NEXT GENERATION

MATH IV

(Textbook)

Next Generation Math IVTextbook

Philippine Copyright 2011 by DIWA LEARNING SYSTEMS INCAll rights reserved. Printed in the Philippines

Editorial, design, and layout by University Press of First Asia

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ISBN 978-971-46-0186-4

AuthorsClarissa Ullero-Collado earnedunitsinMasterofScienceinTeachingmajorinMathematicsfromDeLaSalleUniversityandherbachelor’sdegreeinPhysicsatPhilippineNormalUniversityasaDOSTscholar.Asastudent,shewasconsistentlyawardedastheMathematicianoftheYear.Ms.ColladopreviouslytaughtvariousMathematicsandSciencesubjectsinthetertiarylevel.Asateacher,shehastrainedstudentsforlocalandinternationalcontests.ShewasalsoarecipientoftheExemplaryInstructionalMaterialsawardinherschoolandwasnominatedastheTeacheroftheYear.Currently,sheisthecoordinatorofthegradeschoolMathematicsDepartmentofDeLaSalleSantiagoZobelSchool.

Marwin T. Macalandaobtainedhisbachelor’sdegreeinAppliedMathematicsmajorinActuarialSciencefromtheUniversityofthePhilippinesLosBaños(UPLB).HewasascholaroftheDepartmentofScienceandTechnology(DOST)andarecipientoftheDOSTAcademicExcellenceAward.Mr.MacalandahastaughtvariousMathematicscourses at UPLB and Asia Pacific College, and has been a software engineer in Accenture for two years. At present, heisworkingasanOracletraineratDatabaseQuest,anOraclepartnercompany.

Author, Consultant, and ReviewerLorelei B. Ladao-Saren obtainedhermaster’sdegree inMathematics,withhighdistinction, fromDeLaSalleUniversity(DLSU)–Dasmariñasandherbachelor’sdegreeinStatisticsfromUniversityofthePhilippines–Diliman.SheispresentlypursuingherdoctoratedegreeinMathematicsEducationatthePhilippineNormalUniversity.Ms.Ladao-SarenwasaformerdirectorforResearch,Publication,andCommunityExtensionServicesatWorldCitiColleges. She has also taught Mathematics at Asia Pacific College, Southville Foreign University, and at DLSU–Dasmariñas.ShecurrentlyteachesMathematicsatDLSU–CollegeofSt.Benildeandat theGraduateSchoolofRizalTechnologicalUniversity.

Preface

The Next Generation Math series covers topics and competencies that are aligned with the Basic Education Curriculum (BEC) and the Engineering and Science Education Program (ESEP) of the Department of Education. It is composed of different mathematics disciplines: elementary algebra in first year; intermediate algebra in second year; geometry in third year; and advanced algebra, trigonometry, statistics, and calculus in fourth year. It tries to cover numerous important topics that will satisfy the needs of different groups of learners.

The series supports the constructivist approach to teaching and learning process. Lessons are presented through meaningful activities which are designed to provide you an opportunity to make different connections between concrete situations and mathematics. The activities are designed to develop your skills in problem solving, critical thinking, decision making, and creative thinking through exchange of ideas and your own discovery. Each book in this series provides opportunities for you to discuss, explore, and construct mathematical ideas and interpret new information and knowledge at a different perspective. You will also be able to structure and evaluate your own conjectures and apply previously acquired knowledge and skills.

The series has the following salient features:

• Lessons are inquiry based, enriched with applicable technologies, and integrated with science and real-life applications.

• Emphasis on the development of higher-order thinking skills is evident in the illustrative examples and exercises provided in every lesson. To enhance your mathematics skills, the degree of difficulty of the problems ranges from simple to more challenging ones.

• Exercises include research work to emphasize the importance of research as a tool in satisfying the quest for knowledge and acquiring valuable insights about certain topics.

• Historical notes, application of mathematical ideas in future careers, and pieces of trivia are presented in each chapter.

It is with a sincere desire to provide a useful tool in enhancing appreciation and better understanding of mathematics that the Next Generation Math series was conceptualized.

Table of Contents

Unit I Advanced Algebra

Chapter 1 Relations and Functions

Lesson 1 Introduction to Relations and Functions ........................................................ 2

Lesson 2 Operations on Functions ............................................................................. 11

Lesson 3 Types of Functions and Their Graphs .......................................................... 15

Lesson 4 Inverse Relations and Functions .................................................................. 30

IT Matters ........................................................................................................................... 35

Chapter 2 Linear Functions

Lesson 1 Linear Equations ......................................................................................... 37

Lesson 2 Graphs of Linear Functions ......................................................................... 42

Lesson 3 Problems Involving Linear Functions ........................................................... 48

IT Matters ........................................................................................................................... 52

Chapter 3 Quadratic Functions

Lesson 1 Definitions and Graphs of Quadratic Functions ........................................... 53

Lesson 2 Zeros of Quadratic Functions ...................................................................... 67

IT Matters ........................................................................................................................... 75

Chapter 4 Polynomial Functions

Lesson 1 Polynomial Functions and Important Theorems ........................................... 77

Lesson 2 Rational Zeros of Polynomial Functions ....................................................... 83

Lesson 3 Graphs of Polynomial Functions .................................................................. 90

IT Matters ......................................................................................................................... 105

Chapter 5 Exponential and Logarithmic Functions

Lesson 1 Exponential Functions .............................................................................. 110

Lesson 2 Logarithmic Functions .............................................................................. 119

Lesson 3 Exponential and Logarithmic Equations .................................................... 126

IT Matters ......................................................................................................................... 131

Unit II Trigonometry

Chapter 6 Circular Functions

Lesson 1 Angles and Their Measurements ................................................................ 135

Lesson 2 Definitions and Graphs of Circular Functions ............................................ 144

Lesson 3 Trigonometric Identities ............................................................................. 157

IT Matters ......................................................................................................................... 164

Chapter 7 Inverse Circular Functions

Lesson 1 Definitions and Graphs of Inverse Circular Functions ................................ 166

Lesson 2 Equations Involving Circular and Inverse Circular Functions .................... 177

IT Matters ......................................................................................................................... 185

Chapter 8 Applications of Circular Functions to Triangles

Lesson 1 Solutions of Right Triangles ....................................................................... 187

Lesson 2 Solutions of Oblique Triangles ................................................................... 196

IT Matters ......................................................................................................................... 207

Unit III Statistics

Chapter 9 Introduction to Statistics

Lesson 1 Basic Terms in Statistics ........................................................................... 210

Lesson 2 Data Collection Methods............................................................................ 217

Lesson 3 Summation Notation ................................................................................. 225

IT Matters ......................................................................................................................... 233

Chapter 10 Data Presentation

Lesson 1 Frequency Distribution Table .................................................................... 235

Lesson 2 Graphs and Charts.................................................................................... 249

IT Matters ......................................................................................................................... 262

Chapter 11 Numerical Descriptive Measures

Lesson 1 Measures of Central Location .................................................................... 265

Lesson 2 Measures of Variability .............................................................................. 277

Lesson 3 Other Numerical Measures ........................................................................ 288

IT Matters ......................................................................................................................... 295

Unit IV Introduction to Calculus

Chapter 12 Limits

Lesson 1 Intuitive Notion of a Limit of a Function .................................................... 298

Lesson 2 One-sided Limits ....................................................................................... 313

IT Matters ......................................................................................................................... 329

Chapter 13 The Derivative and Differentiation

Lesson 1 The Derivative of a Function and Basic Theorems on Differentiation .......... 332

Lesson 2 Chain Rule and Higher-order Derivatives ................................................... 340

Lesson 3 Applications of the Derivative .................................................................... 348

IT Matters ......................................................................................................................... 361

Appendix ..........................................................................................................................363

Glossary ..........................................................................................................................367

Bibliography ..........................................................................................................................374

Index ..........................................................................................................................376

Advanced Algebra Unit I

In this unit, you will learn about functions and their real-life applications. In chapter 1, you will fi nd out the difference between a mere relation and a function. It will also discuss the graphs and operations on functions, and inverse relations and functions. Chapters 2 and 3 will deal with linear and quadratic functions.

In chapters 4 and 5, you will study the graphs of higher-degree polynomials and their applications. Some important theorems and the rational zeros of polynomial functions will also be discussed. Moreover, the properties of exponential and logarithmic functions and their applications will be taken up in these last chapters of unit 1.

2 Next Generation Math IV

Chapter 1

Lesson 1

RELATIONS AND FUNCTIONS

Learning Objectives• Defi ne relation and function• Differentiate a function from a relation• Find the domain and range of a function• Perform operations on functions• Solve problems involving the different operations on functions• Use the vertical line test to determine if a relation is a function• Determine the type of function given its graph• Find the inverse of a function• Use the horizontal line test to determine if the inverse of a function is also a function• Solve problems involving inverse relations and functions

Introduction to Relations and Functions

Power Up

Study the given problem and answer the questions that follow.

Joni and her younger sister conducted an experiment on the growth of mongo seeds. They spread the seeds on damp soil, exposed them to enough sunlight, and watered them regularly for fi ve days. At the end of the fi fth day, they measured the height (in centimeters) of the mongo seedlings and recorded the data in the following table.

Day Height of the Mongo Seedlings (cm)

1 3

2 4

3 5

4 6

5 7

1. Identify the dependent variable and the independent variable in the given data. a. What is the least value of the independent variable? b. What is the greatest value of the independent variable? c. What is the least value of the dependent variable? d. What is the greatest value of the dependent variable?

Advanced Algebra 3

2. Create a line graph for the given data. 3. What kind of relationship exists between the number of days and the height of the

mongo seedlings? 4. If Joni will check the height of the mongo seedlings at noon time in day 3, can you

guess the measurement she will get? 5. Based on the graph, can you tell how tall the mongo seedlings would become in day 7?

• A relation can be described as a set of ordered pairs, wherein each ordered pair consists of the abscissa (x-coordinate) and the ordinate (y-coordinate). A relation can also be shown using a table of values, arrow diagrams, graphs, and mathematical sentences.

• The relation y = f(x) means that the elements of the fi rst set constitute the domain, while the elements of the second set constitute the range of the function.

• The domain of a relation is the set of all x-values, while the range is the set of all y-values.

• A function is a special kind of relation where each element of the domain has a distinct value that corresponds with it in the range.

• The vertical line test may be used to determine if the graph of a relation is a function or not. In the vertical line test, if a vertical line is drawn on the graph such that the line will not intersect the graph in two or more points, then the relation is a function.

Example 1: Given the set of ordered pairs {(0, 1), (1, 2), (2, 3), (3, 4), (4, 5), (7, 8)}

a. Determine if the set of ordered pairs is a function or not.b. Write the domain and range of the relation using a table and a mapping diagram.

Solution:a. For every given x-value, there is a corresponding unique y-value. Therefore, the set

of ordered pairs is a function.b. Using a table, the domain and range are as follows:

Domain Range

0 1

1 2

2 3

3 4

4 5

7 8

Walk Through

� Next Generation Math IV

Using a mapping diagram, the domain and range are as follows:

Example 2: An object was dropped from a height h (in feet) at a time t (in seconds). The height of the object is given by the equation h = 10 – 16t2.

a. Complete the table of values below for this relation and determine if the relation describes a function or not.

Domain (t) 0.1 s 0.2 s 0.3 s 0.4 s 0.5 s 0.6 s 0.7 s 0.79 s

Range (h)

b. Draw a graph that will show the relation for the variables h and t. Use the vertical line test to determine whether the relation is a function or not.

c. Is 0.8 s included in the domain of the relation? Why or why not?

Solution:a.

Domain (t) 0.1 s 0.2 s 0.3 s 0.4 s 0.5 s 0.6 s 0.7 s 0.79 s

Range (h) 9.84 ft 9.36 ft 8.56 ft 7.44 ft 6 ft 4.24 ft 2.16 ft 0.0144 ft

The above table of values shows that for every time (t) there is only one corresponding height (h). Therefore, the given relation is a function.

b.

Domain Range

012347

123458

The dotted lines demonstrate the use of the vertical line test. It is shown on the graph that the lines did not pass at least two points on the graph. Therefore, the relation is a function.

10

8

6

4

2

0

h

t

12

0.10.2 0.3 0.4 0.5 0.60.70.79

▪ ▪ ▪ ▪▪

▪▪▪

Advanced Algebra �

c. 0.8 s is not included in the domain of the relation since the height h can never be negative.

Example 3: Consider the table below.

1 2 3 4 5 6 8 10

2 4 ? ? 10 ? 16 20

a. How are the numbers in the second row obtained?b. What relation is described by the entries in the table?c. Supply the missing numbers that will complete the table.d. If the numbers in the first row is x, represent the numbers in the second row in

terms of x.

Solution:a. Since 1 × 2 = 2, 2 × 2 = 4, and 5 × 2 = 10, then the numbers in the second row are

obtained by multiplying the numbers in the first row by 2.b. The table shows the relation “twice the number x” or “double the number x.”c. The numbers that will complete the table are 6, 8, and 12.d. The numbers in the second row are represented by “2x.”

Example 4: Find five ordered pairs that satisfy the relation described by each of the following equations. Then tell whether the relation is a function or not.

a. f x x( ) = + 1

b. y xx

=− 2

c. f x x( ) = − 1

Solution: The set of ordered pairs in each relation may vary.

a. {(0, 1), (1, 2), (–2, 1), (–1, 0), (2, 3)} Since no x-value is repeated, then f(x) is a function.

b. {(0, 0), (1, –1), −

2 1

2, , −

1 1

3, , (3, 3)}

Since no x-value is repeated, then y is a function.

c. {(1, 0), (5, 2), (5, –2), (10, 3), (10, –3), (2, 1), (2, –1), (17, 4), (17, –4)} Since there are x-values repeated, then f(x) is not a function.

Example 5: Find the domain and range of each relation.

a. 3 less x is equal to y

b. y = x2

c. g xx

x xx

( ) =− ≤

< <≥

2 11 5

2 5

,,,

� Next Generation Math IV

d.

f.

e.

–5

–10

–5–10 5 10

5

10

y

x

–5

–10

–5–10 5 10

5

10

y

x

–5

–10

–5–10 5 10

5

10

y

x

Solution:a. 3 less x is equal to y ⇒ 3 – x = y ⇒ D: set of all real numbers R: set of all real numbers

Note that for the domain and range, you can substitute any real number for x and hence get a real number value of y.

b. y = x2 ⇒ D: set of all real numbers R: [0, ∞)

c. g xx

x xx

( ) =− ≤

< <≥

2 11 5

2 5

,,,

⇒ D: set of all real numbers R: {–2, 2} (1, 5)

Advanced Algebra 7

In this relation, the range is determined by excluding the numbers that are not specifi ed in the given defi nition.

d. D: set of all real numbers, R: set of all real numbers

e. D: (–∞, 1], R: [0, ∞)

f. D: set of all real numbers, R: (–∞, 0)

The domain of a given graph can be determined by looking at the values of x from left to right. Similarly, the range of a given graph can be determined by looking at the values of y from bottom to top.

Move Up

I. Determine whether each statement is true or false. 1. The relation y = 2x + 3 is a function.

2. The domain of the relation f (x) = 1 − x is the set of all real numbers. 3. The range of the relation g (x) = x2 – 5 is the set of all positive real numbers. 4. If R = {(–3, –2), (–1, –2), (–1, –4), (0, 5)}, then R is a function. 5. The arrow diagram below describes a relation that is not a function.

6. If h xx( )= 1 , then the range of the function is [0, ∞).

7. The relation described by the graph below is a function.

1

2

1

2

3

4

–10

–5

–5–10 5 10

5

10

y

x

� Next Generation Math IV

8. The relation described in the table below shows that y = x2.

x 0 1 2 3

y 1 2 4 8

9. “y is the absolute value of x” is a function. 10. The table below is an example of a relation that is a function.

x 0 1 2 3

y 1 1 4 8

II. Write the letter that corresponds to the correct answer. If the correct answer is not in the given choices, write E.

1. What is the range of the relation described by y = 3x – 8 if its domain is

{–1, 0, 1}?

a. {11, 8, 5} c. {–11, –8, –5}

b. {–5, 0, 5} d. {0, 3, 5} 2. What is the domain of the relation below?

a. (–∞, ∞) c. (–2, 0) b. [0, ∞) d. [–2, 0]

3. What is the domain and range of the relation described by the graph below? a. domain: (–∞, 0], range: (–∞, +∞) b. domain: (–∞, 0), range: (–∞, +∞) c. domain: (–∞, 10], range: (–10, 10) d. domain: (–10, 10], range: (–∞, 10)

yx x

x

x x

=+ <-- £ <

-( ) ³

ì

í

ïïïïï

î

ïïïïï

1 23 2 0

1 02

,,

,

–5

–5–10 5 10

5

10

y

x

–10

Advanced Algebra �

4. Which mapping diagram does not represent a function?

a. c.

b. d.

5. Which graph does not represent a function?

a. c.

1

2

b

c

13

14

15

y

1

3

5

7

2

4

6

1

3

5

7

2

4

6

y

x

y

x

b. d.

y

y

x

x

10 Next Generation Math IV

6. Which ordered pair satisfies the function y x= +3 45

?

a. 0 15

,

c. (–2, 2) b. (2, 2) d. (7, 20)

7. Which is not a possible value for the domain of the function y xx

= −−

4 251

2

2 ? a. 1 c. 0 b. –1 d. both a and b 8. Which mathematical sentence will describe the relation given below?

x –4 –2 0 2 4

y 4 5 6 7 8

a. y x= +12

3 c. y x= +12

6

b. y x= −12

3 d. y x= −12

6

For numbers 9 and 10, consider the following problem.

The relation of the intensity of light I and the distance from the source of light d

(in feet) is given by the equation I kd

= 2 , where k = 4 530. 9. If the intensity of light is measured in dots per square inch and the source of light is

10 ft, what is the intensity of light? a. 40 dots per square inch c. 50 dots per square inch b. 45.3 dots per square inch d. 51.2 dots per square inch

10. Which approximate distance corresponds to an intensity of 503.5 dots per square inch? a. 1.2 ft c. 2 ft b. 1.5 ft d. 3 ft

III. Analyze and solve each problem carefully.

1. If there exists a relation between the number of tickets sold in a movie house and the amount of money earned, does this relation describe a function? Why or why not?

2. For every deluxe ticket sold, there are 3 premiere tickets sold. The deluxe ticket is 30 pesos cheaper than the premiere ticket. If each premiere ticket costs P150, give an expression that describes the amount of money earned.

3. The distance d a wheel travels (in feet) varies directly with the number of rotations n. If in one rotation the wheel travels 7 ft, how far can the wheel travel after 12 rotations?