11
General Mathematics
Module 4
Exponential Functions
Department of Education ● Republic of the Philippines
General Mathematics – Grade 11
Alternative Delivery Mode
Quarter 1 – Module 4: Exponential Functions
First Edition, 2019
Republic Act 8293, section 176 states that: No copyright shall subsist in any
work of the Government of the Philippines. However, prior approval of the government
agency or office wherein the work is created shall be necessary for exploitation of such
work for profit. Such agency or office may, among other things, impose as a condition
the payment of royalties.
Borrowed materials (i.e., songs, poems, pictures, photos, brand names,
trademarks, etc.) included in this book are owned by their respective copyright holders.
Every effort has been exerted to locate and seek permission to use these materials
from their respective copyright owners. The publisher and authors do not represent
nor claim ownership over them.
Published by the Department of Education – Division of Cagayan de Oro Schools
Division Superintendent: Dr. Cherry Mae L. Limbaco, CESO V
Development Team of the Module
Author/s:
Macapangcat U.Mama Jr.
Reviewers:
Ray O. Maghuyop, EPS-Math
Dr. Shirley A. Merida, PSDS
Illustrators:
Management Team
Chairperson:
Cherry Mae L. Limbaco, PhD, CESO V
Schools Division Superintendent
Co-Chairpersons:
Alicia E. Anghay, PhD, CESE
Asst. Schools Division Superintendent
Lorebina C. Carrasco, OIC-CID Chief
Members
Joel D. Potane, LRMS Manager
Lanie O. Signo, Librarian II
Gemma Pajayon, PDO II
Printed in the Philippines by
Department of Education – Bureau of Learning Resources (DepEd-BLR)
Office Address:
Fr. William F. Masterson Ave Upper Balulang Cagayan de Oro
Telefax:
(08822)855-0048
E-mail Address:
cagayandeoro.city@deped.gov.ph
11
General
Mathematics
Module 4
Exponential Functions
This instructional material was collaboratively developed and reviewed
by educators from public schools. We encourage teachers and other education
stakeholders to email their feedback, comments, and recommendations to
the Department of Education at action@ cagayandeoro.city@deped.gov.ph
We value your feedback and recommendations.
Department of Education • Republic of the Philippines
This page is intentionally blank
Table of Contents
Overview ................................................................................................... 1
Module Content ......................................................................................... 1
Objectives .................................................................................................. 1
General Instructions .................................................................................. 2
Pretest ....................................................................................................... 2
Lesson 1..................................................................................................... 4
Activity 1 .......................................................................................... 4
Activity 2 .......................................................................................... 9
Lesson 2.................................................................................................. 10
Activity 3 ........................................................................................ 13
Lesson 3...................................................................................................14
Activity 4 .........................................................................................20
Enrichment Activity .................................................................................. 21
Summary/Generalizations ....................................................................... 22
Posttest ....................................................................................................23
Glossary ................................................................................................... 26
References ...............................................................................................27
Overview
Have you observed that news stories, gossips or the latest trends in social
media spread rapidly in modern society? With broadcasts televisions and radios,
millions of people hear about important events within hours! Isn’t it amazing? In many
problems, key variables are related by linear models. There are many other important
situations in which variables are related by nonlinear patterns. One example is given
in the exploration, the spread of information. Other examples include spread of
disease, change in population, temperature, bank savings, drugs in the bloodstream,
and radioactivity.
Module Content
In this module, you are expected to learn on how to represent real-life situations
using exponential functions, distinguish between exponential functions, exponential
equations, exponential inequalities, graphs exponential functions and solves problems
involving exponential functions in real-life situations. Can we do it? But before we start,
let us have a short agreement on what you are going to do in order for you to learn on
this module.
What I Need to Know
1.
2.
3.
4.
5.
6.
7.
8.
At the end of this module, you are expected to do the following:
represent real-life situations using exponential functions M11GM-Ie-3
distinguish between exponential function, exponential equation, and
exponential inequality M11GM-Ie-4
solve exponential equations and inequalities M11GM-Ie-f-1
represent an exponential function through its: (a) table of values, (b) graph,
and (c) equation M11GM-If-2
find the domain and range of an exponential function M11GM-If-3
determine the intercepts, zeroes, and asymptotes of an exponential function
M11GM-If-4
graph exponential functions M11GM-Ig-1
solve problems involving exponential functions, equations, and inequalities
M11GM-Ig-2
General Directions
To achieve the objectives of this module, can you try to follow the steps
below? Great!
 First, bring with you a graphing paper or install a graphing calculator
(Geogebra) on your cellular phone.
 Second, take ample of time reading the lessons carefully.
 Third, read and follow instructions honestly.
 Fourth, answer the pre-assessment on the first part of the module.
 Then, do all the given test and exercises thoroughly.
 And lastly, take note of the definitions of the terms that is being highlighted in
through the discussion.
What I Know
GENERAL DIRECTIONS: Read the items carefully. Write your answer on the space
provided before each item.
1. Which of the following functions represent an exponential function?
x
A. f(x) = 2x2 B. (𝑥) = 3𝑥 4
C. (𝑥) = 4𝑙𝑛𝑥
D. 𝑓(𝑥) = 3
2. Which of the following is an exponential equation?
A. f(𝑥) = 𝑥2
B. x2 + y2 = 9
C. 𝑓(𝑥) = 4𝑙𝑛𝑥
D. 128 = 3𝑥
3. It is a function of the form f(𝑥) = b𝑥 where b>0 and not equal to 1.
A. rational
B. linear
C. piece-wise
D. exponential
4. Which among the choices below represents an exponential inequality?
A. f(x) = x 2
B. x 2 + y2 ≥ 9
C. f(x) = 4lnx
5. What value of x can make the equation 5x+1 = 125 true?
A. 1
B. 2
C. 3
D. 125≤ 5
x−3
D. 4
For items 6 and 7, refer to the situation below.
The half-life of a radioactive substance is 72 hours with an initial amount of
55 grams.
6. Give an exponential function that models the situation above.
A. f(x)= (55)
C. f(x) =
B. f(x)= (55)
D. f(x) = (155)
7. How much of the substance remains after 5 days?
A. 17.3 g
B. 17.4 g
C. 17.5 g
x
8. What value of x can the expression 2 be equal to x 2 ?
A. 1
B. 2
C. 3
D. 71.3 g
D. 4
9. What happen to the exponential function as the value of x decreases without
bound?
A.
B.
C.
D.
The function increases without bound.
The function approaches to infinity.
The function approaches to the line x=0.
The function approaches to the line y=0.
x
10. What are the possible values of x in which the relation 3 <9
A. all values of x greater than 4
B. all values of x greater than 6
C. all values of x less than 4
D. all values of x less than 6
x-2
is true?
Answer key on page 25
Your Pre-Test ends here! You may now start learning more about
Representations of Functions and Relations!
3
Lesson
Exponential Functions In Action
1
Imagine that in a certain school, the school head delivers a message of class
suspension due to typhoon. His first goal is to send the message to two of his
constituents and his constituents send the message to another two, and so on. How
much time is needed so that everyone in the school know the announcement?
What’s In
Did you know that exponential function is a phenomenon that exists whenever
a quantity grows or diminishes at a rate proportional to its present value? Such
examples can be observed in businesses such as the compound interest, loans and
mortgages. It can also be used to describe population growth, radioactive decay,
growth of an epidemic and in many other fields of study.
On this lesson, you are going to explore the characteristics and kinds of
exponential expressions such as equations, functions and inequalities.
What’s New
Activity 1: Paper Folding to the Moon
Directions. Your task in this activity is to perform the given steps or procedures below.
Fill in the given table of the number of folds and its corresponding number of partitions.
Write your answers on your journal notebook and submit next session. Enjoy!
Materials: 1 whole sheet of paper, pen, worksheet
Procedures:
1. Get a sheet of paper. Fold the paper once crosswise.
2. Record the number of partitions using Table 1 as a guide.
3. Fold the paper again, this time lengthwise.
Repeat procedure 2 until the 7th fold. Refer to the Table 1.
4
Table 1 Number of Folds and Its Corresponding Parts
Number Number of
“How many times would you have to fold a piece of paper
for it to reach the Moon? How much paper do you need?
of Folds Partitions
As it is observed if you fold a paper in half, the number of
partitions is doubled. Let’s say that a 500 page ream of
bond paper is 5 cm high which means that each bond
paper is 0.01 cm and the mean distance of the moon is
384,400 km from the Earth! Converting that figure into
centimeters and number of pages, we have
3.844 x 10^12 pages away. Amazing right!?
Guide questions:
1. How many partitions will you have by folding the paper 8 times? 10 times? X
times?
2. How many folds you must perform in order to have 512 partitions?
3. What pattern can you observe in Table 1?
Answer key on page 24
Definition
An exponential expression is an expression of the form a b x−c + d, where
b > 0 and b ≠ 1.
An exponential function with base b is the function of the form f(x) = bx or
y= bx
where b > 0 or b ≠ 1.
Example: f(x) = 2x (base is 2)
f(x) = 3x (base 3)
An exponential equation is an equation involving exponential expressions.
Example: 9 = 3x
3x+1 = 27
252x−2 = 125
An exponential inequality is an inequality involving exponential
expressions.
Example: 9 ≥ 3x
3x+1 ≤ 27
Inequality Symbols:
 greater than (>)
 greater than or equal (≥)
 less than (<)
 less than or equal (≤)
5
What is it?
Example 1. Determine whether the given expression is an exponential function,
exponential equation or exponential inequality.
1. f(x) = 2x + 1
2. f(x) = 2x3
exponent)
3. 32 = 2x−2
4. y = e x
(Exponential function with a base 2)
(Not an exponential function since the variable is not in the
(Exponential equation)
(Exponential function with a base e also known as natural
exponential function)
(Exponential inequality)
5. 64 ≤ 2x+3
KINDS OF EXPONENTIAL FUNCTIONS/MODELS
Some of the commonly known real-life applications of exponential functions are the
computations on population growth, exponential decay and compound interest.
A. EXPONENTIAL GROWTH AND DECAY
The rule of exponential growth and decay can be modeled by f(x) = (ab)x where a is the
initial amount, b is the growth factor, and x is the number of intervals (minutes, hours,
days, years, and so on.
The half-life of a radioactive substance is the time it takes for half of the substance to
decay.
Example 2 (Exponential Growth). In the beginning, God created the first man and
woman on Earth. Suppose their number doubles every 2 years. Give an exponential
function that models the situation. How many people on Earth will be after 30 years?
Solution :
Initially,
At t = 0
At t = 2
At t = 4
At t = 6
At t = 8
Number of people on Earth = 1
Number of people on Earth = 21= 2
Number of people on Earth = 22= 4
Number of people on Earth = 23= 8
Number of people on Earth = 24= 16
Have you figured the pattern already? What pattern can you derive from this?
Graph of f(x)=2(x/2)
Figure 1. Amount of population (y) per period of time (x)
6
Given that the initial amount a is 2 and the growth factor is 2 since it uses the term
double (3 if triples and so on.) and x is ranging from 1,2,3,4, and so on until it reaches
to 15.
Thus, The exponential model for this situation is f(x)=2t/2 . After 30 years, the number
of people on Earth is given by (𝑥) = (2)30/2 = (2)15 = 32,768.
Example 3 (Exponential Decay). The half-life of a radioactive substance is 1200
years. If the initial amount of substance is 300 grams, give an exponential model for
the amount remaining after t years. What amount of substance remains after 2400
years?
Solution: Let t = time in years. We use the fact that the mass is halved every 1200
years. Thus we, have:
Initially,
at t = 0
Amount of Substance = 300 g
at t = 1200 years
Amount of Substance = 150 g
at t = 2400 years
Amount of Substance = 75 g
Graph of f(x)=300(1/2)
x
Figure 1. Amount of substance (y) per period of time (x)
Given: a = 300 g , b = ½ (from the definition of half-life), x =
Substituting the given to the exponential model, we have:
f(x) = (ab)x
So the exponential model for this situation is 𝑓 𝑥
3
( )
x
To find the amount of substance remains after 2400 years, let f(x) = (300) (1/2)
𝑓 𝑥
3
( )
(replace x by t/1200)
𝑓 𝑥
3
( )
(replace t by 2400)
f 𝑥
3
( )
(solve for f(x))
Thus, the amount of substance remains after 2400 years is 75 grams.
7
B. COMPOUND INTEREST
If a principal P is invested at an annual rate r, compounded annually, then the
amount after t years is given by
t
A = P(1 + r) .
EXAMPLE 3. Determine the amount of money that will be accumulated if a principal
of ₱10,000 is invested at an annual rate of 5% compounded yearly for 10 years.
Solution
Given: Principal (P ) = ₱10,000
𝑟𝑎𝑡𝑒 (𝑟) = 5% 𝑜𝑟 0.05
𝑡im𝑒 (𝑡) = 10 𝑦𝑒𝑎𝑟𝑠
Find: Amount after 10 years
To solve for A, substitute the values given in the formula. Then,
𝐴 = (1 + 𝑟) = 10,000 (1 + 0.05)10 = ₱16,288.95.
Hence, there will be ₱16,288.95 after 10 years.
C. NATURAL EXPONENTIAL FUNCTION
The natural exponential function is the function (𝑥) = 𝑒𝑥 .
EXAMPLE 4. The population of the Philippines can be approximated by the function
(𝑥) = 20000000𝑒0.0251𝑥 (0 ≤ 𝑥 ≤ 40) where x is the number of years since 1955. Use this
model to approximate the Philippines population during the years of 1955, 1965,
1975 and 2005. Round off answers to the nearest thousand.
Solution.
x=time in years
P(x)
1955 (x=0)
1965 (x=10)
1975 (x=20)
2005 (x=50)
20,000,000
25,706,000
33,040,000
70,156,000
Congratulations! You have finished the lesson. Now let me know
how much you learn! Please turn to the next page and answer the
activity.
8
What’s More
ACTIVITY 2
GENERAL DIRECTIONS: Answer as directed. Write your answers on your Activity
Notebook.
A. Determine whether the given expression is an exponential function, exponential
equation or exponential inequality. Choose your answer among the choices below. (2
points each)
A. Exponential Function
C. Exponential Equation
B. None of these
D. Exponential Inequality
1. 𝒇(𝒙) = 𝒙𝟐 + 𝟔𝒙 – 𝟗 _______________________________
6. 𝟔𝟒 = 𝟐𝒙+𝟐
_______________________________
2. 𝒇(𝒙) = 𝒆𝟐𝒙
_______________________________
7. 𝟖𝟏 = 𝟑𝟐𝒙−𝟏
_______________________________
3. 𝒇(𝒙) = 𝟒𝒍𝒏𝒙
_______________________________
8. 𝒚 ≥ 𝟑𝒙
_______________________________
4. 𝒚 = 𝟐𝒙𝟑
_______________________________
9. 𝟐𝟐𝟓 ≤ 𝟓𝟑𝒙−𝟐 _______________________________
5. 𝒚 = 𝟑𝒙
_______________________________
10. 𝒚 = 𝟑𝒙−𝟒
_______________________________
B. Solve the following problems. Show all your solutions. (5 points each)
1. A population in a barangay starts with 1,000 individuals and triples every 10
years. Give an exponential model to represents the population. What is the
population of the barangay after 20 years?
2. A bank offers a 2% annual interest rate, compounded annually for a certain
amount. Give an exponential model if ₱20,000.00 is invested under this
condition. How much money will there be in the account after 5 years?
3. The half-life of a radioactive substance is 1500 years. If the initial amount of the
substance is 500 grams, give an exponential model for the amount remaining
after t years. What amount of substance remains after 3000 years?
Answer key on page 25
9
Lesson
2
Solving Exponential Equations And
Inequalities
To start our discussion, we will be using some properties of exponential
functions to solve its equations and inequalities.
Take a look at this!
Property 1 (One-to-One Property)
If 𝒙𝟏 ≠ 𝒙𝟐, then 𝑏 𝑥 ≠ 𝑏 𝑥 . Similarly, if 𝒙𝟏 = 𝒙𝟐, then 𝑏 𝑥 = 𝑏 𝑥 .
What is it
EXAMPLE 1
A. Solve the equation 2𝑥 = 8.
Solution By looking at the equation, you must think of the value of x in such a way
that when you raise 2 by that number, the answer is 8. What do you think is the
number? That’s right! That number should be 3 since 23 is 2 x 2 x 2 which is 8.
Thus, x is equal to 3.
B. Solve the equation 4x+1 = 16.
Solution Write both sides with 4 as the base.
+1
4𝑥 = 16
(Copy the given equation)
+1
4𝑥 = 42
(Express the equation having the same base)
𝑥+1 =2
(Use of One-to-one property)
𝑥 = 2−1
(Addition Property of Equality)
𝑥=1
(Solved value of x)
Thus, the value of x is 1 to make the equation true.
C. Solve the equation 125
125
5
x-1
3(x-1)
= 25
=5
x+3
2(x+3)
3(𝑥 − 1) = 2(𝑥 + 3)
x-1
= 25
x+3
.
(Copy the given equation)
(Express the equation having the same base)
(Use of One-to-one property)
10
3𝑥 − 3 = 2𝑥 + 6
(Distributive Property)
3𝑥 − 2𝑥 = 6 + 3
(Combine all the variables in one side and constants on
the other)
𝑥=9
(Solved value of x)
Solution: Note that both 125 and 25 can be expressed with the base 5.
2
D. Solve the equation 9 𝑥 = 3𝑥+3.
Solution: Both sides of the equation can be expressed with the base 3.
2
9 𝑥 = 3𝑥+3
(Copy the given equation)
2
32(𝑥 ) = 3(𝑥+3)
(Express the equation having the same base)
2(𝑥2) = (𝑥 + 3)
(Use of One-to-one property)
2𝑥2 = 𝑥 + 3
(Distributive Property)
2𝑥2 − 𝑥 − 3 = 0
(Equate to 0.)
(2𝑥 − 3)(𝑥 + 1) = 0
(Factor the equation to get the value of x)
(2𝑥 − 3) = 0 or (𝑥 + 1) = 0
(Addition Property of Equality)*
𝑥=
3
2
or 𝑥 = −1
(Solved values of x)
*Addition Property of Equality- adding both sides of the equation with the same
number to make the equation true.
EXAMPLE 2
A. Solve the inequality 3x < 9x-2.
3x < 9x-2
(Copy the given inequality)
3𝑥 < 32(𝑥−2)
(Express the equation having the same base)
𝑥 < 2(𝑥 − 2)
(Since the base 3>1, then the direction of inequality
is retained)
𝑥 < 2𝑥 − 4
(Distributive Property)
4 < 2𝑥 − 𝑥
(Combine similar terms)
4<𝑥
(Solved value of x)
Thus, the solution set to the inequality is {𝑥 ∈ ℝ ∥ 𝑥 > 4}
Solution:
Both 3 and 9 can be written using 3 as the base.
11
Property 2 (Exponential Inequality)
If 𝑏 > 1, then the exponential function 𝑦 = 𝑏𝑥 is increasing for all x. This means that 𝑏𝑥 <
𝑏𝑦 if and only if 𝑥 < 𝑦.
If 0 < 𝑏 < 1, then the exponential function 𝑦 = 𝑏𝑥 is decreasing for all x. This means that 𝑏𝑥
> 𝑏𝑦 if and only if 𝑥 < 𝑦.
x+5
B. Solve the inequality ( )
≥(
3x
)
Solution:
2
= ( ) , then we write both sides of the inequality with the base
Since
( )x+5 ≥ (
)
( )x+5 ≥ ( )
3x
2(3x)
𝑥 + 5 ≤ 2(3𝑥)
𝑥 + 5 ≤ 6𝑥
(Copy the given inequality)
(Express the equation having the same base)
(Since the base
< 1, then the direction of inequality is reversed)
(Distributive Property)
5 ≤ 6𝑥 − 𝑥
(Combine similar terms)
5 ≤ 5𝑥
(Divide both sides by 5)
1≤𝑥
(Solved value of x)
Thus, the solution set to the inequality is {𝑥 ∈ ℝ ∥ 𝑥 ≥ 1}
EXAMPLE 3. Solving Problems Involving Exponential Equations and Inequalities
A. The half-life of Zn-71 is 2.45 minutes. At 𝑡 = 0, there were 𝑦0 grams of Zn-71,
but only 1 of this amount remains after some time. How much time has
256
passed?
Solution: Using exponential models that you have learned previously, we can
determine that after t minutes, the amount of Z n-71 is
𝑦𝑜
12
1
2
𝑡
4
Initial
amount of
substance
time of half-life
t/2.45
y0
t/2.45
t/2.45
4
=
y0
=
=
Copy from the given equation
Cancel y0 since it is common to both sides of the equation
8
Use the one-to-one property
=8
Distributive property
t = 8(2.45)
t = 19.6
Solve for t
Thus, 19.6 minutes have passed since t = 0
We solve the equation y0
t/2.45
=
y0.
What’s More
ACTIVITY 3
DIRECTIONS. Solve the following exponential equations and inequalities.
1. 169x= 13x
4.
2. ( )
5. 4
4
< 4
4
3. ( )
Answer key on page 25
13
Lesson
3
Graphing Exponential Functions
What’s In
Can you still recall on plotting of points in the Cartesian plane?
Let us start! Look at the Cartesian plane below!
QII
QI
QIII
QIV
How many parts are there in the x-y plane? Is it four? No, there are seven parts in
the coordinate plane!
Can you name them?
These are the following:
a.
b.
c.
d.
e.
f.
g.
Quadrant I (x,y)
Quadrant II (-x,y)
Quadrant III (-x,-y)
Quadrant IV (x, -y)
X-intercept (x,0) or (-x,0)
Y-intercept (0,y) or (0, -y)
Origin (0,0)
The x-coordinate is called the abscissa and the y-coordinate is the ordinate.
What do you think is the use of the Cartesian plane? (graphing)
You are right! We can illustrate the graph of a function using the x- and y-plane.
So, let us start with our topic on exponential functions.
14
One way to graph exponential functions is with the use of the table of values to
show the points. Consider the examples below.
What’s More
How do you graph an exponential function?
1.
Sketching the graph of (𝑥) = 2𝑥.
Steps:
1. Construct a table of values of ordered pairs for the given function. The table of
values for 𝑓(𝑥) = 2𝑥 is as follows:
x
-3
-2
-1
0
1
2
3
4
y
1/8
1/4
1/2
1
2
4
8
16
2. Plot the points found in the table and connect using a smooth curve.
a. Plotting of points from the table of vales in Step 1.
15
b. connecting the points through the smooth curve.
What can you observed in the graph above?
That’s right! The graph is increasing. Observe also that for all values of x, it gives a
positive y- values. And also, as the values of x decreases without bound, the function
approaches to the horizontal axis but never actually touch reach the line. Hence, the
line y=0 is called the horizontal asymptote.
Do you know what an asymptote is? Try to read again the underlined sentence above.
Did you get it now?
Asymptotes are line where the graph approaches but never touches. Well, it sounds
like your crush, right? But anyway, are you having fun so far? Let us continue!
How would you call a point that is located exactly on the y-axis?
The point that is on the y-axis is (0, 1). This point is called the y-intercept.
Sketch the graph of (𝑥) =(1/2)x Can you fill in the corresponding values of y?
Just follow the steps above.
1. Construct a table of values of ordered pairs for the given function. The table of
x
values for f(x) = 1 is as follows:
2
x
y
-3
-2
-1
16
0
1
2
3
4
2. Plot the points found in the table and connect using a smooth curve in the
Cartesian plane below.
What can be observed in the graph above? Fill in the blanks.
The graph is
. For all values of x, it gives a
As x increases without bound, the value of the function approaches to
Hence, the line y=0 is called the
.
y-values.
.
How would you call a point that is located exactly on the x-axis? Does exponential
function has this point?
The point where the graph crosses the x-axis is called the x-intercept. And
exponential function has no x-intercept. Why do you think so?
Note that, exponential functions do not have x-intercept since its graph has
horizontal asymptote. This implies that the graph does not intersect the x-axis.
17
REMARKS: In general, the graph of the function depends on the value of the base
(e.i., b>1 or 0<b<1)
0<b<1
b>1
3. Transformation of 𝒇(𝒙) = 𝒃𝒙
We’re almost there! This time let us have a comparison of our graphs to determine
its transformation. What do you know about transformation?
Sketch the graph of f(𝑥) = 2𝑥, 𝑔(𝑥) = 3𝑥 and 𝑕(𝑥) = 4𝑥 in one plane.
Observe the graph above and together, let us analyze its behavior.




All the graphs are increasing since b>1.
The y-intercept of f(𝑥) = 2𝑥 is
.
𝑥
The y-intercept of f (𝑥) = 3 is
.
The y-intercept of f (𝑥) = 4𝑥 is 1.
18



The line y=0 is the horizontal asymptote.
The functions f, g and h have no zero. This means that there is no x-values
that makes the function 0.
The base determines the steepness of the graph. Observe that in every 1 unit
change in x, f(𝑥) = 2𝑥 increases by 2 times, 𝑔(𝑥) = 3𝑥 and 𝑕(𝑥) = 4𝑥 increases by
4 times.
Definition.
Let b be a positive number not equal to 1. The transformation of an exponential
function with the base the base b is the form (𝑥) = 𝑎 𝑏𝑥 𝑐 + 𝑑. It is defined as the
process where the graph of the function changes position without changing its
shape or size.
Transformations Involving Exponential Functions
Type of
Transformation
Equation
Horizontal Translation
𝒇(𝒙) = 𝒃𝒙 + 𝒄
Vertical Translation
Reflection
Description
𝒇(𝒙) = 𝒃𝒙 + 𝒄
𝒇(𝒙) = −𝒃𝒙
𝒇(𝒙) = 𝒃−𝒙
Vertical Stretching or
Shrinking
𝒇(𝒙) = 𝒄𝒃𝒙
Shifts the graph of (𝒙) = 𝒃𝒙 c
units to the left, if c>0 and to
the right, if c<0.
Shifts the graph (𝒙) = 𝒃𝒙 c units
upward, if c>0 and downward, if
c<0.
Reflects the graph of 𝒇(𝒙) = 𝒃𝒙
about the x-axis.
Reflects the graph of 𝒇(𝒙) = 𝒃𝒙
about the y-axis.
Multiplying y-coordinates of
(𝒙) = 𝒃𝒙 by c.
Stretches the graph of 𝒇(𝒙)=𝒃𝒙
if c>1 and shrinks if c<1.
19
ACTIVITY 4
DIRECTIONS. Consider the given exponential function. Sketch the graph of
x+2
𝑕(𝑥) = 2 . Determine its domain, range, x-intercept and asymptote.
20
Enrichment Activity
Read the following situation and write a reflection based on the questions
below.
Radioactive Substances
In July 2002, National Geographic ran an article about the problems
that America faces with its ever-growing amount of nuclear waste. Currently
the United States has over 77,000 tons of waste. Environmentalists talk about how the
radioactive material will be dangerous for thousands of years because of its long halflife. In fact, it will take 240,000 years for plutonium 239 to become safe!
When scientists talk about half-life, they are referring to how long it will take for
half of a sample to decay. In the case of nuclear waste, it refers to how long it takes
for half of the radioactive material to turn into lead.
Waste Material
In the Philippines, about 35,580 tons of garbage daily and on average each
person produces 0.5 and 0.3 kg of garbage in urban and rural areas, respectively.
Imagine this rate after 10 years. Moreover as the population increases, the amount of
garbage produce also increase.
Bacteria
The most common example is the growth of bacteria colonies. Bacteria multiply
at an alarming rate. If we assume that bacteria can double every hour and if we start
with just a single bacteria, then after one day there will be over 16 million bacteria!
Your Task
In your own community, conduct a survey on the possible application of
exponential functions. Create a situation similar to the above situations and predict
what will happen after a certain time. What value can be obtained from this situation?
What can you do to improve/prevent a situation like this?
Reflect on the following.
1. What is the importance of exponential growth and decay in the life of human
beings?
2. How do you know whether the exponential function is growth or decay?
21
Let us summarize…
An exponential equation is an equation involving exponential expressions.
An exponential inequality is an inequality involving exponential expressions.
The transformation of an exponential function with the base the base b is the form
𝑓(𝑥) = 𝑎 ∙ 𝑏𝑥−𝑐 + 𝑑.
Asymptotes are line where the graph approaches but never touches.
The natural exponential function is the function (𝑥) = 𝑒𝑥.
Transformations Involving Exponential Functions
Type of
Transformation
Equation
Description
Horizontal
Translation
𝒇(𝒙) = 𝒃𝒙+𝒄
Vertical
Translation
𝒇(𝒙) = 𝒃𝒙 + 𝒄
Reflection
𝒇(𝒙) = −𝒃𝒙
Reflects the graph of 𝒇(𝒙) = 𝒃𝒙
about the x-axis.
𝒇(𝒙) = 𝒃−𝒙
Reflects the graph of 𝒇(𝒙) = 𝒃𝒙
about the y-axis.
𝒇(𝒙) = 𝒄𝒃𝒙
Multiplying y-coordinates of 𝒇(𝒙) =
𝒃𝒙 by c.
Vertical
Stretching or
Shrinking
Shifts the graph of (𝒙) = 𝒃𝒙 c units
to the left, if c>0 and to the right, if
c<0.
Shifts the graph (𝒙) = 𝒃𝒙 c units
upward, if c>0 and downward, if
c<0.
Stretches the graph of (𝒙) = 𝒃𝒙 if
c>1 and shrinks if c<1.
22
Posttest
GENERAL DIRECTIONS: Read the items carefully. Write your answer on the space
provided before each item.
1. Which of the following functions represent an exponential function?
A. (𝑥) = 4𝑥2
D. 𝑓(𝑥) = 2𝑥+1
C. 𝑓(𝑥) = 𝑙𝑛 𝑥
2. It is a function of the form (𝑥) = 𝑏𝑥 where b>0 and not equal to 1.
A. Rational
C. Piece-wise D. Exponential Function
B. Linear
3. Which among the choices below represents an exponential inequality?
A. (𝑥) = 𝑥 2
3x+3
B. 𝑥2 + 𝑦2 ≥ 9
C. 𝑓(𝑥)= 4𝑙𝑛𝑥 D. 125 ≤ 25
4. What value of x can make the equation 5𝑥+1 = 125 true?
A. 2
B. 3
C. 4
D. 5
For items 5 and 6, refer to the situation below.
The half-life of a radioactive substance is 24 hours with an initial amount of 100
grams.
5. Give an exponential function that models the situation above.
A. f(x)= (100)
B. f(x)= (100)
(24/t)
C. f(x)=
(t/24)
(t/24)
D. f(x)= (100)
(t/72)
6. How much of the substance remains after 5 days?
A. 3.125 g
B. 7 g
C. 12.5 g
D. 50 g
7. John and Peter are solving (0.6)x-3 > (0.36)-x-1. Shown below are their
solutions. Who get the correct answer?
John
Peter
A. John
(0.6)𝑥−3 > (0.62)-x-1
(0.6)𝑥−3 > (0.62)-x-1
(0.6)𝑥−3 > (0.6)2(-x-1)
(0.6)𝑥−3 > (0.6)2(-x-1)
(0.6)𝑥−3 > (0.6)−2x-2
(0.6)𝑥−3 > (0.6)−2x-2
x-3<-2x-2
x-3<-2x-2
3x>1
3x<1
x>
X<
B. Peter
C. Both John and Peter
23
D. Neither John nor Peter
8. Determine the amount of substance remaining after 12 hours in situation
number 5-6.
A. 7.05 g
B. 7.5 g
C. 70.71 g
D. 71.70 g
2x3  4 x2
9. Solve for x : 16
A.
B.
C.
D.
4
10. Which of the following best describes the graph of an exponential function
at the left?
A. The function is decreasing, define for all values of x
and as the function approaches 0, x increases without
bound.
B. The function is increasing, defined for all values of
x, and as the function approaches 0, y increases
without bound.
C. The function is increasing, defined for all x values,
attains only positive y-values and the line y=0 is the
horizontal asymptote.
D. The function is decreasing, defined for all x values
and attains x and y values.
Answer key on page 25
24
ACTIVITY 2
25
POST TEST
10.B
9. D
9. C
8. D
8. C
7. C
7. B
6. C
6. A
5. A
5. B
4. B
4. A
3. A
3. D
2. A
2. D
1. B
1. D
PART A
PRETEST
1. D
2. D
3. D
4. D
5. B
6. B
7. A
8. B
9. C
10.A
10.A
5.
ACTIVITY 3
ACTIVITY 2
2. 22,081.62
1. [-3/2, ∞)
1. 9000
2. -3
PART B
1. 0
2. -2
3. (-∞, 1/3)
ACTIVITY 1
1. 256; 1024
2. 9
3.
3. f(x)= 500(1/2)x/1500 ;
125 grams
2 raised to
the number
of folds
KEY ANSWERS
GLOSSARY OF TERMS
Exponential Decay and Growth. The exponential growth and decay can be modeled
by (𝑥) = (𝑎𝑏)𝑥 where a is the initial amount, b is the growth factor, and x is the number
of intervals. The half-life of the substance is the time it takes for half of the substance
to decay.
Exponential Equation. An exponential equation is an equation involving exponential
expressions.
Exponential Expression. An exponential expression is an expression of the form
𝑎 ∙ 𝑏𝑥 − 𝑐 + 𝑑, where 𝑏 > 0 and 𝑏 ≠ 1.
Exponential Function. An exponential function with base b is the function of the form
f(𝑥) = 𝑏𝑥 or 𝑦 = 𝑏𝑥 where 𝑏 > 0 𝑜𝑟 𝑏 ≠ 1.
Exponential Inequality. An exponential inequality is an inequality involving
exponential expressions.
Exponential Transformation. Let b be a positive number not equal to 1. The
transformation of an exponential function with the base the base b is the form f(𝑥)=𝑎∙𝑏𝑥−𝑐+𝑑.
It is defined as the process where the graph of the function changes position without changing its
shape or size.
26
References
A. Books / Manuals / Other Printed Materials
Crisologo, L., Hao, L., Miro, E., Palomo, E., Ocampo, S., and Tresvalles, R. General
Mathematics Teacher’s Guide. Department of Education- Bureau of Learning
Resources, Ground Floor Bonifacio Bldg, DepEd Complex Meralco Avenue,
Pasig City, Philippines 1600. Lexicon Press Inc. 2016. blr.lrpd@deped.gov.ph.
B. Websites
Eisegel. "Paper Folding To The Moon | Scienceblogs". 2009. Scienceblogs.Com.
https://scienceblogs.com/startswithabang/2009/08/31/paper-folding-to-themoon.
Nykamp
DQ,
“The
exponential
function.”
Insight. http://mathinsight.org/exponential_function.
From Math
https://www.youtube.com/watch?v=xZn4f1eIl3g
file:///C:/Users/Intel/Downloads/Applications_of_Exponential_Functions_Student.pdf
http://teachtogether.chedk12.com/teaching_guides/view/14
https://www.math-exercises.com/equations-and-inequalities/exponential-equations-andinequalities
https://www.math-exercises.com/equations-and-inequalities/exponential-equations-and-inequalities
https://www.youtube.com/watch?v=dV1mUjnrGY4
https://courses.lumenlearning.com/waymakercollegealgebra/chapter/characteristics-of-graphs-ofexponential-functions/
https://www.google.com/search?q=find+the+domain+and+range+of+an+exponential+function.&spell=1&
sa=X&ved=2ahUKEwjm98yvytXpAhWOUt4KHcmOBXIQBSgAegQIDRAm&biw=1366&bih=646#kpvalbx
=_SSvPXpvsENXmwQPjuLvgBg54
https://www.youtube.com/watch?v=gVGlA4jdQso
https://www.youtube.com/watch?v=CwclkRAJmm8
https://www.youtube.com/watch?v=A7tB_ycw_Z0
https://www.youtube.com/watch?v=6FeKywTfAw0
Mobile Application(s)
International Geogebra Institute. Wolfauser 90, 4040 Linz, Austria. “GeoGebra
Calculator”.
Google
Store,
Version
5.0.366.0-3D
(2017).
http://www.geogebra.org/. Accessed on October 12, 2019.
27
For inquiries and feedback, please write or call:
Department of Education – Bureau of Learning Resources (DepEd-BLR)
DepEd Division of Cagayan de Oro City
Fr. William F. Masterson Ave Upper Balulang Cagayan de Oro
Telefax:
((08822)855-0048
E-mail Address:
cagayandeoro.city@deped.gov.ph