments. The culminating lesson is an application of the rules of symbolic logic, as
learners are taught to write their own justifications to mathematical and real-life
statements.
This Teaching Guide is intended to be a practical resource for teachers. It includes
activities, explanations, and assessment tools. While the beginning teacher may use
this Teaching Guide as a script, more experienced teachers can use this resource as a
starting point for writing their own lesson plans. In any case, it is hoped that this
resource, together with the Teaching Guide for other subjects, can support teachers in
achieving the vision of the K to 12 Program.
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vi
Hour 1 Hour 2 Hour 3 Hour 4
Week a Lesson 1 Lesson 1, 2 Lesson 3 Lesson 3
Week b Lesson 4 Lesson 5, 6 Lesson 6 Lesson 7
Week c Lesson 7 Lesson 8 Lesson 8 Review/Exam
Week d Lesson 9 Lesson 10 Lesson 10 Lesson 11
Week e Lesson 11 Review/Exam Lesson 12 Lesson 12, 13
Week f Lesson 14 Lesson 14 Lesson 15 Lesson 15
Week g Lesson 16 Lesson 16 Review/Exam Review/Exam
Week h Lesson 17 Lesson 17 Lesson 18, 19 Lesson 19, 20
Week i Lesson 20 Lesson 21 Lesson 21 Lesson 21
Week j Lesson 22 Lesson 22 Review/Exam Review/Exam
First Quarter
Hour 1 Hour 2 Hour 3 Hour 4
Week a Lesson 23 Lesson 24 Lesson 25 Lesson 25, 26
Week b Lesson 26 Lesson 27 Lesson 27 / Review Review/Exam
Week c Lesson 28 Lesson 28 Lesson 29 Lesson 29
Week d Lesson 29 Lesson 30 Lesson 30 Review/Exam
Week e Lesson 31 Lesson 31 Lesson 32 Lesson 33
Week f Lesson 34 Lesson 35 Lesson 35 Review/Exam
Week g Lesson 36 Lesson 36 Lesson 37 Lesson 37
Week h Lesson 38 Lesson 39 Lesson 39 Lesson 39
Week i Lesson 40 Lesson 40 Lesson 40 Lesson 41
Week j Lesson 41 Lesson 41 Review/Exam Review/Exam
Second Quarter
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K to 12 BASIC EDUCATION CURRICULUM SENIOR HIGH SCHOOL CORE SUBJECT
K to 12 Senior High School Core Curriculum General Mathematics December 2013 Page 1 of 5
Grade: 11 Semester: First Semester Core Subject Title: General Mathematics No. of Hours/Semester: 80 hours/semester Prerequisite (if needed): Core Subject Description: At the end of the course, the students must know how to solve problems involving rational, exponential and logarithmic functions; to solve business-related problems; and to apply logic to real-life situations.
CONTENT CONTENT STANDARDS PERFORMANCE STANDARDS LEARNING COMPETENCIES CODE
Functions and Their Graphs
The learner demonstrates
understanding of...
1. key concepts of functions.
The learner is able to...
1. accurately construct mathematical models to
represent real-life
situations using
functions.
The learner...
1. represents real-life situations using functions, including piece-wise functions.
M11GM-Ia-1
2. evaluates a function. M11GM-Ia-2
3. performs addition, subtraction, multiplication, division, and composition of functions
M11GM-Ia-3
4. solves problems involving functions. M11GM-Ia-4
2. key concepts of rational functions.
2. accurately formulate and solve real-life problems
involving rational
functions.
5. represents real-life situations using rational functions. M11GM-Ib-1 6. distinguishes rational function, rational equation, and
rational inequality. M11GM-Ib-2
7. solves rational equations and inequalities. M11GM-Ib-3 8. represents a rational function through its: (a) table of
values, (b) graph, and (c) equation. M11GM-Ib-4
9. finds the domain and range of a rational function. M11GM-Ib-5 10. determines the:
(a) intercepts
(b) zeroes; and
(c) asymptotes of rational functions
M11GM-Ic-1
11. graphs rational functions. M11GM-Ic-2 12. solves problems involving rational functions,
equations, and inequalities. M11GM-Ic-3
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K to 12 BASIC EDUCATION CURRICULUM SENIOR HIGH SCHOOL CORE SUBJECT
K to 12 Senior High School Core Curriculum General Mathematics December 2013 Page 2 of 5
CONTENT CONTENT STANDARDS PERFORMANCE STANDARDS LEARNING COMPETENCIES CODE
3. key concepts of inverse functions, exponential
functions, and
logarithmic functions.
3. apply the concepts of inverse functions,
exponential functions,
and logarithmic functions
to formulate and solve
real-life problems with
precision and accuracy.
1. represents real-life situations using one-to one functions.
M11GM-Id-1
2. determines the inverse of a one-to-one function. M11GM-Id-2 3. represents an inverse function through its: (a) table of
values, and (b) graph. M11GM-Id-3
4. finds the domain and range of an inverse function. M11GM-Id-4 5. graphs inverse functions. M11GM-Ie-1 6. solves problems involving inverse functions. M11GM-Ie-2 7. represents real-life situations using exponential
functions. M11GM-Ie-3
8. distinguishes between exponential function, exponential equation, and exponential inequality.
M11GM-Ie-4
9. solves exponential equations and inequalities. M11GM-Ie-f-1 10. represents an exponential function through its: (a) table
of values, (b) graph, and (c) equation. M11GM-If-2
11. finds the domain and range of an exponential function. M11GM-If-3 12. determines the intercepts, zeroes, and asymptotes of
an exponential function. M11GM-If-4
13. graphs exponential functions. M11GM-Ig-1 14. solves problems involving exponential functions,
equations, and inequalities. M11GM-Ig-2
15. represents real-life situations using logarithmic functions.
M11GM-Ih-1
16. distinguishes logarithmic function, logarithmic equation, and logarithmic inequality.
M11GM-Ih-2
17. illustrates the laws of logarithms. M11GM-Ih-3 18. solves logarithmic equations and inequalities. M11GM-Ih-i-1 19. represents a logarithmic function through its: (a) table
of values, (b) graph, and (c) equation. M11GM-Ii-2
20. finds the domain and range of a logarithmic function. M11GM-Ii-3 21. determines the intercepts, zeroes, and asymptotes of
logarithmic functions. M11GM-Ii-4
22. graphs logarithmic functions. M11GM-Ij-1 23. solves problems involving logarithmic functions,
equations, and inequalities.
M11GM-Ij-2
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K to 12 BASIC EDUCATION CURRICULUM SENIOR HIGH SCHOOL CORE SUBJECT
K to 12 Senior High School Core Curriculum General Mathematics December 2013 Page 3 of 5
CONTENT CONTENT STANDARDS PERFORMANCE STANDARDS LEARNING COMPETENCIES CODE
Basic Business Mathematics
The learner demonstrates
understanding of...
1. key concepts of simple and compound interests,
and simple and general
annuities.
The learner is able to...
1. investigate, analyze and solve problems involving
simple and compound
interests and simple and
general annuities using
appropriate business and
financial instruments.
24. illustrates simple and compound interests. M11GM-IIa-1 25. distinguishes between simple and compound interests. M11GM-IIa-2 26. computes interest, maturity value, future value, and
present value in simple interest and compound interest
environment.
M11GM-IIa-b-1
27. solves problems involving simple and compound interests.
M11GM-IIb-2
28. illustrates simple and general annuities. M11GM-IIc-1 29. distinguishes between simple and general annuities. M11GM-IIc-2 30. finds the future value and present value of both simple
annuities and general annuities. M11GM-IIc-d-1
31. calculates the fair market value of a cash flow stream that includes an annuity.
M11GM-IId-2
32. calculates the present value and period of deferral of a deferred annuity.
M11GM-IId-3
2. basic concepts of stocks and bonds.
2. use appropriate financial instruments involving
stocks and bonds in
formulating conclusions
and making decisions.
33. illustrate stocks and bonds. M11GM-IIe-1 34. distinguishes between stocks and bonds. M11GM-IIe-2 35. describes the different markets for stocks and bonds. M11GM-IIe-3 36. analyzes the different market indices for stocks and
bonds. M11GM-IIe-4
37. interprets the theory of efficient markets. M11GM-IIe-5 3. basic concepts of
business and
consumer loans.
3. decide wisely on the appropriateness of
business or consumer
loan and its proper
utilization.
38. illustrates business and consumer loans. M11GM-IIf-1 39. distinguishes between business and consumer loans. M11GM-IIf-2 40. solves problems involving business and consumer loans
(amortization, mortgage). M11GM-IIf-3
Logic The learner demonstrates understanding of...
1. key concepts of propositional logic;
syllogisms and
fallacies.
The learner is able to...
1. judiciously apply logic in real-life arguments.
41. illustrates a proposition. M11GM-IIg-1 42. symbolizes propositions. M11GM-IIg-2 43. distinguishes between simple and compound
propositions. M11GM-IIg-3
44. performs the different types of operations on propositions.
M11GM-IIg-4
45. determines the truth values of propositions. M11GM-IIh-1 46. illustrates the different forms of conditional
propositions. M11GM-IIh-2
47. illustrates different types of tautologies and fallacies. M11GM-IIi-1
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K to 12 BASIC EDUCATION CURRICULUM SENIOR HIGH SCHOOL CORE SUBJECT
K to 12 Senior High School Core Curriculum General Mathematics December 2013 Page 4 of 5
CONTENT CONTENT STANDARDS PERFORMANCE STANDARDS LEARNING COMPETENCIES CODE
48. determines the validity of categorical syllogisms. M11GM-IIi-2 49. establishes the validity and falsity of real-life arguments
using logical propositions, syllogisms, and fallacies. M11GM-IIi-3
2. key methods of proof and disproof.
2. appropriately apply a method of proof and
disproof in real-life
situations.
50. illustrates the different methods of proof (direct and indirect) and disproof (indirect and by
counterexample).
M11GM-IIj-1
51. justifies mathematical and real-life statements using the different methods of proof and disproof.
M11GM-IIj-2
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(x, y) (x, y)
x
y
-
f = {(1, 2), (2, 2), (3, 5), (4, 5)}
g = {(1, 3), (1, 4), (2, 5), (2, 6), (3, 7)}
h = {(1, 3), (2, 6), (3, 9), . . . , (n, 3n), . . .}
f h x
y g (1, 3) (1, 4)
x y
f g
-
h
f g x 2 X y 2 Yh X
y x = 7 y = 11 13 x = 2
y = 17 19
x = a (a, b) (a, c)
y x = a
-
(a) (b) (c) (d) (e)
x
y x
x y
y = 2x+ 1
y = x2 2x+ 2
x2 + y2 = 1
y =px+ 1
y =2x+ 1
x 1
-
y = bxc+ 1 bxc
x y x = 0 y +1
1
x
R
R
[1, 1]
[1,+1)
(1, 1) [ (1,+1)
R
y f(x) y
x f
f(x) = 2x+ 1
q(x) = x2 2x+ 2
g(x) =px+ 1
r(x) =2x+ 1
x 1F (x) = bxc+ 1 bxc
C x
40
40 C(x) = 40x
-
A x
A = xy x
x + 2y = 100 y = (100 x)/2 = 50 0.5xA(x) = x(50 0.5x) = 50x 0.5x2
300
1
m
t(m)
t(m) =
(300 0 < m 100300 +m m > 100
8.00
1.50
d
-
F (d)
F (d) =
(8 0 < d 48 + 1.5bdc d > 4
bdc dd b4.1c = b4.9c = 4
25
0
0
100
100
100
T (x)
T (x)
-
15
1, 000
400
f(x) =
8 3
700
-
f(x) = 700dx4
e x 2 N
150
130 110
100
f(x) =
8>>>>>>>>>>>:
150x 0 x 20
130x 21 x 50
110x 51 x 100
100x x > 100
x 2 N
-
x
f a
a f f(a)
x = 1.5
f(x) = 2x+ 1
q(x) = x2 2x+ 2
g(x) =px+ 1
r(x) =2x+ 1
x 1F (x) = bxc+ 1 bxc
1.5 x
f(1.5) = 2(1.5) + 1 = 4
q(1.5) = (1.5)2 2(1.5) + 2 = 2.25 3 + 2 = 1.25
g(1.5) =p1.5 + 1 =
p2.5
r(1.5) =2x+ 1
x 1 =2(1.5) + 1
(1.5) 1 =3 + 1
0.5= 8
F (1.5) = bxc+ 1 = b1.5c+ 1 = 1 + 1 = 2
g(4) r(1) g r
4 g(x)r(x)
f q
f(3x 1) q(2x+ 3)
-
f(3x 1) x f(x) = 2x+ 1 (3x 1)
f(3x 1) = 2(3x 1) + 1 = 6x 2 + 1 = 6x 1
q(3x+ 3) x q(x) = x2 2x+ 2 (2x+ 3)
q(2x+ 3) = (2x+ 3)2 2(2x+ 3) + 2 = (4x2 + 12x+ 9) 4x 6 + 2 = 4x2 + 8x+ 5
f(x) = x 2
f(0) 2
f(3)
f(1)
f() 2
f(x+ 1) x 1
f(3x) 3x 2
f(x) =4
x
f(1) 4
f(2) 2
f(1) 4
f(p2) 2
p2
f(1/x) 4x
f(2x) 2/x
f(x) =px 3
f(3) 0
f(4) 1
f(12) 3
f(x 3)px 6
f
1
1 x
3x 21 x
f(x2 + 4x+ 7)px2 + 4x+ 4 |x+ 2|
-
200
25
C(x) = 25x+200 x C(x)
2700 3950
t s(t) = 5t2 + 100g = 10m/s2
-
1
3
2
5
1
3
+
2
5
=
5
15
+
6
15
=
5 + 6
15
=
11
15
1
x 32
x 5
(x 3)(x 5) (x2 8x+ 15)
1
x 3 +2
x 5 =x 5
x2 8x+ 15 +2(x 3)
x2 8x+ 15 =x 5 + 2x 6x2 8x+ 15
=
3x 11x2 8x+ 15
10
21
15
8
10
21
158
=
2 53 7
3 52 2 2 =
6 2 5 6 3 56 3 7 6 2 2 2 =
25
28
-
x2 4x 5x2 3x+ 2
x2 5x+ 6x2 3x 10
x2 4x 5x2 3x+ 2
x2 5x+ 6x2 3x 10 =
(x+ 1)(x 5)(x 2)(x 1)
(x 2)(x 3)(x 5)(x+ 2)
=
(x+ 1)
(x 5)(x 2)(x 3)
(x 2)(x 1)(x 5)(x+ 2)
=
(x+ 1)(x 3)(x 1)(x+ 2) =
x2 2x 3x2 + x 2
f g
f + g (f + g)(x) = f(x) + g(x)
f g (f g)(x) = f(x)g(x)
f g (f g)(x) = f(x) g(x)
f/g (f/g)(x) = f(x)/g(x)
x g(x) = 0
-
f(x) = x+ 3
p(x) = 2x 7
v(x) = x2 + 5x+ 4
g(x) = x2 + 2x 8
h(x) = x+ 72 x
t(x) = x 2x+ 3
(v + g)(x)
(f p)(x)
(f + h)(x)
(p f)(x)
(v/g)(x)
(v + g)(x) =x2 + 5x+ 4
+
x2 + 2x 8
= 2x2 + 7x 4
(f p)(x) = (x+ 3) (2x 7) = 2x2 x 21
(f + h)(x) = (x+ 3) +x+ 7
2 x = (x+ 3) 2 x2 x +
x+ 7
2 x =(x+ 3)(2 x) + (x+ 7)
2 x =
=
6 x x2 + x+ 72 x =
13 x2
2 x =13 x2
2 x 11 =
x2 13x 2
(p f)(x) = (2x 7) (x+ 3) = 2x 7 x 3 = x 10
(v/g)(x) = (x2 + 5x+ 4) (x2 + 2x 8) = x2
+ 5x+ 4
x2 + 2x 8
f(x) = 2x+ 1 q(x) = x2 2x+ 2 r(x) = 2x+ 1x 1
f1
(x) = x2 + 3
q(x) f(x)
x2 + 3
q(x) + f(x) = (x2 2x+ 2) + (2x+ 1)
= x2 + 3
= f1
(x)
f1
(x) = q(x) + f(x)
f2
(x) = x24x+1
-
q(x) f(x) x2 4x+ 1
q(x) f(x) = (x2 2x+ 2) (2x+ 1)
= x2 4x+ 1
= f2
(x)
f2
(x) = q(x) f(x)
f3
(x) =2x2 + x
x 1
2x2 + x
x 1 x 1 r(x) =2x+ 1
x 1f(x) r(x)
f(x) + r(x) = 2x+ 1 +2x+ 1
x 1
=
(2x+ 1)(x 1)x 1 +
2x+ 1
x 1
=
(2x+ 1)(x 1) + (2x+ 1)x 1
=
(2x2 x 1) + (2x+ 1)x 1
=
2x2 + x
x 1= f
3
(x)
(f + g)(x) = f(x) + g(x)
f1
(x) = q(x) + f(x) = (q + f)(x)
f2
(x) = q(x) f(x) = (q f)(x)f3
(x) = f(x) + r(x) = (f + r)(x)
g1
(x) = 2x3 3x2 + 2x + 2
-
2x3 3x2 + 2x+ 2 f(x) q(x)
f(x) q(x) = (2x+ 1)(x2 2x+ 2)
= (2x)(x2 2x+ 2) + (x2 2x+ 2)
= (2x3 4x2 + 4x) + (x2 2x+ 2)
= 2x3 3x2 + 2x+ 2
= g1
(x)
g1
(x) = f(x) q(x)
g2
(x) = x 1
r(x) =2x+ 1
x 1 x 1 2x + 1f(x) r(x)
f(x)
r(x)= (2x+ 1) 2x+ 1
x 1
= (2x+ 1) x 12x+ 1
=
2x+ 1
2x+ 1 (x 1)
= x 1
= g2
(x)
g2
(x) =f(x)
r(x)
g3
(x) =1
x 1
g3
(x) =1
x 1 r(x) =2x+ 1
x 12x+ 1 r(x) f(x) = 2x+ 1
r(x)
f(x)=
2x+ 1
x 1 (2x+ 1)
=
2x+ 1
x 1 1
2x+ 1
=
1
x 1= g
3
(x)
g3
(x) =r(x)
f(x)
-
f(x) = 2x+ 1
q(x) = x2 2x+2 (2x+1)2 2(2x+1)+ 2
f g (f g)
(f g)(x) = f(g(x)).
f(x) = 2x+ 1 q(x) = x2 2x+ 2
r(x) = 2x+ 1x 1
g(x) =px+ 1
F (x) = bxc+ 1
(g f)(x)
(g f)(x) = g(f(x))
=
pf(x) + 1
=
p(2x+ 1) + 1
=
p2x+ 2
(q f)(x) (f q)(x)
-
(q f)(x) = q(f(x))
= [f(x)]2 2 [f(x)] + 2
= (2x+ 1)2 2(2x+ 1) + 2
= (4x2 + 4x+ 1) (4x+ 2) + 2
= 4x2 + 1
(f q)(x) = f(q(x))
= 2(x2 2x+ 2) + 1
= 2x2 4x+ 5
(q f)(x) (f q)(x)
(f r)(x)
(f r)(x) = f(r(x))
= 2r(x) + 1
= 2
2x+ 1
x 1
+ 1
=
4x+ 2
x 1 + 1
=
(4x+ 2) + (x 1)x 1
=
5x+ 1
x 1
(F r)(5)
(F r)(5) = F (r(5))
= br(5)c+ 1
=
2(5) + 1
5 1
+ 1
=
11
4
+ 1 = 2 + 1 = 3
-
f g f + g f gf g f/g g/f
f(x) = x+ 2 g(x) = x2 4x2 + x 2 x2 + x+ 6 x3 + 2x2 4x 8 1
x 2 x 2
f(x) =px 1 g(x) = x2 + 4px 1 + x2 + 4
px 1 x2 4
px 1(x2 + 4)
px 1
x2 + 4x 2
f(x) =x 2x+ 2
g(x) =1
xx 2x+ 2
+
1
x
x 2x+ 2
1x
x 2x(x+ 2)
x(x 2)x+ 2
x+ 2
x(x 2)
f(x) =1
x+ 2g(x) =
x 2x
1
x+ 2+
x 2x
1
x+ 2 x 2
x
x 2x(x+ 2)
x
(x+ 2)(x 2)(x+ 2)(x 2)
x
f(x) =1
x2g(x) =
px
1
x2+
px
1
x2px
px
x21
x2px
x2px
f(x) = x2 +3x g(x) = x 2f g g f f f g g x2 x 2 x2 + 3x 2 x4 + 6x3 + 12x2 + 9x x 4
(f g) = x(g f) = x
f(x) = 3x 2 g(x) = 13
(x+ 2)
f(x) =x
2 x g(x) =2x
x 1f(x) = (x 1)3 + 2 g(x) = 3
px 2 + 1
-
p(x) = anxn+ an1x
n1+ an2x
n2+ + a
1
x+ a0
a0
, a1
, . . . , an 2 R an 6= 0 na0
, a1
, a2
, . . . , an
an anxn a0
100, 000
y
x y =100, 000
x
x
y
750 g(x)
g(x) =100, 000
x+ 750
x
y
https://www.desmos.com/calculatorhttps://www.geogebra.org/download
-
f(x) =p(x)
q(x)p(x) q(x)
q(x) q(x) 6 0f(x) x q(x) 6= 0
v v(t)
t
v t
t
v
v(t) =10
tv t
c(t) =5t
t2 + 1t c(t)
t = 1, 2, 5, 10
t
c(t)
-
a b c
x y
http://illuminations.nctm.org/Lesson.aspx?id=1968
-
x
y
y =a
x b + c
x P (x)
-
P (x) =2x2 + 800
x
b(t) =50t
t+ 10 t 20
t b(t)
t = 1, 2, 5, 10, 15, 20
-
x2 + 3x+ 2
x+ 4
1
3x2
x2 + 4x 32
px+ 1
x3 1
1
x+ 2x 2
1
(x+ 2)(x 2)
f(x) =p(x)
q(x)p(x) q(x)
q(x)
2
x 3
2x=
1
5
5
x 3 2
xf(x) =
x2 + 2x+ 3
x+ 1
y =x2 + 2x+ 3
x+ 1
x
x y
-
15px 1
5x4 6x7 + 1 5 x3
x
y = 5x3 2x+ 18
x 8 = x
2x 1px 2 = 4
x 1x+ 1
= x3
y =7x3 4
px+ 1
x2 + 3
6x 5x+ 3
0
-
x+ 1
2x= 10
x+ 1
2x 10
x2
x 3
2x=
1
5
10x 10x
-
10x
2
x
10x
3
2x
= 10x
1
5
20 15 = 2x
5 = 2x
x =5
2
xx
x+ 2 1
x 2 =8
x2 4
x
x+ 2 1
x 2 =8
(x 2)(x+ 2)
(x2)(x+2)
(x 2)(x+ 2) xx+ 2
(x 2)(x+ 2) 1x 2 = [(x 2)(x+ 2)]
8
(x 2)(x+ 2)
(x 2)x (x+ 2) = 8
x2 3x 10 = 0
x2 3x 10 = 0
(x+ 2)(x 5) = 0
x+ 2 = 0 x 5 = 0
x = 2 x = 5
x = 2 x = 5
-
x
x
12 + x 25 + x
12 + x
25 + x= 0.6
25 + x
12 + x
25 + x= 0.6
12 + x = 0.6(25 + x)
12 + x = 0.6(25) + 0.6x
x 0.6x = 15 12
0.4x = 3
x = 7.5
x
60%
v = dt
v = dt t =dv
v v + 10 5v5
v+10
4
3
5
v+
5
v + 10=
4
3
3v(v + 10)
5
v+
5
v + 10=
4
3
-
3v(v + 10) 5v+ 3v(v + 10) 5
v + 10= 3v(v + 10) 4
3
15(v + 10) + 15v = 4v(v + 10)
30v + 150 = 4v2 + 40v
4v2 + 10v 150 = 0
2v2 + 5v 75 = 0
(2v + 15)(v 5) = 0
v = 152
v = 5
v
(a, b) {x|a < x < b} a b
[a, b] {x|a x b} a b
[a, b) {x|a x < b} a b
(a, b] {x|a < x b} a b
(a,1) {x|a < x}a
[a,1) {x|a x}a
(1, b) {x|x < b}b
(1, b] {x|x b}b
(1,1)R
-
x
2x
x+ 1 1
2x
x+ 1 1 0
2x (x+ 1)x+ 1
0
x 1x+ 1
0
x = 1 x = 1x = 1
x = 1
1
-
1 1x 1x+ 1
x < 1 1 < x < 1 x > 1x = 2 x = 0 x = 2
x 1 +x+ 1 + +x 1x+ 1
+ +
x < 1 x 1
1
{x 2 R|x < 1 x 1}(1,1) [ [1,1)
3
x 2 2x = 2 x = 1
2
x = 1 x = 3
2(x+ 1) + + +x + +x 2 +2(x+ 1)
x(x 2) + +
{x 2 R|x < 1 0 < x < 2}
x
h
x h h x
8 = x2h
h x
h =8
x2
h > x
8
x2> x
-
8
x2> x
8
x2 x > 0
8 x3
x2> 0
(2 x)(x2 + 2x+ 4)x2
> 0
x = 16 x = 28 x = 0 x = 4
4
x < 0 0 < x < 2 x > 2
x = 1 x = 1 x = 32 x + + x2 + 2x+ 4 + + +
x2 + + +(2x)(x2+2x+4)
x2 + +
0 < x < 2
x < 0
x
x
-
1120
x1600
x+ 4
1600
x+ 4 1120
x 10
1600
x+ 4 1120
x 10
160
x+ 4 112
x 1
160
x+ 4 112
x 1 0
160x 112(x+ 4) (x2 + 4x)x(x+ 4)
0
160x 112x 448 x2 4xx(x+ 4)
0
x2 44x+ 448x(x+ 4)
0
(x 16)(x 28)x(x+ 4)
0
x = 16 x = 28 x = 0 x = 4
x < 4 4 < x < 0 0 < x < 16 16 < x < 28 x > 28x = 5 x = 1 x = 10 x = 20 x = 30
x 16 + +x 28 +x + + +x+ 4 + + + +(x 16)(x 28)
x(x+ 4)+ + +
-
4 < x < 0 16 < x < 28
x = 16 x = 28
3
x+ 1=
2
x 3
2x
x+ 1+
5
2x= 2 5
x2 10x 1 =
14 5xx 1 4,1
x2 4xx 2 =
14 9xx 2 7
(x+ 3)(x 2)(x+ 2)(x 1) 0 (1,3] [ [2,1)
(x+ 4)(x 3)(x 2)(x2 + 2) 0 [4, 2) [ [3,1)
x+ 1
x+ 3 2 (1,5] [ (3,1)
x 2x2 3x 10 < 0 (1,2) [ (2, 5)
x3 + x
6 + x
-
t
-
f(x) =p(x)
q(x)p(x) q(x)
q(x) q(x) 6 0f(x) x q(x) 6= 0
s = dt
s = 100t
-
x
s(x)
s(x) =100
x
s = dt
x
x s(x) s(x) =100
x
x
s(x)
s(x) =100
x
https://www.libreoffice.org/download/libreoffice-fresh/http://docs.google.com
-
x
-
f(x) =x 1x+ 1
x x 10 10
x 10 8 6 4 2f(x) 1
1
E F
-
f(x) =x 1x+ 1
E F
f(x) =x 1x+ 1
x = 1E F
x = 1
E F
x 1
-
f(x) =x2 3x 10
x
x
x = 0 f
6 x 10 x 6= 0
x 5 4 3 2 1f(x) 6 4.5 2.67 12 6
x 3 4 5 6 7 8 9 10
f(x) 3.33 1.5
-
x = 0
x 1 x 1
x = 2 x = 5x x
-
x
p
p(x) =12 + x
25 + x
p(x)
x
p(x)
25 + x
P (t) =
60(t+ 1)
t+ 6
P t bc
-
t = 5
P (5) =
60(5 + 1)
5 + 6
= b32.726c = 32
P (x)
t
P (t)
t
I
V R I = VR
R
I
f(x) =x 3x+ 4
6 x 2 xx
x 6 5 4 3 2 1f(x) 6 2.5 1.33 0.75 0.4 0.167
f x = 4
f(x) =x2 + x 6x2 + x 20
x = 4,5
6 x 2 x
-
x 6 5 4 3 2 1f(x) 0.75 0.22 0.3 0.3 0.22 0
x = 3, 2
-
x
f(x)
x
x
y x = 0
x y
x
-
f(x) =x 2x+ 2
f(x) {x 2 R | x 6= 2}
x = 2 x = 2f(x) x
x f(x) y 1
x x
x 2 x = 2 x = 2 f(x)x
y f(0) f(0) = 22
= 1
f(x) x
x
x = 2 x 2x < 2 2 x 2 x > 2 2+
x 2
x 3 2.5 2.1 2.01 2.001 2.0001 x 2
f(x) f(x)
f(x) ! +1 x ! 2 f(x)x 2
x 2+
x 1 1.5 1.9 1.99 1.999 1.9999 x 2+
f(x) 3 7 39 399 3999 39999 f(x)
-
f(x) ! 1 x ! 2+ f(x)x 2
x 2x = 2
x = a f f
x a
a
x = a
-
x
f(x) x x! +1
f(x) x! +1
x x! +1f(x) f(x) 1
f(x) x x! 1f(x) x! 1
x 5 10 100 1000 10000 x! 1f(x) f(x) 1+
x f(x)
y = 1
y = b f f(x)
b x x! +1 x! 1
y = b b
-
x 2x! +1 x! 1
x
x = 2x = 2
x < 2 2 < x < 2 x > 2x = 3 x = 0 x = 3
x 2 +x+ 2 + +x 2x+ 2
+ +
x x x
x x = 2
y
f(x) < 1 x! +1 f(x) > 1 x! 1(2, 0)
f(x) x ! 2 f(x)x! 2+
f(x)
-
y =x 2x+ 2
y = 1 f(x)
-
f(x) (1, 1) [ (1,+1)
f(x) =4x2 + 4x+ 1
x2 + 3x+ 2
x x! 1 x! +1
x
x = 1000 4x2+4x+1 4, 004, 001
4x2 4, 000, 000
x x2 + 3x + 2 x2
x f(x)4x2
x2= 4 f(x)
x y = 4
f(x) =2x2 5
3x2 + x 7
2x2 53x2 + x 7
2x2
3x2=
2
3
x y = 23
f(x) =3x+ 4
2x2 + 3x+ 1
3x+ 4
2x2 + 3x+ 13x
2x2=
3
2xx x
3
2x0 y = 0
f(x) =4x3 1
3x2 + 2x 5
x4x3 1
3x2 + 2x 54x3
3x2=
4x
3
x4x
3
x
y
-
n m
n < m y = 0
n = m y = ab ab
n > m
x
y
y
y
x = 0
xx
x
f(x) =3x2 8x 32x2 + 7x 4
x
f(x) (1,4) [ (4, 12
) [ (12
,+1)
f(x)
f(x) =3x2 8x 32x2 + 7x 4 =
(3x+ 1)(x 3)(2x 1)(x+ 4)
y f(0) = 0 0 30 + 0 4 =
3
4
x 3x+ 1 = 0) x = 13
x 3 = 0) x = 3
-
2x 1 = 0) x = 12
x+ 4 = 0) x = 4
y = 3
2
f(x)
x
4,13
, 12
3
x < 4 4 < x < 13 13 < x 3
x = 10 x = 2 x = 0 x = 1 x = 103x+ 1 + + +x 3 +2x 1 + +x+ 4 + + + +(3x+ 1)(x 3)(2x 1)(x+ 4) + + +
x x x x x
-
y
x
-
R
x y
x
-
f(x) =2
x+ 1
f(x) =2
x2 + 2x+ 1
f(x) =3x
x+ 3
f(x) =2x+ 3
4x 7
f(x) =(4x 3)(x 1)(2x+ 1)(x+ 1)
f(x) =(5x 2)(x 2)(3x 4)(x+ 2)
f(x) =x2 x+ 6x2 6x+ 8
f(x) =x2 4x 5
x 4
f(x) =x 1
x3 4x
f(x) =x2 9x2 + 4
N(t) t
N(t) =75t
t+ 5t 0.
N
N(t) t!1t ! 1 N(t) ! 75
c
t
c(t) =20t
t2 + 2t 0.
c(t) c(t) t!1t!1 c(t)! 0
x = 3 x = 3 y = 1 x 5 y5
9
f(x) =(x 5)2(x2 + 1)
(x+ 3)(x 3)(x2 + 5)
-
p =5125000V 2 449000V + 19307
125V 2(1000V 43)p V
p V
V = 0 V = 43/1000
p = 0
p
V
V = 0.043
-
TC =5
9
(TF 32)
TF =9
5
TC + 32
F
C
-
200
F 93.33C 120C 248F
f x1
x2
f f(x1
) 6= f(x2
)
y x
2
-
d
F (d) =
(8.00 0 < d 4(8.00 + 1.50 bdc) d > 4
bdc d
F (3) = 8
F (3) = F (2) = F (3.5) = 8 F
y x
https://www.world-airport-codes.com
-
y x 2
y = x24
-
y = 2x 1
x 4 3 2 1y 9 7 5 3 1
y x
x y
x 9 7 5 3 1y 4 3 2 1
x
y
-
x 4 3 2 1y 1 1 1 1 0
y
x y = 1 x = 1, 2, 3, 4
x y
x 1 1 1 1 0y 4 3 2 1
x = 1 y
x y
x y
f A B f
f1 B A f1(y) = x
f(x) = y y B
x y
-
y
x
y = f(x)
x y
y x
f(x) = 3x+ 1
y = 3x+ 1
x y x = 3y + 1
y x
x = 3y + 1
x 1 = 3yx 13
= y =) y = x 13
f(x) = 3x+ 1 f1(x) =x 13
f(f1(x)) f1(f(x))
-
f(x) f1(x)
f1(x) f(x)
f(f1(x)) = x x f1
f1(f(x)) = x x f
g(x) = x3 2
y = x3 2x y x = y3 2
y x
x = y3 2
x+ 2 = y3
3px+ 2 = y =) y = 3
px+ 2
g(x) = x3 2 g1(x) = 3px+ 2
f(x) =2x+ 1
3x 4
y =2x+ 1
3x 4x y x =
2y + 1
3y 4y x
x =2y + 1
3y 4x(3y 4) = 2y + 1
3xy 4x = 2y + 1
3xy 2y = 4x+ 1 y
-
y
y(3x 2) = 4x+ 1
y =4x+ 1
3x 2
f(x) f1(x) =4x+ 1
3x 2
f(x) = x2 + 4x 2
y = x2 + 4x 2
x y x = y2 + 4y 2y x
x = y2 + 4y 2
x+ 2 = y2 + 4y
x+ 2 + 4 = y2 + 4y + 4
x+ 6 = (y + 2)2
px+ 6 = y + 2
px+ 6 2 = y =) y =
px+ 6 2
y = px+ 6 2 x
y x = 3 y 1 5f(x) = x2 + 4x 2
f(x) = |3x|
y = |3x|
-
f1 f
f(1) = f(1) = 3 x 1 y f
y = |4x|x y x = |4y|
y x
x = |4y|
x =p(4y)2 |x| =
px2
x2 = 4y2
x2
4
= y2
r
x2
4
= y =) y = r
x2
4
x = 2 y = 1 y = 1 y = q
x24
f(x) = |3x|
k(t) = 59
(t32)+273.15t
k = 59
(t 32) + 273.15k t
t k
k =5
9
(t 32) + 273.15
k 273.15 = 59
(t 32)9
5
(k 273.15) = t 329
5
(k 273.15) + 32 = t =) t = 95
(k 273.15)
t(k) = 95
(k 273.15) k
-
f(x) =1
2
x+ 4 f1(x) = 2x 8
f(x) = (x+ 3)3 f1(x) = 3px 3
f(x) =3
x 4 f1
(x) =4x+ 3
x
f(x) =x+ 3
x 3 f1
(x) =3x+ 3
x 1
f(x) =2x+ 1
4x 1 f1
(x) =x+ 1
4x 2
f(x) = |x 1|
-
y = x
y = x
y = x
y = x
http://artforkidshub.com/5-free-symmetry-art-activity/
-
y = x
y = x
x y
f f1 f1(f(x)) = x f1(x)
y f(x) x
xf(x)! y f
1(y)! x
y = x
y = f1(x) y = f(x) = 2x + 1
{x |2 x 1.5} f(x)
-
y = 2x+ 1 y = x
{y 2 R | 3 y 4}
f1(x) =x 12
f1(x) = [3, 4]
f1(x) = [2, 1.5]
-
f(x) f1(x)
[2, 1.5] [3, 4][3, 4] [2, 1.5]
f(x) =1
x
f(x) =1
xy = x
y = x f(x) f1(x) = f(x)
-
f1(x) = f(x) =1
x
f(x) = 3px+ 1
f(x) y = x
-
f(x) = 3px+ 1 y = x3 1
f1(x) = x3 1
-
f(x) =5x 1x+ 2
f(x) = (1, 2) [ (2,1)
f(x) = (1,5) [ (5,1)
x = 2
y = 5
y = x
-
x y
y = x
x = 5y = 2
f(x) f1(x)
(1, 2) [ (2,1) (1,5) [ (5,1)(1,5) [ (5,1) (1, 2) [ (2,1)
y = x
y = x
f1
1(x) = f(x)
-
f(x) = (x+ 2)2 3 2 = 3(x+ 2)2
2
x 0
x 0
x y x =3(y + 2)2
2
, y 0y x
-
x =3(y + 2)2
2
2x
3
= (y + 2)2
r2x
3
= y + 2 y 2 r
2x
3r2x
3
2 = y =) y =r
2x
3
2
x = 54
f1(54) =
r2(54)
3
2 =r
108
3
2 =p36 2 = 6 2 = 4
t
d
t(d) =
12.5
d
3
t =
12.5
d
3
d t
d t
t =
12.5
d
3
3pt =
12.5
d
d =12.53pt
d(t) =12.53pt
t = 6.5 d(6.5) =22.53p6.5
= 12.06
12.06
-
f(x) = x2 + 1
{0, 0.5, 1, 1.5, 2, 2.5, 3}
x 0 0.5 1 1.5 2 2.5 3
f(x) 1 1.25 2 3.25 5 7.25 10f
x 1 1.25 2 3.25 5 7.25 10
f1(x) 0 0.5 1 1.5 2 2.5 3
A = {(4, 4), (3, 2), (2, 1), (0,1), (1,3), (2,5)}
A1 = {(4,4), (2,3), (1,2), (1, 0), (3, 1), (5, 2)}
f(x) = 2px 2 + 3
[2,1)
[3,1) [2,1)
f(x) =3x+ 2
x 4(1, 3) [ (3,1)
(1, 4) [ (4,1)
x = a f1(a) = 0 1/2
-
w l
w = (3.24 103)l2
l 0
l =p
w/(3.4 103)
-
n
s n = 2s
b f(x) = bx y = bx
b > 0 b 6= 1
x = 3, 2, 1, 0, 1, 2 3
y =1
3
xy = 10x y = (0.8)x
-
x 3 2 1
y =1
3
x 13
1
9
1
27
y = 10x1
1000
1
100
1
10
y = (0.8)x
f(x) = 3x f(2) f(2) f1
2
f(0.4) f()
f(2) = 32 = 9
f(2) = 32 = 13
2
=
1
9
f
1
2
= 3
1/2=
p3
f(0.4) = 30.4 = 32/5 =5p3
2
=
5p9
3.14159 3
f() = 3 33.14 33.14159
3
3
b
b
g(x) = a bxc + d
a c d
-
t t = 0
t
t = 0
t = 100 20(2)
t = 200 = 20(2)2
t = 300 = 20(2)3
t = 400 = 20(2)4
y = 20(2)t/100
y T y0
y t y = y0
(2)
t/T
t
t = 0
t = 10
t = 20
t = 30
-
y = 101
2
t/10
100, 000 6%
t
t = 0 = 100, 000
t = 1 = 100, 000(1.06) = 106, 000
t = 2 = 106, 000(1.06) = 112, 360
t = 3 = 112, 360(1.06) 119, 101.60t = 4 = 119, 101.60(1.06) 126, 247.70t = 5 = 26, 247.70(1.06) 133, 822.56
y = 100, 000(1.06)t
P r
t A = P (1 + r)t
y = 100000(1.06)t t t
t = 8 y = 100, 000(1.06)8 159, 384.81 t = 10 y = 100, 000(1.06)10 179, 084.77 200, 000
-
e 2.71828 ee
e
f(x) = ex
T t
T = 170165e0.006t t
t
T
50, 000
A = 50, 000(1.044)t t
t = 18 A = 50, 000(1.044)18 108, 537.29100, 000
A = 20, 000(1.05)t t
t = 10, A = 20, 000(1.05)10 32, 577.8932, 577.89
-
y = 1001
2
x/250x = 0
x = 500 y = 1001
2
500/250
= 100
1
2
2
= 25
y = 1, 000(3)x/80 x = 0
x = 100 y = 1, 000(3)100/80 = 3, 948.22 3, 948
10, 000
A = 10, 000(1.02)t t
A = 10, 000(1.02)12 = 12, 682.42
12, 682.42
-
4
x1= 16x y = 2x 2x 26
a bxc + d b > 0b 6= 1
f(x) = bx b > 0
b 6= 1
7
2xx2=
1
343
5
2x 5x+1 0f(x) = (1.8)x y =
(1.8)x
x
x y
f(x) = 2x3
f(x) = 2x
y = ex
2
2
(5
x+1) = 500
625 5x+8
-
a 6= 0a0 = 1
an =1
an
r s
aras = ar+s
ar
as= ars
(ar)s = ars
(ab)r = arbrab
r=
ar
br
-
49 = 7
x+1
7 = 2x+ 3
3
x= 3
2x1
5
x1= 125
8x = x2 9x2 = 3x3 + 2x 12x+ 3 > x 12
x2 > 8
x1
6= x2
bx1 6= bx2 bx1 = bx2 x1
= x2
4
x1= 16
4
x1= 16
4
x1= 4
2
x 1 = 2
x = 2 + 1
x = 3
4
x1= 16
(2
2
)
x1= 2
4
2
2(x1)= 2
4
2(x 1) = 4
2x 2 = 4
2x = 6
x = 3
-
x = 3 431 = 42 =
16
125
x1= 25
x+3
125
x1= 25
x+3
(5
3
)
x1= (5
2
)
x+3
5
3(x1)= 5
2(x+3)
3(x 1) = 2(x+ 3)
3x 3 = 2x+ 6
x = 9
9
x2= 3
x+3
(3
2
)
x2= 3
x+3
3
2x2= 3
x+3
2x2 = x+ 3
2x2 x 3 = 0
(2x 3)(x+ 1) = 0
2x 3 = 0 x+ 1 = 0
x =3
2
x = 1
-
b > 1 y = bx x
bx < by x < y
0 < b < 1 y = bx x
bx > by x < y
bm < bn
m < n m > n b
3
x < 9x2
3
x < (32)x2
3
x < 32(x2)
3
x < 32x4
3 > 1
x < 2x 4
4 < 2x x
4 < x
(4,+1] x = 5 x = 4
1
10
x+5
1
100
3x
1
100
=
1
10
2
1
10
1
10
x+5
1
100
3x
1
10
x+5
1
10
2
3x
1
10
x+5
1
10
6x
-
1
10
< 1
x+ 5 6x
5 6x x
5 5x
1 x
[1,+1) x = 1 x = 0 1
y0
1
256
t
y = y0
1
2
t/2.45y0
1
2
t/2.45=
1
256
y0
1
2
t/2.45=
1
256
1
2
t/2.45=
1
2
8
t
2.45= 8
t = 19.6
t = 0
(0.6)x3 > (0.36)x1
-
(0.6)x3 > (0.36)x1
(0.6)x3 > (0.62)x1
(0.6)x3 > (0.6)2(x1)
(0.6)x3 > (0.6)2x2
x 3 > 2x 2
3x > 1
x >1
3
(0.6)x3 > (0.36)x1
(0.6)x3 > (0.62)x1
(0.6)x3 > (0.6)2(x1)
(0.6)x3 > (0.6)2x2
x 3 < 2x 2
3x < 1
x (0.6)2x2 x 3 > 2x 2x > 1
3
x
16
2x3= 4
x+2 x = 83
1
2
2x
= 2
3x 1
2
2
1 x = 3
4
2x+7 322x329
6
,+1
2
5
5x1
254
25
4
2
5
2(1,1
5
]
x
7
x+4= 49
2x1 x = 2
4
x+2= 8
2x x = 12
3
5x+2
=
3
2
2x
27
,+1
-
f(x) = bx b > 1
f(x) = bx 0 < b < 1
f(x) = 2x
f(x)
x 4 3 2 1
f(x)1
16
1
8
1
4
1
2
-
f(x) = 2x f(x) = 2x
x
y x
y = 0
g(x) =
1
2
x
g(x)
x 3 2 1
f(x)1
2
1
4
1
8
1
16
-
g(x) =
1
2
x
g(x) =
1
2
x
x
x y = 0
b > 1 0 < b < 1 f(x) = bx
b > 1 0 < b < 1
-
f(x) = bx b > 0 b 6= 1
R
(0,+1)
y x
y = 0 x
f(x) = 2x g(x) = 3x
x 4 3 2 1f(x)
g(x)
x 4 y 1f(x) g(x)
y
f(x) =
1
2
xg(x) =
1
3
x
x 4 3 2 1f(x)
g(x)
x 4 y 1f(x) g(x)
-
y
f(x) = 5x y
5
x=
1
5
x=
1
5
x
-
x 3
x 4 y1
y = 2x
y = 2x
y = 2xy = 3 2x
y = 25
2xy = 2x + 1
y = 2x 1y = 2x+1
y = 2x1
-
x 3 2 1y = 2x
y = 2x
y = 2x 0.125 .25 .5 1 2 4 8y
1
y = 3 2xy
y = 25
2xy
2/5
y = 2x + 1
y = 2x 1 0.875 0.75 0.5
y = 2x+1
y = 2x1
y = 2x y = 2x y = 2x
y
x 3 2 1y = 2x
y = 2x 0.125 0.25 0.5 1 2 4 8y = 2x
y y = 2x yy = 2x y = 2x y = 2x
x
y = 2x x y = 2x x y = 2x
y = 2x y
-
y = f(x) x y = f(x)
y = f(x) y y = f(x)
y = 2x y = 3(2x) y = 0.4(2x)
y
x 3 2 1y = 2x
y = 3(2x)
y = 0.4(2x)
y y = 3(2x) y
y = 2x y y = 0.4(2x)
y y = 2x
-
R
y y y = 3(2x)
y y = 0.4(2x)
y = 0
(0,+1)
a > 0 y = af(x) y
y = f(x) a
a > 1 0 < a < 1 y = f(x)
y = 2x y = 2x 3 y = 2x + 1
y
x 3 2 1y = 2x
y = 2x 3 2.875 2.75 2.5 2 1y = 2x + 1
-
R
y = 2x + 1 (1,+1) y = 2x 3 (3,+1)
y y
y = 2x
y = 2x y = 0
y = 2x + 1 y = 1 y = 2x 3y = 3
d y = f(x) + d d
d > 0 d d < 0 y = f(x)
y = 2x y = 2x2 y = 2x+4
y
-
x 3 2 1y = 2x
y = 2x2
y = 2x+4
R
(0,+1)
y x = 0 y
y = 2x+4 24 = 16 y y = 2x2 22 = 0.25
y = 0
c y = f(x c) cc > 0 c c < 0
y = f(x)
-
f(x) = a bxc + d
b b > 1 0 < b < 1
|a| ax
d d > 0 d d < 0
c c > 0 c c < 0
y = bx
y
y = 3x 4
y =
1
2
x+ 2
y = 2x5
y = (0.8)x+1
y = 2
1
3
x
y = 0.25(3x)
y = 2x3 + 1
y =
1
3
x1 2
-
x
2
4
= x 43 = x 51 = x16
12= x
1
5
1
4
-
5
x= 625
3
x=
1
9
7
x= 0
10
x= 100, 000
x
x b logb x b
x log3
81 = 4 3
4
= 81
log
2
32 = 5 2
5
= 32
log
5
1 = 0 5
0
= 1
log
6
1
6
= (1) 61 = 1
6
log
2
32
log
9
729
log
5
5
log 1216
log
7
1
log
5
1p5
4
1p5
1/2
-
a b b 6= 1 ab logb a b
logb a= a logb a
b a
log
2
32 = 5 2
5
= 32
log
9
729 = 3 9
3
= 729
log
5
5 = 1 5
1
= 5
log
1/2 16 = 41
2
4= 16
log
7
1 = 0 7
0
= 1
log
5
1p5
= 12
5
1/2=
1p5
logb a = c
bc = a
b c c = logb a
logb a a log2(8)
logb x log51
125
= 3 53 = 1125
log x log10
x
e
e
ln lnx loge x
-
5
3
= 125
7
2=
1
49
10
2
= 100
2
3
2
=
4
9
(0.1)4 = 10, 000
4
0
= 1
7
b= 21
e2 = x
(2)2 = 4
log
5
125 = 3
log
7
1
49
= 2
log 100 = 2
log 23
4
9
= 2
log
0.1 10, 000 = 4log
4
1 = 0
log
7
21 = b
b = 3 73 6= 21 b7
1.5645
lnx = 2
logm = n
log
3
81 = 4
log
p5
5 = 2
log 34
64
27
= 3
log
4
2 =
1
2
log
10
0.001 = 3ln 8 = a
-
10
n= m
3
4
= 81
(
p5)
2
= 53
4
3=
64
27
4
1/2= 2
10
3= 0.001 103 =
1
1, 000ea = 8
10
31
log 10
31
= 31
R =2
3
log
E
10
4.40
E 104.40
10
12
-
E = 1012 R =2
3
log
10
12
10
4.40=
2
3
log 10
7.6
log107.6 = 7.6
10
7.6log 10
7.6= 7.6
R =2
3
(7.6) 5.1
10
12/104.40 = 107.6 39810717
http://www.phivolcs.dost.gov.ph/index.php?option=com_content&task=view&id=45&Itemid=100
-
D = 10 logI
10
12
I 2 1012 2
10
6
-
60 85
90 100
10
6 m2
D = log10
6
10
12 = 10 log 106
log 10
6
10
6
log 10
6
= 6
D = 10(6) = 60
10
6
10
12 = 106
= 100, 000
-
[H+]
pH = log[H+]
pH = log1
[H+]
10
5 log 105 log 105
10
5log 10
5= 5
log 105 = (5) = 5
-
log
3
243 5
log
6
1
216
3
log
0.25 16 2
49
x= 7 log
49
7 = x
6
3=
1
216
log
6
216 = 3
10
2
= 100 log 100 = 2
log 112
4
121
= 2
11
2
2=
4
121
ln 3 = y ey = 3
log 0.001 = 3 103 = 0.001
-
log
3
(x 2) = 5 lnx 9 y = log 12x
logx 2 = 4 lnx2 > (lnx)2 g(x) = log
3
x
x
-
x y
logx 2 = 4
log2
x = 4
log2
4 = x
logx 2 = 4 x
log2
x = 4 4
log2
4 = x x
log2 x = 4
-
f(x) = 3x
x f(x)
41
g(x) = log3
x
x
x g(x)1
81
1
3
log
2
x = 4 logx 16 = 2 log 1000 = x
log
2
x = 4
2
4
= x
x = 16
logx 16 = 2
x2 = 16
x2 16 = 0(x+ 4)(x 4) = 0x = 4,+4
x = 4
log 1000 = x
-
log 1000
3 = xx = 3
log
3
81
3
4
= 81
log
4
1 log
3
3 log
4
4
2
log
4
3 = 0 4
0
= 1
log
3
3 = 1 3
1
= 3
log
4
4
2
= 2 4
2
= 4
2
-
b x b > 0 b 6= 1
logb 1 = 0
logb bx= x
x > 0 blogb x = x
logb 1 b b?
= 1
logb bx b bx
x
logb x b x b
x
log
2
14
2
3
= 8 2
4
= 16
2
3.8074 14.000
2
log2 14= 14
logb 1 = 0 logb bx= x blogb x = x
log 10
ln e3
log
4
64
log
5
1
125
5
log5 2
log 1
log 10 = log
10
10
1
= 1
ln e3 = loge e3
= 3
log
4
64 = log
4
4
3
= 3
log
5
1
125
= log
5
5
3= 3
-
5
log5 2=2
log 1 = 0
10
2 m2
D = 10 log
I
I0
D = 10 log
10
2
10
12
D = 10 log 1010
D = 10 10D = 100
pH = log[H+]3.0 = log[H+]3.0 = log[H+]10
3.0= 10
logH+
10
3.0= [H+]
10
3.0
log
7
49
log
27
3
1
3
ln e
-
log
7
7
3 78
log
7
7
3
+ log
7
7
8
log
7
49
7
log
7
49 log7
7
log
7
7
5
5 log7
7
log
2
2
4
2
10
log
2
2
4 log2
2
10 6
log
3
(27 81) log3
27 + log
3
81
b > 0, b 6= 1 n 2 R u > 0, v > 0
logb(uv) = logb u+ logb v
logb
uv
= logb u logb v
logb(un) = n logb u
-
logb(uv) = logb u+ logb v log77
3 78= log
7
7
3
+ log
7
7
8
logb
uv
= logb u logb v
log
7
49
7
= log
7
49 log7
7
log
2
2
4
2
10
= log
2
2
4 log2
2
10
logb(un) = n logb u
log
7
7
5
= 5 log7
7
log
3
(27 81) = log3
27 + log
3
81
r = logb u s = logb v u = br v = bs
logb(uv) = logb(brbs)
) logb(uv) = logb br+s
) logb(uv) = r + s
) logb(uv) = logb u = logb v
r = logb u u = br un = brn
un = brn
) logb(un) = logb(brn)
) logb(un) = rn
) logb(un) = n logb u
logb un
(logb u)n n
u n logb u
-
log
2
(5 + 2) 6= log2
5 + log
2
2
log
2
(5 + 2) 6= (log2
5)(log
2
2)
log
2
(5 2) 6= log2
5 log2
2
log
2
(5 2) 6= log2 5log
2
2
log
2
(5
2 2) 6= 2 log2
(5 2)
log(ab2)
log
3
3
x
3
ln[x(x 5)]
log(ab2) = log a+ log b2 = log a+ 2 log b
log
3
3
x
3
= 3 log
3
3
x
= 3(log
3
3 log3
x) = 3(1 log3
x) = 3 3 log3
x
ln[x(x 5)] = lnx+ ln(x 5)
log 2 + log 3
2 lnx ln ylog
5
(x2) 3 log5
x
2 log 5
log 2 + log 3 = log(2 3) = log 6
2 lnx ln y = ln(x2) ln y = lnx2
y
log
5
(x2) 3 log5
x = log5
(x2) log5
(x3) = log5
x2
x3
= log
5
1
x
= log
5
(x1) = log5
x
2 log 5
2 = 2(1)
= 2(log 10) 1 = logb b
= log 10
2 n logb u = logb un
= log 100
2 log 5 = log 100 = log100
5
= log 20
-
log
3
729 6
log
9
729 3
log
27
729 2
log
1/27 729 2log
729
729 1
log
81
729 3/2
log
81
729
81
1/4= 3 3
6
= 729 (81
1/4)
6
= 729
81
6/4= 81
3/2= 729 log
81
729 =
3
2
log
3
n 729 =6
n
a b x a 6= 1, b 6= 1logb x =
loga x
loga b
log
3
n 729 =6
n
log
3
n 729 =log
3
729
log
3
3
n=
6
n.
log
8
32
log
243
1
27
log
25
1p5
-
log
8
32 =
log
2
32
log
2
8
=
5
3
log
243
1
27
=
log
3
1
27
log
3
243
=
35
log
25
1p5
=
log
5
1p5
log
5
25
=
1/22
= 14
log
6
4
log 122 e
log
6
4 =
log
2
4
log
2
6
=
2
log
2
6
log 122 =
ln 2
ln
1
2
=
ln 2
ln 1 ln 2 =ln 2
0 ln 2 =ln 2
ln 2 = 1
log
x3
2
3 log x log 2
ln(2e)2 (2e)2 = 4e3 2 ln 2 + 2
log
4
(16a) 2 + log4
a
log(x+ 2) + log(x 2) log(x2 4)
2 log
3
5 + 1 log
3
75
2 ln
3
2
ln 4 ln
9
16
(log
3
2)(log
3
4) = log
3
8
(log
3
2)(log
3
4) = log
3
6
-
log 2
2
= (log 2)
2
log
4
(x 4) = log4 xlog
4
4
3 log
9
x2 = 6 log9
x
3(log
9
x)2 = 6 log9
x
log
3
2x2 = log3
2 + 2 log
3
x
log
3
2x2 = 2 log3
2x
log
5
2 0.431
log
5
8, log5
1
16
, log
5
p2, log
25
2 log
25
8
1.2920,1.7227, 0.2153, 0.2153, 0.6460
log 6 0.778 log 4 0.602
log
6
4, log 24, log4
6, log
2
3
log
3
2
0.7737, 1.3802, 1.2925,0.1761, 0.1761
b x logb x
-
b
x
logb x
f(x) = logb x logb u = logb v u = v
ab = 0 a = 0 b = 0
x
log
4
(2x) = log4
10
log
3
(2x 1) = 2logx 16 = 2
log
2
(x+ 1) + log2
(x 1) = 3log x2 = 2
(log x)2 + 2 log x 3 = 0
log
4
(2x) = log4
10
2x = 10
x = 5
-
log
4
(2 5) = log4
(10)
log
3
(2x 1) = 22x 1 = 32
2x 1 = 92x = 10
x = 5
log
3
(2 (5)1) = log3
(9) = 2
logx 16 = 2
x2 = 16
x2 16 = 0(x+ 4)(x 4) = 0 a2 b2 = (a+ b)(a b)x = 4, 4
log
4
(16) = 2 4 log4(16)
log
2
(x+ 1) + log2
(x 1) = 3log
2
[(x+ 1)(x 1)] = 3 logb u+ logb v = logb(uv)(x+ 1)(x 1) = 23
x2 1 = 8x2 9 = 0(x+ 3)(x 3) = 0 a2 b2 = (a+ b)(a b)x = 3, 3
log
2
(3 + 1) log
2
(31
3 log2
(3 + 1) = log
2
(2)
x = 3
log x2 = 2
log x2 = 2
x2 = 102
x2 = 100
-
x2 100 = 0(x+ 10)(x 10) = 0x = 10, 10
log(10)
2
= 2 log(10)
2
= 2
log x2 = 2
log x2 = log 102 2 = 2(1) = 2(log 10) = log 102
x2 = 102
x2 100 = 0(x+ 10)(x 10) = 0x = 10, 10
log(10)
2
= 2 log(10)
2
= 2
logb un= n logb u
log x2 = 2
2 log x = 2
log x = 1
x = 10
log x2 = 2 log x x > 0
(log x)2 + 2 log x 3 = 0
log x = A
A2 + 2A 3 = 0(A+ 3)(A 1) = 0A = 3 A = 1
log x = 3 log x = 1x = 103 =
1
1000
x = 10
log
1
1000
log 10
x 2x = 3
2
x= 3
-
log 2
x= log 3
x log 2 = log 3 logb un= n logb u
x =log 3
log 2
1.58496
x log 12x
1
8
1
4
1
2
x log2 x
1
8
1
4
1
2
x log 12x
1
8
1
4
1
2
123
x log2
x
1
8
31
4
21
2
1
1
2
log 12x x log 1
2x
log
2
x x log2
x
-
logb x
0 < b < 1 x1
< x2
logb x1 > logb x2
b > 1 x1
< x2
logb x1 < logb x2
< > b
x
log
3
(2x 1) > log3
(x+ 2)
log
0.2 > 32 < log x < 2
log
3
(2x 1) > log3
(x+ 2)
2x 1 > 0 x+ 2 > 02x 1 > 0 x > 1
2
x+ 2 > 0 x > 2x > 1
2
x > 12
x 2
log
3
(2x 1) > log3
(x+ 2)
2x 1 > x+ 2x > 3 x
) x > 3(3,+1)
log
0.2 > 3
x > 0
-
3 15
3 = log 15
1
5
3
log 15x > log 1
5
1
5
3
0.2 =1
5
log 15x > log 1
5
1
5
3x 0
2 log 102 log 102
log 10
2 < log x < log 102
log 10
2 < log x log x < log 102
10
2 < x x < 102
1
100
< x < 100
1
100
, 100
EB ENEBEN
7.2 =2
3
log
EB10
4.46.7 =
2
3
log
EN10
4.4
-
EB 7.2
3
2
= log
EB10
4.4
10.8 = logEB10
4.4
10
10.8=
EB10
4.4
EB = 1010.8 104.4 = 1015.2
EN 6.7
3
2
= log
EN10
4.4
10.05 = logEN10
4.4
10
10.05=
EN10
4.4
EB = 1010.05 104.4 = 1014.45EBEN
=
10
15.2
10
14.45= 10
0.75 5.62
n
n+ 1
E1
E2
n n+1E
2
E1
n =2
3
log
E1
10
4.4n+ 1 =
2
3
log
E2
10
4.4
E1
3
2
n = logE
1
10
4.4
10
3n/2=
E1
10
4.4
E1
= 10
3n/2 104.4 = 103n2 +4.4
E2
3
2
(n+ 1) = logE
2
10
4.4
10
3(n+1)/2=
E2
10
4.4
E2
= 10
3(n+1)/2 104.4 = 103(n+1)
2 +4.4
E2
E1
=
10
3(n+1)2 +4.4
10
3n2 +4.4
= 10
32 31.6
-
A = P (1 + r)n A
P r n
A = 2P r = 2.5% = 0.025
A = P (1 + r)n
2P = P (1 + 0.025)n
2 = (1.025)n
log 2 = log(1.025)n
log 2 = n log(1.025)
n =log 2
log 1.025 28.07
log ln
ln 2
ln 1.025
P (x) = 20, 000, 000 e0.0251x x x = 0
P (x) = 200, 000, 000
200, 000, 000 = 20, 000, 000 e0.0251x
10 = e0.0251x
ln 10 = ln e0.0251x
ln 10 = 0.0251x(ln e)
ln 10 = 0.0251x
x =ln 10
0.0251 91
-
f(t) = Aekt
t A k
f(0) = 5, 000 f(90) = 12, 000
f(0) = Aek(0) = A = 5, 000
f(90) = 5, 000ek(90) = 12, 000) e90k = 125
ln e90k = ln12
5
) 90k = ln 125
) k 0.00973f(t) = 5, 000 e0.00973t
f(180) = 5, 000 e0.00973(180) 28, 813
y =ex + ex
2
y = 4
http://newsinfo.inquirer.net/623749/philippines-welcomes-100-millionth-babyhttp://mathworld.wolfram.com/Catenary.htmlhttp://mathworld.wolfram.com/Catenary.html
-
4 =
ex + ex
2
8 = ex + ex
8 = ex + ex
8 = ex +1
ex8ex = e2x + 1
e2x 8ex + 1 = 0u = ex u2 = e2x u2 8u+ 1 = 0
u = 4p15
u = ex 4 +p15 = ex 4
p15 = ex
ln(4 +
p15) = x ln(4
p15) = x
y = 4
x ln(4 +p15) ln(4
p15) 4.13
log
5
(x 1) + log5
(x+ 3) 1 = 0
log
3
x+ log3
(x+ 2) = 1
3
x+1= 10
1 log 3log 3
1.0959
lnx > 1 e 2.7183 (e,+1)
-
log
0.5(4x+ 1) < log0.5(1 4x) (0, 1/4)
log
2
[log
3
(log
4
x)] = 0
f(x) = bx
f(x) = bx
y = f(x) = bx
y = bx
x = by x y
y = logb x
f1(x) = logb x
-
y = log2
x
y = log2
x
x 116
1
8
1
4
1
2
y 4 3 2 1
y = log2 x y = log2 x
x > 0
x
x = 0
-
y = 2x y = log2
x
y = x
y = log 12x
y = log 12x
x 116
1
8
1
4
1
2
y 4 3 2 1 1 2 3
-
y = log 12x y = log 1
2x
x > 0
x
x = 0
y = log2
x y = log 12x
y = logb x b
y = logb
x(b > 1) y = logb
x(0 < b < 1)
y = logb x b > 1 0 < b < 1
-
{x 2 R|x > 0}x logb x
x y
x = 0 y
y = x
y = bx y = logb
x(b > 1) y = bx y = logb
x(0 < b < 1)
y = log2
x
y = log 12x
y = 2 log2
x
x
y = 2 log2
x y = 2 log2
x
y
-
x 116
1
8
1
4
1
2
log
2
x 4 3 2 1y = 2 log
2
x 8 6 4 2
{x|x 2 R, x > 0}
{y|y 2 R}
x = 0
x
y = log3
x 1
y = log3
x >
1y = log
3
x (1, 0), (3, 1) (9, 2)
(1,1), (3, 0) (9, 1)
-
{x|x 2 R, x > 0}
{y|y 2 R}
x = 0
x
x x
y = 0
0 = log
3
x 1log
3
x = 1
x = 31 = 3
y = log0.25(x+ 2)
y = log0.25 x 0 < 0.25 < 1
y = log0.25[x (2)]
2y = log
0.25 x (1, 0), (4,1), (0.25, 1)(1, 0), (2,1), (1.75, 1)
-
{x|x 2 R, x > 2}x + 2 log
0.25(x + 2) x
2{y|y 2 R}
x = 2x 1
f(x) = a logb(x c) + d
b b > 1 0 < b < 1
a ax
f(x) = a logbx d d > 0 dd < 0 c c > 0 c
c < 0
-
y = logb x y
y = logx(x+ 3)
y = log 13(x 1)
y = (log5
x) + 6
y = (log0.1 x) 2
y = log 25(x 4) + 2
y = log6
(x+ 1) + 5
log 2 0.3010 log 3 0.4771 log 5 0.6990 log 7 0.8451
2
1/3
5
1/4
2
1/3
5
1/4
n =2
1/3
5
1/4log n =
1
3
log 2 14
log 5
log n
log n 13
(0.3010) 14
(0.6990) 0.0744
0.0744 n n
log n 0.0744
n 100.0744
-
t
P r
I Is Ic
F t
-
r
t Pt
r
tt t
-
Is = Prt
Is
P
r
t
-
P = 1, 000, 000
r = 0.25% = 0.0025
t = 1
Is
Is = Prt
Is = (1, 000, 000)(0.0025)(1)
Is = 2, 500
P = 50, 000
r = 10% = 0.10
t =9
12
Is
M t =M
12
Is = Prt
Is = (50, 000)(0.10)
9
12
Is = (50, 000)(0.10)(0.75)
Is = 3, 750
-
P r t
P =Isrt
=
1, 500
(0.025)(4)P = 15, 000
r =IsPt
=
4, 860
(36, 000)(1.5)r = 0.09 = 9%
t =IsPr
=
275
(250, 000)(0.005)t = 0.22
Is = Prt = (500, 000)(0.125)(10)
Is = 625, 000
-
r = 7% = 0.07
t = 2
IS = 11, 200
P
P =Isrt
=
11, 200
(0.07)(2)
P = 80, 000
P = 500, 000
Is = 157, 500
t = 3
r
r =IsPt
=
157, 500
(500, 000)(3)
r = 0.105 = 10.5%
P
r = 5% = 0.05
Is =1
2
P = 0.5P
t
-
t =IsPr
=
0.5P
(P )(0.05)
t = 10
t r
F
F = P + Is
F
P
Is
Is Prt
F = P + Prt
F = P (1 + rt)
F = P (1 + rt)
F
P
r
t
P = 1, 000, 000, r = 0.25% = 0.0025
F
F
-
Is P F = P + Is.
F = P (1 + rt)
t = 1
Is = Prt
Is = (1, 000, 000)(0.0025)(1)
Is = 2, 500
F = P + Is
F = 1, 000, 000 + 2, 500
F = 1, 002, 500
F
F = P (1 + rt)
F = (1, 000, 000)(1 + 0.0025(1))
F = 1, 002, 500
t = 5
Is = Prt
Is = (1, 000, 000)(0.0025)(5)
Is = 12, 500
F = P + Is
F = 1, 000, 000 + 12, 500
F = 1, 012, 500
F = P (1 + rt)
F = (1, 000, 000)(1 + 0.0025(5))
F = 1, 012, 500
-
P r t I
P r t I
1
2
I F
-
I F P F r = 9.5%
I = 9, 500 P = 300, 000 t = 5 I = 16, 250 F = 1, 016, 250
1
4
-
P
r P r
-
P (1 + r) = P (1 + r) 100, 000 1.05 = 105, 000P (1 + r)(1 + r) = P (1 + r)2 105, 000, 1.05 = 110, 250P (1 + r)2(1 + r) = P (1 + r)3 110, 250 1.05 = 121, 550.63P (1 + r)3(1 + r) == P (1 + r)4 121, 550.63 1.05 = 127, 628.16
(1 + r) 1 + r
r
F = P (1 + r)t
P
F
r
t
Ic
Ic = F P
P = 10, 000
r = 2% = 0.02
t = 5
F
Ic
F = P (1 + r)t
F = (10, 000)(1 + 0.02)5
-
F = 11, 040.081
Ic = F PIc = 11, 040.81 10, 000Ic = 1, 040.81
F
P = 50, 000
r = 5% = 0.05
t = 8
F
Ic
F = P (1 + r)t
F = (50, 000)(1 + 0.05)8
F = 73, 872.77
Ic = F PIc = 73, 872.77 50, 000Ic = 23, 872.77
F
-
P = 10, 000
r = 0.5% = 0.005
t = 12
F
F
F = P (1 + r)t
F = (10, 000)(1 + 0.005)12
F = 10, 616.78
F t r
F = P (1 + r)t
P
P (1 + r)t = F
P (1 + r)t
(1 + r)t=
F
(1 + r)t
P =F
(1 + r)t
P = F (1 + r)t
P
P =F
(1 + r)t= F (1 + r)t
P
F
r
t
-
F = 50, 000
r = 10% = 0.1
t = 7
P
P
P =F
(1 + r)t
P =50, 000
(1 + 0.1)7
P = 25, 657.91
F = 200, 000
r = 1.1% = 0.011
t = 6
P
P
P =F
(1 + r)t
P =200, 000
(1 + 0.011)6
P = 187, 293.65
P r t Ic
-
P r t Ic F
Fc = 23, 820.32 Ic = 3, 820.32
Fc = 25, 250.94 Ic = 250.94
Fc = 90, 673.22 Ic = 2, 673.22
P = 89, 632.37
tt
r Ict
-
1
2
1
2
1
2
1
2
1
2
-
1
2
1
2
1
2
1
2
1
2
m i(m)
j
j =i(m)
m=
n
n = tm =
-
r
i(m) j
r, i(m), j
i(m)
i(1) = 0.02
0.02
1
= 0.02 = 2%
i(2) = 0.02
0.02
2
= 0.01 = 1%
i(3) = 0.02
0.02
4
= 0.005 = 0.5%
i(12) = 0.02
0.02
12
= 0.0016 = 0.16%
i(365) = 0.02
0.02
365
F = P (1 + j)t
j = i(m)
m t mt
-
F = P
1 +
i(m)
m
!mt
F =
P =
i(m) =
m =
t =
F = P (1 + j)t
F = P
1 +
i(m)
m
!mt
ji(m)
m
t mt
-
P = 10, 000
i(4) = 0.02
t = 5
m = 4
F
P
j =i(4)
m=
0.02
4
= 0.005
n = mt = (4)(5) = 20 .
F = P (1 + j)n
= (10, 000)(1 + 0.005)20
F = 11, 048.96
Ic = F P = 11, 048.96 10, 000 = 1, 048.96
P = 10, 000
i(12) = 0.02
t = 5
m = 12
F
P
-
j =i(12)
m=
0.02
12
= 0.0016
n = mt = (12)(5) = 60 .
F = P (1 + j)n
= (10, 000)(1 + 0.0016)60
F = 11, 050.79
Ic = F P = 11, 050.79 10, 000 = 1, 050.79
P = 50, 000
i(12) = 0.12
t = 6
m = 12
F
-
F = P
1 +
i(12)
m
!tm
F = (50, 000)
1 +
0.12
12
(6)(12)
F = (50, 000)(1.01)72
F = 102, 354.97
P =F
1 +
i(m)
m
!mt
F =
P =
i(m) =
m =
t =
F = 50, 000
t = 4
i(2) = 0.12
P
j =i(2)
m=
0.12
2
= 0.06
-
n = tm = (4)(2) = 8
P =F
(1 + j)n
P =50, 000
(1 + 0.06)8=
50, 000
(1.06)8= 31, 370.62
F = 25, 000
t = 21
2
i(4) = 0.10
P
j =i(4)
m=
0.10
4
= 0.025
n = tm = (21
2
)(4) = 10
P =F
(1 + j)n
P =25, 000
(1 + 0.025)10=
25, 000
(1.025)10= 19, 529.96
-
i(m)
F
F
-
3%
-
i(m) x =1
j=
m
i(m)m = x(i(m))
m ! 1 x = mi(m)
! 1 m
x =m
i(m)
F = P
1 +
i(m)
m
!mt
F = P
1 +
1
x
xi(m)t
F = P
1 +
1
x
xi(m)t
x
1 +
1
x
x
x ! 11 +
1
x
xe
P i(m)
F t
F = Pei(m)t
-
P = 20, 000
i(m) = 0.03
t = 6
F
F = Pei(m)t
F = Pei(m)t
= 20, 000e(0.03)(6) = 20, 000e0.18 = 23, 944.35
-
P = 3, 000
F = 3, 500
i(12) = 0.25% = 0.0025
m = 12
j =i(12)
m=
0.0025
12
t
F = P (1 + j)n
3, 500 = 3, 000
1 +
0.0025
12
n
3, 500
3, 000=
1 +
0.0025
12
n
-
n
log
3, 500
3, 000
= log
1 +
0.0025
12
n
log(1.166667) = n log
1 +
0.0025
12
n = 740.00
t = nm =740
12
= 61.67
1, 000 300 12%
F = 1, 300
m = 2
i(2) = 0.12
j =i(2)
2
=
0.12
2
= 0.06
n t
F = P (1 + j)n
1, 300 = 1, 000(1 + 0.06)n
1.3 = (1.06)n
log(1.3) = log(1.06)n
log(1.3) = n log(1.06)
n =log 1.3
log(1.06)= 4.503
300
n = 5 t = nm =5
2
= 2.5
1, 000 300
-
n
n = 5 n = 4.503 t
F = 15, 000
P = 10, 000
t = 10
m = 2
n = mt = (2)(10) = 20
i(2)
F = P (1 + j)n
15, 000 = 10, 000(1 + j)20
15, 000
10, 000= (1 + j)20
1.5 = (1 + j)20
(1.5)120
= 1 + j
(1.5)120 1 = j
j = 0.0205
j =i(m)
m
0.0205 =i(2)
2
i(2) = (0.0205)(2)
i(2) = 0.0410 4.10%
-
F = 2P
t = 10 s
m = 4
n = mt = (4)(10) = 40
i(4)
F = P (1 + j)n
2P = P (1 + j)n
2 = (1 + i)40
(2)
1/40= 1 + j
(2)
1/40 1 = j
j = 0.0175 1.75%
j =i(4)
m
0.0175 =i(4)
4
i(4) = (0.0175)(4)
i(4) = 0.070 7.00%
-
i(1)
i(4) = 0.10
m = 4
i(1)
F1
= F2
P (1 + i(1))t = P
1 +
i(4)
m
!mt
P
(1 + i(1))t =
1 +
i(4)
m
!mt
1
t
(1 + i(i)) =
1 +
0.10
4
4
i(i) =
1 +
0.10
4
4
1 = 0.103813 10.38%
-
j
j = (1.025)4 1
F1
= F2
t t = 1
P t
i(12) = 0.12 m = 12 P t
i(1) = m = 1 P t
F1
F2
F1
= F2
P
1 +
i(1)
1
!(1)t
= P
1 +
i(12)
12
!12t
1 +
i(1)
1
!=
1 +
0.12
12
12
i(1) = (1.01)12 1
i(1) = 0.126825%
i(2) = 0.08 m = 2 P t
i(4) = m = 4 P t
-
F1
F2
F1
= F2
P
1 +
i(4)
4
!(4)t
= P
1 +
i(2)
2
!(2)t
1 +
i(4)
4
!4
=
1 +
0.08
2
2
1 +
i(4)
4
!4
= (1.04)2
1 +
i(4)
4
=