2. Next Generation Math IV
Textbook
Philippine Copyright 2011 by DIWA LEARNING SYSTEMS INC
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ISBN 978-971-46-0186-4
Authors
Clarissa Ullero-Collado earned units in Master of Science in Teaching major in Mathematics from De La Salle
University and her bachelor’s degree in Physics at Philippine Normal University as a DOST scholar. As a student,
she was consistently awarded as the Mathematician of the Year. Ms. Collado previously taught various Mathematics
and Science subjects in the tertiary level. As a teacher, she has trained students for local and international contests.
She was also a recipient of the Exemplary Instructional Materials award in her school and was nominated as the
Teacher of the Year. Currently, she is the coordinator of the grade school Mathematics Department of De La Salle
Santiago Zobel School.
Marwin T. Macalanda obtained his bachelor’s degree in Applied Mathematics major in Actuarial Science from the
University of the Philippines Los Baños (UPLB). He was a scholar of the Department of Science and Technology
(DOST) and a recipient of the DOST Academic Excellence Award. Mr. Macalanda has taught various Mathematics
courses at UPLB and Asia Pacific College, and has been a software engineer in Accenture for two years. At present,
he is working as an Oracle trainer at Database Quest, an Oracle partner company.
Author, Consultant, and Reviewer
Lorelei B. Ladao-Saren obtained her master’s degree in Mathematics, with high distinction, from De La Salle
University (DLSU)–Dasmariñas and her bachelor’s degree in Statistics from University of the Philippines–Diliman.
She is presently pursuing her doctorate degree in Mathematics Education at the Philippine Normal University. Ms.
Ladao-Saren was a former director for Research, Publication, and Community Extension Services at World Citi
Colleges. She has also taught Mathematics at Asia Pacific College, Southville Foreign University, and at DLSU–
Dasmariñas. She currently teaches Mathematics at DLSU–College of St. Benilde and at the Graduate School of
Rizal Technological University.
3. 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.
4. 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
5. 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
6. 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
7. Advanced Algebra Unit I
In this unit, you will learn about functions and their real-life applications. In chapter 1,
you will find 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.
8. Chapter 1
RELATIONS AND FUNCTIONS
Learning Objectives
• Define 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
Lesson 1 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 five days. At the end of the fifth 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?
2 Next Generation Math IV
9. 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?
Walk Through
• 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 first 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
Advanced Algebra 3
10. Using a mapping diagram, the domain and range are as follows:
Domain Range
0 1
1 2
2 3
3 4
4 5
7 8
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. h
12
10
▪ ▪ The dotted lines demonstrate
▪ the use of the vertical line test. It
8
▪ is shown on the graph that the
6 ▪ lines did not pass at least two
points on the graph. Therefore,
4 ▪ the relation is a function.
2 ▪
0 ▪ t
0.1 0.2 0.3 0.4 0.5 0.6 .7 0.79
0
Next Generation Math IV
11. 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
x
b. y =
x −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 , −1, 1 , (3, 3)}
2
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
−2, x ≤ 1
c. g ( x ) = x , 1 x 5
2, x ≥ 5
Advanced Algebra
12. y y
d. 10 e. 10
5 5
x x
–10 –5 5 10 –10 –5 5 10
–5 –5
–10 –10
y
f. 10
5
x
–10 –5 5 10
–5
–10
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, ∞)
−2, x ≤ 1
c. g ( x ) = x , 1 x 5 ⇒ D: set of all real numbers
2, x ≥ 5 R: {–2, 2} (1, 5)
Next Generation Math IV
13. In this relation, the range is determined by excluding the numbers that are not
specified in the given definition.
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.
1
1 2
2 3
4
1
6. If h (x ) = , then the range of the function is [0, ∞).
x
7. The relation described by the graph below is a function.
y
10
5
x
–10 –5 5 10
–5
–10
Advanced Algebra 7
14. 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?
ìx + 1, x -2
ï
ï
ï
y = ï3, -2 £ x 0
í
ï
ï
ï(x - 1)2 , x ³ 0
ï
ï
î
a. (–∞, ∞) c. (–2, 0)
b. [0, ∞) d. [–2, 0]
3. What is the domain and range of the relation described by the graph below?
y
a. domain: (–∞, 0], range: (–∞, +∞)
10
b. domain: (–∞, 0), range: (–∞, +∞)
c. domain: (–∞, 10], range: (–10, 10)
d. domain: (–10, 10], range: (–∞, 10) 5
x
–10 –5 5 10
–5
–10
Next Generation Math IV
15. 4. Which mapping diagram does not represent a function?
1
1 b 2
a. c. 3
2 c 4
5
6
7
1
13 2
y 3
b. 14 d. 4
5
15 6
7
5. Which graph does not represent a function?
a. y c. y
x x
y y
b. d.
x x
Advanced Algebra
16. 3x + 4
6. Which ordered pair satisfies the function y = ?
5
a. 1 c. (–2, 2)
0, 5
b. (2, 2) d. (7, 20)
4x 2 − 25
7. Which is not a possible value for the domain of the function y = ?
x2 − 1
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
1 1
a. y= x +3 c. y= x +6
2 2
1 1
b. y= x −3 d. y = x −6
2 2
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
k
(in feet) is given by the equation I = 2 , where k = 4 530.
d
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?
10 Next Generation Math IV
