Senior High School
General
Mathematics
Quarter 1- Module 6:
Lesson 1: Solving Exponential Equations and Inequality
Lesson 2: Graphing Exponential Function
Core Subject SHS- General Math (Grade 11)
Alternative Delivery Mode
Quarter 1 - Module 6: Solving Exponential Equations and Inequality, representing an
exponential function through its: (a) table of values, (b) graph, and (c) equation,
finding the domain and range of an exponential function and determining the
intercepts, zeroes and asymptotes of an exponential function
First Edition, 2020
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, stories, poems, pictures, photos, brand names,
trademarks, etc.) included in this module 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
Secretary: Leonor Magtolis Briones
Undersecretary: Diosdado M. San Antonio
Writer:
Editors:
Reviewers:
Development Team of the Module
Cherry Ann S. Dueñas
Henry D. Espina Jr.
Dr. Clavel D. Salinas
Dr. Arlene Buot (Moderator)
Cherry Ann S. Duenas
Cherry Ann S. Duenas
Illustrator:
Layout Artist:
Management Team:
Dr. Marilyn S. Andales, CESO V
Dr. Leah B. Apao
Dr. Ester A. Futalan
Dr. Cartesa M. Perico
Dr. Mary Ann P. Flores
Mr. Isaiash T. Wagas
Dr. Clavel D. Salinas
Schools Division Superintendent
Assistant Schools Division Superintendent
Assistant Schools Division Superintendent
Assistant Schools Division Superintendent
CID Chief
Education Program Supervisor - LRMDS
PSDS/ SHS Division Coordinator
Printed in the Philippines by ______________________________________
Department of Education – Region VII, Division of Cebu Province
Office Address:
Telefax:
E-mail Address:
IPHO Bldg., Sudlon, Lahug, Cebu City, Philippines
(032) 255 - 6405
cebu.province@deped.gov.ph
ii
Senior High School
General
Mathematics
Quarter 1- Module 6:
Lesson 1: Solving Exponential Equations and Inequality
Lesson 2: Graphing Exponential Function
iii
Points to Ponder!
Introductory Message
For Educators:
Learning is a constant process. Amidst inevitable circumstances, Department of Education
extends their resources and looks for varied ways to cater your needs and to adapt to the new system
of Education as a fortress of Learning Continuity Plan. One of the probable solutions is the use of
Teacher-made Educational Modules in teaching.
You are reading the General Mathematics- Grade 11: First Quarter Alternative Delivery
Mode (ADM) Module on solving exponential equations and inequalities (M11GM-Ie-f-1),
representing an exponential function through its: (a) table of values, (b) graph, and (c) equation
(M11GM-If-2), finding the domain and range of an exponential function (M11GM-If-3), and
determining the intercepts, zeroes, and asymptotes of an exponential function (M11GM-If-4) as
written and found in the K-12 Most Essential Learning Competencies.
The creation of this module is a combined effort of competent educators from different levels
and various schools of Department of Education-Cebu Province. In addition, this module is
meticulously planned, organized, checked and verified by knowledgeable educators to assist you in
imparting the lessons to the learners while considering the physical, social and economical restraints
in teaching process.
The use of Teacher-made Educational Module aims to surpass the challenges of teaching in a
new normal education set-up. Through this module, the students are given independent learning
activities, which embodies in the Most Essential Learning Competencies based from the K-12
Curriculum Competencies, to work on in accordance to their capability, efficiency and time. Thus,
helping the learners acquire the prerequisite 21st Century skills needed with emphasis on utmost effort
in considering the whole well-being of the learners.
As the main source of learning, it is your top priority to explain clearly on how to use this
module to the learners. While using this module, learner’s progress and development should be
recorded verbatim to assess their strengths and weaknesses while doing these activities at home.
Moreover, you are anticipated to persuade learners to comply and to finish the modules on or before
the scheduled time.
iv
For the Learners:
As a significant stakeholder of learning, Department of Education researched and explored on
innovative ways to address your needs with high consideration on social, economic, physical and
emotional aspects of your well-being. To continue the learning process, DepEd comes up with an
Alternative Delivery mode of teaching using Teacher-Made Educational Modules.
You are reading the General Mathematics- Grade 11: First Quarter Alternative Delivery
Mode (ADM) Module on solving exponential equations and inequalities (M11GM-Ie-f-1),
representing an exponential function through its: (a) table of values, (b) graph, and (c) equation
(M11GM-If-2), finding the domain and range of an exponential function (M11GM-If-3), and
determining the intercepts, zeroes, and asymptotes of an exponential function (M11GM-If-4) as
written and found in the K-12 Most Essential Learning Competencies.
This module is especially crafted for you to grasp the opportunity to continue learning even at
home. Using guided and independent learning activities, rest assured that you will be able to take
pleasure as well as to deeply understand the contents of the lesson presented; recognizing your own
capacity and capability in acquiring knowledge.
This module has the following parts and corresponding icons:
WHAT I NEED TO KNOW
The first part of the module will keep you on tract on
the Competencies, Objectives and Skills expected for
you to be developed and mastered.
WHAT I KNOW
This part aims to check your prior knowledge on the
lesson to take.
WHAT’S IN
WHAT’S NEW
This part helps you link the previous lesson to the
current one through a short exercise/drill.
The lesson to be partaken is introduced in this part of
the module creatively. It may be through a story, a
song, a poem, a problem opener, an activity, a situation
or the like.
v
WHAT IS IT
A brief discussion of the lesson can be read in this part.
It guides and helps you unlock the lesson presented.
WHAT’S MORE
A comprehensive activitiy/es for independent practice
is in this part to solidify your knowledge and skills of
the given topic.
This part of the module is used to process your learning
WHAT I HAVE LEARNED and understanding on the given topic.
A transfer of newly acquired knowledge and skills to a
real-life situation is present in this part of the module.
WHAT I CAN DO
This activity assesses your level of mastery towards the
topic.
ASSESSMENT
In this section, enhancement activities will be given for
ADDITIONAL ACTIVITIES you to further grasp the lessons.
This contains answers to all activities in the module.
ANSWER KEYS
At the end of this module you will also find:
References
Printed in this part is a list of all reliable and valid resources used
in crafting and designing this module.
vi
In using this module, keep note of the fundamental reminders below.
1. The module is government owned. Handle it with care. Unnecessary marks are
prohibited. Use a separate sheet of paper in answering all the given exercises.
2. This module is organized according to the level of understanding. Skipping one
part of this module may lead you to confusion and misinterpretation.
3. The instructions are carefully laden for you to understand the given lessons. Read
each item cautiously.
4. This is a Home-Based class, your reliability and honour in doing the tasks and
checking your answers are a must.
5. This module helps you attain and learn lessons at home. Make sure to clearly
comprehend the first activity before proceeding to the next one.
6. This module should be returned in good condition to your teacher/facilitator once
you completed it.
7. Answers should be written on a separate sheet of paper or notebook especially
prepared for General Mathematics subject.
If you wish to talk to your teacher/educator, do not hesitate to keep in touch with him/her for
further discussion. Know that even if this is a home-based class, your teacher is only a call away.
Good communication between the teacher and the student is our priority to flourish your
understanding on the given lessons.
We do hope that in using this material, you will gain ample knowledge and skills for you to
be fully equipped and ready to respond the demands of the globally competitive world. We are
confident in you! Keep soaring high!
vii
WHAT I NEED TO KNOW
Good day dear learner!
This module is solely prepared for you to access and to acquire lessons
befitted in your grade level. The exercises, drills and assessments are carefully made
to suit your level of understanding. Indeed, this learning resource is for you to fully
comprehend the steps on how to Solve Exponential Equations and Inequalities
(M11GM-Ie-f-1), to represent an exponential function through its: (a) table of values,
(b) graph, and (c) equation (M11GM-If-2), to find the domain and range of an
exponential function (M11GM-If-3), and to determine the intercepts, zeroes, and
asymptotes of an exponential function (M11GM-If-4). Independently, you are going
to go through this module following its proper sequence. Although you are going to
do it alone, this is a guided lesson and instructions/directions on how to do every
activity is plotted for your convenience.
PERFORMANCE STANDARD:
In this module, the learner is able to apply the concepts of exponential functions, and to
formulate and solve real-life problems with precision and accuracy.
MOST ESSENTIAL LEARNING COMPETENCY:
The learner must be able to solve exponential equations and inequalities (M11GM-Ie-f-1), to
represent an exponential function through its: (a) table of values, (b) graph, and (c) equation
(M11GM-If-2), to find the domain and range of an exponential function (M11GM-If-3), and to
determine the intercepts, zeroes, and asymptotes of an exponential function (M11GM-If-4).
LESSON AND COVERAGE:
LESSON
INTENDED LEARNING OUTCOMES
The learner must be able to:
Solving Exponential Equation
Solving Exponential Inequality
•
recall the different laws of exponent;
•
solve exponential equations; and,
•
observe accuracy
equations.
•
give the range of values that satisfy the
exponential inequality;
•
solve exponential inequality; and,
•
show perseverance in solving exponential
1
in
solving
exponential
inequalities.
Graph of Exponential Function
Lesson 1
•
represent an exponential function through its
(a) table of values, (b) graph, and (c) equation;
•
find the domain and range of an exponential
function;
•
determine the intercepts, zeroes,
asymptotes of an exponential function;
•
graph exponential functions; and,
•
display awareness on the effect of rapid
increase and decrease of exponential function.
Solving Exponential Equations and Inequalities
MOST ESSENTIAL LEARNING COMPETENCY:
The learner must be able to solve exponential equations and inequalities (M11GM-Ie-f-1)
Learning Outcome(s):
After going through this module, the learners are expected to:
•
differentiate exponential equations from exponential inequalities;
•
solve exponential equations and inequalities; and,
•
show perseverance in solving exponential equations and inequalities.
Lesson Outline:
1. Solve exponential equations
2. Solve exponential inequalities
2
and
WHAT I KNOW
Pre-Assessment:
Directions: Find out how much you already know about solving exponential equations and
inequalities. Choose the letter of the best answer. Take note of the items that you were not able to
answer correctly and find the right answer as you go through this module.
1. What is the value of x that satisfy the equation 2x =16?
a. 4
b. 8
c. 12
d. 16
2. What should be the first step to solve the value of x in the equation 8x = 26?
a. Subtract 26 to both side of the equation.
b. Equate the exponents on both sides.
c. Expand 26 and make it equal to 64.
d. Express 8 with 2 as a base.
3. What is the value of x: 62x = 614
a. x=6
b. x=7
c. x=12
4. Solve for the value of x: 51-2x = 25
a. 2
b. -2
c. x= 1/2
d. x=- 1/2
5. What is the value of x in the equation 33 = 34x+2 ?
a. x=1/4
b. x=-1/4
c. x=1/2
d. x=-1/2
6. Solve for x: 2x = 4x+1
a. x=-2
c. x=-1/2
d. x=1/2
7. Solve the exponential equation: ¼ = 162x-5
a. -11/4
b. 6
c. 9/4
d. 8
8. What is the value of x in the equation 3x-4 = 9x+28 ?
a. -52
b. -24
c. -32
d. -60
9. What is the value of x in 165x=64x+7 ?
a. 4
b. -4
c. 3
d. -3
10. Solve: 98-x=27x-3
a. x=5
b. x=2
b. x=-5
c. x=1/5
d. x=16
d. x=-1/5
11. What should be the value of x in 122x-10 to make the whole expression equal to 1?
a. 1
b. 10
c. -5
d. 5
3
12. Which of the following DOES NOT satisfy the inequality 53x-1 >25?
a. 1
b. 2
c. 3
13. What is the range of x that satisfy the inequality 22x+4 <23x+8
a. x>4
b. x<4
c. x>-4
d. 4
d. x<-4
14. What is the range that satisfies the inequality 3x+5 ≥ 37?
a. x>2
b. x<2
c. x≥2
d. 0<x<2
15. What is the range that satisfy the inequality 4x<166?
a. x>10
b. x<10
c. x>12
d. x<12
WHAT’S IN
LET US RECALL!
A. Directions: Read each statement below and then write A under the column RESPONSE if you
agree with the statement. Otherwise, write D.
RESPONSE
STATEMENT
1. Any number raised to zero is equal to one (1).
2. An expression with a negative exponent CANNOT be written as an expression
with a positive exponent.
3. 2-3 is equal to 1/8.
4. Laws of exponents may be used to simplify expressions with rational exponents.
5.
6. 304-2 = 16
7.
may be written as (32x3y5)2 where x≠0 and y≠0
8. (-16)2/3 = -16
9. The exponential expression
10. 32 • 40 + 11/2 • 50 = 11
is equivalent to
.
B. Directions: Determine whether the following is an exponential function, an exponential
equation, an exponential inequality or none of these.
1. 49x = 72
2. 3 < 9x
3. y = 81x
4.3(15x) = 45
5. 3 ≥ 9x-1
6. y = 1.25x
4
WHAT’S NEW
ACTIVITY 1
Directions: Use your knowledge in simplifying expressions to guide you to the end of the maze. Use
any coloring material to color the arrows that leads your way through the maze. Afterwards, write the
correct sequence of the letter in the boxes below.
M
A
ACTIVITY 2
Direction: Fill in the box with the correct exponent/base to make the equation true.
1. 2
=8
6. 2
=
2. 3
= 81
7. (1/2) = ¼
3. 4
= 256
8. 4
4. 5
=1
9. (1/3)
5. 7
= 49
10. 8
= 1/64
=9
=4
5
WHAT IS IT
Solving Exponential Equations
In the previous activities, you recall how to simplify expressions with exponents and to
distinguish exponential equation, inequality and function. Now you are ready to start learning about
how to solve equations involving them.
But before examples are given on how to solve exponential equations and inequalities, it is
necessary to recall the different Laws of Exponents.
Assume that a and b are nonzero real numbers, and m and n are any integers.
1. Zero Property:
b0 = 1
2. Negative Property:
or
(bm)(bn) = bm+n
3. Product Property:
4. Quotient Property:
5. Power of Power Property:
(bm)n = bmn
6. Power of Product Property:
(ab)m = ambm
7. Power of a Quotient Property:
In solving exponential equation, one uses the fact about exponential function such that:
If b x = b y , then x = y
This means that when the two sides of the equation have the same base (in this case b), then
you can equate its exponents (x and y).
Example 1: Solve the equation 4x-1 = 16
Solution. Write both sides with 4 as the base.
4x-1 = 16
4x-1 = 42
16 can be expressed as 42. Substitute 42 to 16.
x-1=2
Since both side of the equation is of the same base, you may equate the exponents.
x=2+1
Simplify
x=3
Alternate Solution. We can also write both sides with 2 as the base.
4x-1 = 16
(22)x-1 = 24
4 can be expressed as 22 while 16 can be expressed as 24.
22(x-1) = 24
2(x - 1) = 4
Equate the exponent.
6
2x - 2 = 4
Simplify and solve for the unknown.
2x = 4 + 2
2x = 6
Divide both sides by 2.
x=3
You can check that x = 3 is a solution by substituting it back to the original equation:
4x-1 = 16
43-1 = 16
42 = 16
16=16
This example shows that there may be more than one way to solve an exponential equation.
The important thing is to write both sides using the same base.
Example 2: Solve the equation 125x-1 = 25x+3
Solution. Both 125 and 25 can be written using 5 as the base.
125x-1 = 25x+3
(53)x-1 = (52)x+3
125 can be expressed as 53 while 25 can be expressed as 52.
53x-3 = 52x+6
Use Distributive Property of Multiplication to simplify the exponent.
3x-3 = 2x+6
Equate the exponents.
3x-2x = 6+3
Simplify.
x=9
Example 3: Solve the equation
Solution. Both 9 and 3 can be written using 3 as the base.
9 can be expressed as 32.
Simplify the exponent.
2
2x = x+3
2x2 – x - 3=0
(2x -3)(x + 1) = 0
2x - 3 = 0
x+1=0
2x =3
x=-1
x=3/2
by factoring
equate each factor to 0
Alternative solution: To solve 2x2-x-3=0, you may also use the quadratic formula
a is the numerical coefficient of the quadratic term, b is the numerical coefficient of the linear term
and c is the constant term.
2x2-x-3=0
a = 2 ; b = -1 ; c = -3
7
substitute the value of a, b and c in the formula
simplify
this means that the equation has two roots
take the positive root and simplify
take the negative root and simplify
Thus, the answer is still 3/2 and -1.
Solving Exponential Inequalities
Exponential inequalities can be solved using the following property.
Property of Exponential Inequalities
If b > 1, then the exponential function y=b x is increasing for all x. This means that bx < by if and
only if x < y.
If 0 < b < 1, then the exponential y=bx is decreasing for all x. This means that b x > by if and only if
x < y.
When both sides of an inequality have the same base, you may apply the key facts directly.
You may also check by substituting any values that falls on the given range if it satisfies the
inequality.
Example 4: What values of x satisfy the following inequality: 22x+3 > 23x ?
Solution. Since both sides of the inequality have the same base, then we can proceed to the next steps.
22x+3 > 23x
Note that the base is 2
2x + 3 > 3x
no changing of the direction of the inequality because the base 2 > 1
3> 3x-2x
3 > x or x < 3 or (-∞,3)
This means that all number less than 3 will satisfy the inequality.
To check, you may use any value of x less than 3 and substitute it to the inequality.
Since 0 is less than 3, then substituting it to the original inequality, must satisfy the inequality;
22x+3 > 23x
22(0)+3 > 23(0)
substituting 0 to x to check if it satisfies the inequality
23 > 2 0
simplifying
8>1
CHECKED!
8
Note to learners:
1. You should be careful when solving exponential inequalities such as bm < bn. The resulting direction of
the inequality (m < n or m > n) is based on whether the base b is greater than 1 or less than 1. This means
that when the base b is less than 1, you should change the direction of the inequality (example, if the
original inequality symbol used is greater than (>), you need to change it into less than (<).
2. When dividing an inequality by negative numbers, the direction of the symbol will also change.
Example 5: Solve the inequality
Solution. Since the base ½ is less than 1, the given inequality implies 3x ≤ 2x + 3.
note that the base is ½, which is less than 1
reverse the inequality symbol
simplify
3x ≤ 2x + 3
3x-2x ≤ 3
x≤3
In many inequalities, the bases are different but can be rewritten in terms of the same base.
Example 6: Solve the inequality 3x < 9x-2.
Solution. Since 9 can be expressed as 32, then,
3x < (32)x-2
substitute 9 with 32
3x < 32(x-2)
use distributive property of multiplication
x
2x-4
3 <3
simplify the exponent
Since the base 3 is greater than 1, then this inequality is equivalent to
x < 2x - 4
the direction of the inequality is retained
4 < 2x – x
simplify
4 < x or
x>4
Thus, the solution set is x>4 or (4+∞). (You can verify that x = 5 and 6 are solutions by substituting it
to the given, but 4 and 3 are not.)
WHAT’S MORE
INDEPENDENT ACTIVITY 1
Directions: Solve for the value of x in the given exponential equations. Write the letter that
corresponds to your answer from the box below.
a. x= -8
b. x=-3
c. x= 0
d. x=1
e. x=2
f. x=17/8
1. 2x-3 = 22x+5
2. 35x-1 = 92x
9
g. x=11/4
h. x=4/11
3.
4. 9x+4 = 272x-1
5. 162x-3 = 32
Hint: Express ½ with a base of 2
INDEPENDENT ACTIVITY 2
Directions: Solve for the range of values of x of the given exponential inequalities. Write the letter
that corresponds to your answer from the box below.
a. x > -2
b. x < -2
c. x > -13/6
d. x < -13/6
e. x > -1/5
f. x < -1/5
g. x ≥ -7/10
h. x ≤ -7/10
i. x > 7/2
j. x < 7/2
6. 42x-1 > 46
7. 53x+4 < 252x+3
8. 91-5x ≤ 273
9.
10.
Hint: Express
with a base of
WHAT I HAVE LEARNED
Directions: Answer the questions that follow.
1. In your own words, how can you differentiate an exponential equation from an exponential
inequality?
____________________________________________________________________________
____________________________________________________________________________
2. How can you solve for the unknown value in an exponential equation and inequality if it has
the same base?
____________________________________________________________________________
____________________________________________________________________________
3. How can you solve for the unknown value in an exponential equation and inequality if it
does not have the same base?
____________________________________________________________________________
____________________________________________________________________________
4. In exponential inequality, when do you reverse the direction of the inequality?
____________________________________________________________________________
____________________________________________________________________________
10
WHAT I CAN DO
ERROR ANALYSIS
Test I. Directions: Read and analyze the given situation. Defend your answer.
John and Peter are solving (0.6)x-3 > (0:36)-x-1. Did anyone get the correct solution?
If not, spot the error or errors.
Conclusion/Realization:
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
ASSESSMENT
Test I. Directions: Solve the following exponential equations. Write your solution in a
separate sheet of paper. You will be graded according to the following rubrics:
SCORE
4
3
2
1
DESCRIPTORS
Solution shows thorough and appropriate mathematical concepts and the
correct final answer is obtained.
There is a minor error in the solution but the final answer is obtained.
Several steps of the solution contain errors.
The equation is not solved at all
1. 32x = 9
2. 53x+2 = 25x-1
3. 16x+3 = 643
4. 41-x = (1/4)2x-3
11
5. (1/243)2x = 811-x
Test II. Directions: Solve the following exponential inequality. Write your solution in a
separate sheet of paper. You will be graded according to the following rubrics:
SCORE
4
DESCRIPTORS
Solution shows thorough and appropriate mathematical concepts and the
correct final answer is obtained.
There is a minor error in the solution but the final answer is obtained.
Several steps of the solution contain errors.
The equation is not solved at all
3
2
1
1. 32x-1 < 95
2. 53x-2 > 1252x+1
3. 25-x ≤ 5-x+3
4. 52x+4 < (1/25)3
5. 16-2 < (1/4)5-2x
Lesson 2
Graphing Exponential Function
MOST ESSENTIAL LEARNING COMPETENCY:
The learner must be able to represent an exponential function through its: (a) table of values,
(b) graph, and (c) equation (M11GM-If-2), to find the domain and range of an exponential function
(M11GM-If-3), and to determine the intercepts, zeroes, and asymptotes of an exponential function
(M11GM-If-4).
Learning Outcome(s):
After going through this module, the learners are expected to:
•
represent an exponential function through its (a) table of values, (b) graph, and (c)
equation;
•
find the domain and range of an exponential function;
•
determine the intercepts, zeroes, and asymptotes of an exponential function;
•
graph exponential functions; and,
•
display awareness on the effect of rapid increase and decrease of exponential
function.
12
Lesson Outline:
1. Graphing Exponential Function
2. Graphing Transformation of Exponential Functions
In the previous lesson, you learned how to solve exponential equations and inequalities. But
before you learn how to graph exponential functions, test your prior knowledge first by answering the
given questions below.
WHAT I KNOW
PRE-TEST
Directions: Find out how much you already know about graphing exponential functions. Choose the
letter of the best answer. Take note of the items that you were not able to answer correctly and find
the right answer as you go through this module.
1. Which sequence below represents an exponential function?
a. {2, 6, 10, 14, 18,…}
c. {4, 8, 24, 96,…}
b. {3, 5, 9, 16, 24,…}
d. {256, 64, 16, 4,…}
2. What is the function as shown by the table below:
x
-2
0
y
a. y= 8(3)
x/2
72
b. y=8(1/3)
8
x
2
4
8/9
c. y=24(3)
x
8/81
d. y=6(1/3)x
3. Which of the following is NOT true to all exponential function?
a. It is a one-to-one function.
b. There is no vertical asymptote.
c. The domain is the set of all real numbers.
d. The range is the set of all positive real numbers.
4. Which of the following function is the reflection of y = 2x along the x-axis?
a. y=2-x
b. y= -2x
c. y= -2-x
d. y=(1/2)x
5. Which of the following is an exponential decay function?
i. y=2-x
ii. y= -2x
iii. y = -2-x
a. i and ii
b. ii and iii
c. i and iii
13
d. all of the above
6. Find the domain of the function y= 2(3)x?
a. All negative numbers
b. All positive numbers
c. All real numbers
d. All integers
7. All of the following is true for both functions y=3x and y=3-x EXCEPT
a. They have the same range.
c. They are both increasing.
b. They have the same y-intercept.
d. They have the same horizontal asymptote.
8. Which of the following function is the vertical shift of y=3x, five units upward?
a. y=3x+5
b. y=3x-5
c. y=3x + 5
d. y=3x-5
9. What is the range of an exponential function y=bx, given that b > 0 and b≠1?
a. y>0
b. y<0
c. y≥0
d. y≤0
10. What is the equation of the horizontal asymptote of the function y=5x-3?
a. x=0
b. y=0
c. x=1
d. y=1
11. Which of the following graph belongs to y=-2x
a.
b.
c.
d.
c. 4
d. 6
12. What is the range of the given graph below?
a. (-∞, -5)
b. (-5, +∞)
c. (-∞, -5]
d. [-5, +∞)
13. What is the y-intercept of the function y=2(3)x+1?
a. 0
b. 2
14. Which of the following statement is NOT true about the function y=3x+2 ?
a. The range of the function is the set of all positive numbers.
b. Its graph intersects the y-axis at (0,9)
c. The graph is asymptotic to the y-axis.
d. There is no x-intercept.
14
15. Which of the statement below is NOT true about the exponential functions y=-2x and y=2x?
a. y=-2x is a reflection of y=2x along the x-axis.
b. y=-2x and y=2x have the same domain.
c. Their horizontal asymptote is at y=0.
d. Their y-intercept is at (0,1)
WHAT’S IN
CROSSWORD PUZZLE
Directions: Read the clues and fill in the word that fits the description of each item. The letters of your
answer must correspond to the number of boxes available.
DOWNWARD
1. Exponential ________________; a
pattern of data/graph that increases
exponentially at a consistent rate.
2. the set of permissible values of y
3. __________________ Asymptote:
the value of x on a graph in which
function approaches but does not
actually reach.
4. a line on a graph of a function
representing a value toward which the
function may approach, but does not
actually reach.
5. the point in which the graph crosses either the x or the y-axis
6. the set of permissible values of x
7. Exponential ________________; a pattern of data/graph that decreases exponentially at a
consistent rate.
8. __________________ Asymptote: the value of y on a graph in which function approaches but does
not actually reach.
9. the point where the graph of a function intersects the x-axis of the coordinate system.
10. the point where the graph of a function intersects the y-axis of the coordinate system.
15
WHAT’S NEW
How important is saving money to you? Why is it important to save
money?
It is important to save money to help protect you in the event of
financial emergency. When you have cash set aside for emergencies, it will
provide you with safety net when you need it the most.
Are you familiar with the different money challenges on social media? People are posting
pictures of their savings after a certain number of weeks. You can actually do the same!
ACTIVITY 1:
Suppose you are doing a 10-day challenge wherein you will put P1 in your piggy bank on the
first day, P2 on the second day, P4 on the third day and double the amount in the next days until the
10th day.
Fill in the table below with how much would you have to put in your piggy bank after certain
number of days. Afterwards, graph the table below and answer the guide questions that follows.
Number of
1
2
3
4
5
6
7
8
9
10
days (x)
Amount (y)
1
2
4
Construct the graph from the given table:
Question:
1. If the pattern continues, how much will you have to put in
your piggy bank on the 10th day? _____________
2. How much will your total savings be after 10 days?
_____________________
3. What have you observe with the graph?
_________________________________________________
_________________________________________________
16
WHAT IS IT
Graph of Exponential Function
In the previous activity, notice that the graph increases rapidly. This graph is an example of a
graph of exponential function.
The most basic exponential function is in the form y = bx where b is a positive number (just
like example 1 below). In this case, the graph increases rapidly at a constant rate and is called
exponential growth. However, an exponential function can also be of the form y = bx where 0 < b < 1
(example 2). In this case, the graph decreases rapidly at a constant rate. This is called the exponential
decay.
The graph of an exponential function is a necessary tool in describing its behavior and
characteristics: its intercepts, asymptotes, and zeroes. A graph can also provide insights as to real-life
situations that can be modeled by exponential functions which will be discussed in the next module.
You can graph exponential function by plotting a few points by filling up the table of values
first. To fill in the table of values, substitute the value of x to the function and get the value of the
corresponding y.
Example 1: Sketch the graph of y=2x
Solution.
Step 1. Fill in the table of values.
To fill in the table of values, you need to substitute the value of x to the given function in
order to get the corresponding value of y.
Example:
Given that x = -4
If x = 1
y=2x = 2-4 = 1/16
y=2x = 21 = 2
If x = -3,
If x = 2,
y=2x = 2-3 = 1/8
y=2x = 22 = 4
If x = -2
If x = 3
x
-2
y=2 = 2 = ¼
y=2x = 23 = 8
If x = -1
If x = 4
x
-1
y=2 = 2 = ½
y=2x = 24 = 16
If x = 0
y=2x = 20 = 1
The table of values for y is as follows:
17
Step 2. Plot the points found in the table and connect them using a smooth curve.
It can be observed that the function is defined for all values of x (then, the domain is the set
of all real numbers) and attains only positive y-values (thus, the range is the set of all positive number
or y > 0). Notice also that the graph has the x-axis as an asymptote on the left, and increases very fast
on the right. That is, the line y = 0 is a horizontal asymptote.
Example 2: Sketch the graph of y= (1/2)x
Note that y= (1/2)x can also be written as y=(2-1)x or y=2-x
Step 1. The table of values is shown below:
Step 2. Sketch the graph using the points
from the table of values.
It can be observed that the function is defined for all values of x and attains only positive yvalues. The graph has the x-axis as an asymptote on the right but decreasing rapidly on the right. That
is, the line y = 0 is a horizontal asymptote.
In general, depending on whether b > 1 or 0 < b < 1, the graph of f(x) = b x has the following
behavior:
b>1
0 < b <1
18
Properties of Exponential Functions
The following properties of f(x) = bx, where b > 1 and 0 < b < 1where b ≠ 1, can be observed
from the graph:
f(x) = bx, where b>1
f(x) = bx, where 0<b<1
Domain
set of Real Numbers R
set of Real Numbers R
Range
(0, +∞) or y>0
(0, +∞) or y>0
x-intercept
No x-intercept
No x-intercept
y-intercept
1 or at (0,1)
1 or at (0,1)
x-intercept
No x-intercept
No x-intercept
Horizontal Asymptote
x-axis
x-axis
or y=0
or y=0
Vertical Asymptote
No Vertical Asymptote
No Vertical Asymptote
Characteristic
Increasing
Decreasing
Notice that both functions, have the same Domain, Range, intercepts and asymptote. The only
difference is that if the base is more than 1, it is increasing to the right whereas if the base is between
0 and 1, it is decreasing to the right.
Let us review the different terms used to describe a graph of a function:
1. Domain-is the set of permissible values of x. The domain of ALL exponential function is the set of Real
Numbers R.
2. Range- is the set of permissible values of y.
3. y-intercept- is the point where the graph of a function intersects the y-axis of the coordinate system. To
get the y-intercept, one needs to set x= 0 and solve for the value of y.
4. x- intercept - is the point where the graph of a function intersects the x-axis of the coordinate system. To
get the x-intercept, one needs to set y= 0 and solve for the value of x
5. Horizontal Asymptote- is a y-value on a graph in which function approaches but does not actually
reach.
6. Vertical Asymptote- is a x-value on a graph in which function approaches but does not actually reach.
All exponential function has no vertical asymptotes because there is no restriction in the values of x.
Solved Examples
Example 3. Graph the functions f(x) = 3x and g(x) = 4x in the same coordinate plane. Indicate the
domain, range, y-intercept, and horizontal asymptote. Compare the two graphs.
Solution. For both these functions, the base is greater than 1. Thus, both functions are increasing. The
following table of value will help complete the sketch:
-2
-1
0
1
2
x
f(x)=3
1/9
1/3
1
3
9
f(x)=4x
1/16
¼
1
4
16
19
For both functions:
Domain: Set of all real numbers
Range: Set of all positive real numbers
(since the graph does not cross the x-axis)
x-intercept: There is no x-intercept
y-intercept: 1
Horizontal Asymptote: y = 0 (the graph
does not touch the x-axis)
Vertical Asymptote: There is no vertical
asymptote.
y-intercept
The two graphs have the same domain, range, y-intercept, and horizontal asymptote.
However, the graph of g(x) = 4x rises faster than does f(x) = 3x as x increases, and is closer to the xaxis if x < 0.
Example 4. Compare the graphs of y=0.5x, y=5x and y=2
We can observe from the graph that although changing the
base changes the shape of the graph, the other properties of the
graph (domain, range, intercepts, and asymptotes) does not
change.
Graphing Transformations of Exponential Functions
A. Reflecting Graphs
Compare the graph of y=2x, y=2-x and y=-2x below:
Comparing y=2x and y=-2x,
Solution. Substitute the value of x in the given function to get the value of y.
If x = -3
If x = -2
If x = -1
If x = 0
If x = 1
y = 2-3
y = 2-2
y = 2-1
y = 20
y = 21
y = 2x
y = 1/8 =
y = 1/4 =
y = 1/2 =
y=1
y=2
0.125
0.25
0.5
y = -2-3
y = -2-2
y = -2-1
y = -20
y = -21
y = −2x
y = -1/8 =
y = -1/4 =
y = 1/2 =
y = -1
y = -2
-0.125
-0.25
-0.5
20
If x = 2
y = 22
y=4
If x = 3
y = 23
y=8
y = -22
y = -4
y = -23
y = -8
The summary of the table of values from the solution is given below:
x
y=2
x
y = −2
x
−3
−2
−1
0
1
2
3
0.125
0.25
0.5
1
2
4
8
−0.125
−0.25
−0.5
−1
−2
−4
−8
Observe the table of values of y = 2x and y = -2x.
Notice that the y-coordinate of each point on the graph
of y = -2x is the negative y-coordinate of the graph of y = 2x. The
graphs of y = 2x and y = -2x are given at the right:
Observe also that y = -2x reflects y=2x about the x-axis.
Comparing y=2x and y=2-x,
Solution. Substitute the value of x in the given function to get the value of y.
If x = -3
If x = -2
If x = -1
If x = 0
If x = 1
x
-3
-2
-1
0
y=2
y=2
y=2
y = 21
y=2 y=2
y = 1/8 =
y = 1/4 =
y = 1/2 =
y=1
y=2
0.125
0.25
0.5
y = 2−x y = 2-(-3)
y = 2-(-2)
y = 2-(-1)
y = 2-(0)
y = 2-(1)
y = 23 = 8
y = 2 2 = 4 y = 21 = 2
y = 20 = 1
y = 2-1 = ½
= 0.5
If x = 2
y = 22
y=4
If x = 3
y = 23
y=8
y = 2-(2)
y = 2-2 = ¼
= 0.25
y = 2-(3)
y = 2-3 =
1/8 =
0.125
The summary of the table of values from the solution is given below:
x
y = 2x
−3
0.125
−2
0.25
−1
0.5
0
1
y = 2−x
8
4
2
1
1
2
2
4
3
8
0.5 0.25 0.125
Sketching the graphs of y = 2x and y = 2-x.
The value of y= 2-x at x is the same as the
value of y = 2x at –x. Thus, the graph of y = 2-x
reflects the graph of y=2x about the y-axis.
21
In general,
Reflection
The graph of y = -bx is the reflection about the x-axis of the graph of y = bx.
The graph of y = b-x is the reflection about the y-axis of the graph of y = bx.
In other words, replacing x with –x reflects the graph across the y-axis; replacing y with –y
reflects it across the x-axis.
So, when we replace x with –x (from y =2x to y =2-x), its graph reflects each other along the
y-axis. But when we replace y with –y (from y = 2x to –y=2x which is the same as y=-2x), its graph
reflects each other along the x-axis.
Observations:
Range
y-intercept
Horizontal asymptote
y = 2x
y>0
1
y=0
y = -2x
y<0
-1
y=0
y = 2-x
y>0
1
y=0
B. Stretching and Shrinking
Example 2. Use the graph of y = 2x to graph the functions y = 3(2x) and y = 0.4(2x).
Solution. Some y-values are shown on the following table.
x
−3
−2
−1
0
1
2
3
y = 2x
x
y = 3(2)
y = 0.4(2)x
0.125
0.25
0.5
1
2
4
8
0.375
0.05
0.75
0.1
1.5
0.2
3
0.4
6
0.8
12
1.6
24
3.2
The y=coordinate of each point on the graph of
y = 3(2 ) is 3 times the y-coordinate of each point on
y = 2x. Similarly, the y=coordinate of each point on the
graph of y = 0.4(2x) is 0.4 times the y-coordinate of each
point on y = 2x.
x
Observations:
y = 2x
y = 3(2x)
y =0.4(2x)
Range
y>0
y>0
y>0
y-intercept
1
3
0.4
Horizontal asymptote
y=0
y=0
y=0
*Note that the y-intercepts were also multiplied correspondingly. The y-intercept of y = 3(2x) is 3, and
the y-intercept of y = 0.4(2x) is 0.4.
22
In general,
Vertical Stretching or Shrinking
Let c be a positive constant. The graph of y = c ∙ bx can be obtained from the graph of
y = bx by multiplying each y-coordinate by c. The effect is vertical stretching (if c > 1) or
shrinking (if c < 1) of the graph of y = f(x).
C. Vertical and horizontal shifts
Example 3. Use the graph of y = 2x to graph y = 2x – 3 and y = 2x +1
Solution. Some y-values are shown on the following table.
−3
−2
−1
0
1
2
3
y=2
y = 2x-3
0.125
-2.875
0.25
-2.75
0.5
-2.5
1
-2
2
-1
4
1
8
5
y = 2x+1
1.125
1.25
1.5
2
3
5
9
x
x
The graphs of these functions are shown below:
Observe that the graph of y = 2x + 1, is just shifting the
graph of y = 2x, one unit upward. Whereas, the graph
of y = 2x – 3, shifted the graph of y = 2x, three units
downward.
Observations:
Range
y-intercept
Horizontal asymptote
y = 2x
y>0
1
y=0
y = 2x + 1
y>1
2
y=1
y = 2x - 3
y > -3
-2
y = -3
In general,
Vertical Shifts
Let k be a real number. The graph of y = bx + k is a vertical shift of k units up (if k > 0) or k
units down (if k < 0) of the graph of y = f(x).
Example 4. Use the graph of y = 2x to graph y = 2x–2 and y = 2x+4 .
Solution. Some y-values are shown on the table in the next page.
23
−3
−2
−1
0
1
2
3
x
0.125
0.25
0.5
1
2
4
8
x-2
0.031
0.063
0.125
0.25
0.5
1
2
x+4
2
4
8
16
32
64
128
x
y=2
y=2
y=2
The graphs of these functions are shown below:
Observe that the graph of y = 2x+4, is just
shifting the graph of y = 2x, four units to the left.
Whereas, the graph of y = 2x-2, shifted the graph of
y = 2x, two units to the right.
Observations:
Range
y-intercept
Horizontal asymptote
y = 2x
y>0
1
y=0
y = 2x+4
y>0
16
y=0
y = 2x-2
y>0
1/4
y=0
In general,
Horizontal Shifts
Let k be a real number. The graph of y = bx+k is a horizontal shift of k units to the right (if
k > 0) or k units to the left (if k < 0) of the graph of y = bx.
WHAT’S MORE
INDEPENDENT ACTIVITY 1
Directions: Match the given exponential function in Column A with its table of values/graph in
Column B.
COLUMN A
COLUMN B
x
0
1
2
3
1. f(x) =3x
a.
y
-1
-3
-9
-27
2. f(x) = -3x
b.
24
x
y
-2
9
0
1
2
1/9
4
1/81
3. f(x) = 4(3) x
x
y
-2
-8/9
-1
-2/3
0
0
1
2
d.
x
y
-1
4/3
0
4
1
12
2
36
e.
x
y
-2
1/81
-1
1/27
0
1/9
1
1/3
4. f(x) = 3x+1
c.
5. f(x) = 3x-2
6 f(x) = 3x – 1
7. f(x) = 3x + 4
f.
g.
h.
25
INDEPENDENT ACTIVITY 2
Directions: Fill in the table of value of the given exponential function and sketch its graph. Describe
the graph of the function afterwards.
y = 3x – 5
8. Fill in the table of value
x
-2
-1
0
1
2
y
9. Sketch the graph
10. Describe the graph
Domain:
_______________________
Range:
_______________________
x-intercept:
_______________________
y-intercept:
_______________________
Horizontal Asymptote:
_______________________
Vertical Asymptote:
_______________________
WHAT I HAVE LEARNED
Directions: Answer the questions that follow.
1. What can you say about the domain of an exponential function?
2. Does an exponential function have vertical asymptote? Why?
3. What have you observe with the graph and equation of the functions which are reflection of
each other upon the x and y-axis?
4. What have you observe with the graph and equation of a function when you stretch or
shrink it?
5. What have you observe with the graph and equation of a function when you shift it
upward/downward?
6. What have you observe with the graph and equation of a function when you shift it to the
left/right?
26
WHAT I CAN DO
A version of infographic (right)
explains the potential impact of social
distancing during the pandemic. It
illustrates that reducing the exposure to
each infected person can have a dramatic
effect on the total number of infections a
short time later. The numbers rely on the
mathematical concept of exponential
growth.
Directions:
1. Fill in the table of values from the given infographic:
Now
1
1
1
No Action
50% exposure reduction
75% exposure reduction
10 days from now
60 days from now
2. Sketch the graphs
10
20
30
40
50
60
Number of Days
Answer the questions:
a. In your own calculation, if social distancing is not practiced, how many people will possibly get
infected after 100 days from observation date?
b. How did you arrive with your answer?
c. If you are part of the government, what should you do to prevent further the transmissions of
COVID-19 infections in your place?
27
ASSESSMENT
Directions: Do the following activity. Write your answer in your notebook/ graphing paper.
Each function will be rated according to the rubric below.
(a) Complete the table of values for the given functions below. Write your solution in your
pad paper. Afterwards, (b) sketch the graph and (c) label the domain, range, y-intercept and
horizontal asymptote of each graph.
x
y
-2
-1
0
1
2
1. f(x) = 6x
2. f(x) = -5x
3. f(x)= 3x+2
4. f(x) = 2(5)x
5. f(x) = 4x – 5
RUBRICS:
5
Table of
values
Graph
Analysis of the
graph
4
All
There is one
corresponding error
in
values of y are solving
the
solved
corresponding
correctly and y-coordinate.
accurately.
All points in
There is one
the graph are point that is
accurately
not correctly
plotted in the plotted in the
coordinate
cartesian
plane
plane.
3
There are two
errors
in
solving
the
corresponding
y-coordinate.
2
There are 3 or
4 errors in
solving
the
corresponding
y-coordinate.
1
The table of
values is not
solved
correctly.
There are two
points that are
not correctly
plotted in the
cartesian
plane.
The graph of
the function is
not correctly
plotted in the
coordinate
plane.
The graph is
accurately
labelled and
analyzed
There are two
errors in the
analysis of the
graphs
There are 3 or
4 points that
are not
correctly
plotted in the
cartesian
plane.
There are
three errors in
the analysis of
the graph
There is one
error in the
analysis of the
graph.
28
The analysis of
the graph is
not correctly
labelled
ANSWER KEYS
Lesson 1:
What I Know
1. A
2. D
3. B
4. D
5. A
6. A
7. C
8. D
9. C
10. A
11. D
12. A
13. C
14. C
15. D
What’s In
A.
1. A
2. D
3. A
4. A
5. A
6. D
7. A
8. D
9. D
10. D
B.
1. Exponential Equation
2. Exponential Inequality
3. Exponential Function
4. None of these
5. Exponential Inequality
6. Exponential Function
What’s New
Activity 1
A-E-F-I-H-D-G-K-L-M
Activity 2
1. 3
2. 4
3. 4
4. 0
5. 2
6. ½
7. 2
8. -3
9. -2
10. 2/3
Activity 2
3. D
4. B
5. A
6. C
7. C
8. C
9. A
10. B
11. A
12. B
13. D
14. C
15. D
What’s More
Activity 1
1. A
2. D
3. B
4. G
5. F
Activity 2
6. I
7. A
8. G
9. D
10. E
What’s In
What I Have Learned
Answers may vary
D
What I Can Do
G
R
O M
W
T
H
H
X
I
Y
Assessment:
Test I
1. x = 1
2. x = -4
3. x = 3
4. x = 3/2
5. x = 4/3
6. x = 3/2
7. x = 1/6
8. x = 2
9. x = -2/3
10.x = -3
Test II
1. x > 1
2. x < 11/2
3. x > 3
4. x < -5/3
5. x > -19/3
6. x ≥ -3
7. x < -3
8. x < -5
9. x > ½
10. x < 5/4
O
A
I
D
R
N
T
I
N
I
N
T
E
R
C
E
P
T
R
A
N
G
E
C
A
Z
O
N
A
S
Y
M
P
T
O
T
E
A
V
E
R
T
I
C
A
L
What I Can Do
R
C
E
P
T
E
R
C
E
P
T
What’s New
x
y
4
8
What I Have Learned
Answers may vary
5 6 7
8
9
16 32 64 128 256
No Action
50% Reduction
75% Reduction
Now
1
1
1
a. 9 537
b. Answer may vary
c. Answer may vary
Assessment:
1. 229
263
2. the graph increases
rapidly
What’s More
Activity 1
1. H
2. A
3. D
4. G
5. E
6. C
7. F
Lesson 2:
What I Know:
1. D
2. B
29
10 days 60 days
2.5
406
1.25
15
0.625
2.5
References
Most Essential Learning Competencies (MELCS)
SHS General Mathematics LM pages 83-98
SHS General Mathematics TG pages 95-111
Google Internet Information about the lessons:
https://www.brilliant.org/wiki/exponential-inequalities/
Activities in the Module:
Mathematics Learner’s Material 9 Module 4: Zero Exponents, Negative Integral Exponents,
Rational Exponents, and Radicals page 232
https://theconversation.com/amp/coronavirus-is-growing-exponentially-heres-what-thatreally-means-134591
https://www.publicdomainpictures.net/en/view-image.php?image=32921&picture=peso-billsbackground
30
For inquiries or feedback, please write or call:
Department of Education: Region VII, Division of Cebu Province
Office Address: IPHO Bldg., Sudlon, Lahug, Cebu City, Philippines
Telefax:
(032) 255 - 6405
Email Address: cebu.province@deped.gov.ph