CONTACT US
If you have questions regarding the content of this module, please contact any of the following
persons or offices for clarification. Please channel questions to rightful persons/offices.
A. Teacher
For Grade 9 – Excellence
Name
Email Address
Phone Number
Facebook
For Grade 9 - Prudence
Name
Email Address
Phone Number
:
:
:
:
RENE B. TINQUILAN, LPT, MAED
renetinquilan@gmail.com
+639063673530
https://web.facebook.com/rene.posneg
:
:
:
JUDY-ANN B. MORALES, LPT
juannmorente@gmail.com
+639100500034
:
:
:
:
RALPH H. CELESTE, DBA
Ralphceleste23@gmail.com
+639091914949/+639356521482
:
:
:
:
MARICAR B. LEPORNIO, PhD
smcbdo.principal@gmail.com
+639494022293
www.facebook.com /maricar.lepornio
:
:
:
:
ANNIE M. MADUAY
anniemaduay@gmail.com
+639094401026/+639759219372
www.facebook.com /annie.maduay
B. Senior High School Focal
Name
Email Address
Phone Number
Facebook
C. School Principal
Name
Email Address
Phone Number
Facebook
D. Reproduction In-Charge
Name
Email Address
Phone Number
Facebook
SY 2021-2022 |Mathematics|Grade 9 | First Quarter
Page | 1
The Marian Way
God listens to true prayers. During this trying time, we
encourage our Marians to religiously ask for the guidance of our
Almighty.
The plague dramatically breaks out borders to borders. Millions of people died, and
overwhelming numbers of infected people caused fear and panic amongst us. Let us continue to
pray and practice the Ignacian-Marian way. Together, we will survive and heal as one.
Vision
Dynamic and Holistically
developed individuals
actively witnessing the
gospel values in the
community.
Mission
We commit ourselves to:
1. promote the total formation of persons through
quality instruction and integration of activities
2. form vibrant, responsible community leaders
inspired by the virtues of Mo. Ignacia and imbued
with the Gospel values;
3. provide development programs for the acquisition
of appropriate skills, promotion of positive attitudes
and enhancement of personal discipline;
4. foster the development of multiple and holistic
competencies to ensure work effectiveness.
GOAL STATEMENT
St. Mary’s College is a Catholic School that is an instrumentality of the Congregation of the
Religious of the Virgin Mary that aims to provide within its community of students and
personnel Catholic values. Its goal is to provide an educational program and environment
animated by Catholic doctrine, beliefs, teachings, traditions, and practices, the exercise of
which is protected by, among others, Article III, Section 5 of the 1987 Philippine Constitution.
In order for us to approximate our vision and live our mission, we dedicate to produce
graduates who are God-fearing, capable of independent learning and critical thinking,
enabling them to respond successfully by continuing education in a technologically advanced
world and to serve the society, promoting justice and peace and protecting the youth against
harassment and immorality.
Page | 2
SY 2021-2022 |Mathematics |Grade 9 | First Quarter
QUALITY POLICY
We, at the St. Mary’s College, commit to provide quality Catholic Ignacian Marian education
to mold students to be Ignacian Marian leaders of faith, excellence, and service wherever
they are at all times. We commit to collaboratively comply and maintain an effective quality
management system by periodically reviewing and validating the processes and services in
line with the quality objectives and standards for continual improvement.
SUBJECT OUTLINE
Grade/Year Level
: Grade 9
Quarter
: First
Subject Title
: Mathematics 9
Time Frame
: Nine (9) Weeks
Content Standard
The learner demonstrates understanding of key concepts of quadratic equations, inequalities
and functions, and rational algebraic equations.
Performance Standard
The learner is able to investigate thoroughly mathematical relationships in various situations,
formulate real-life problems involving quadratic equations, inequalities and functions, and
rational algebraic equations and solve them using a variety of strategies.
Subject Requirements

Accomplished Worksheets

Accomplished Performance tasks
Topics
Quadratic Equations
Subtopics

Quadratic Equations in one variable

Solving quadratic equations by:
 Factoring
 Extracting Square Roots
 Completing the Square
 Quadratic Formula
Nature of Roots of Quadratic
Equations

Roots of Quadratic Equations

Writing Equations from Solutions
SY 2021-2022 |Mathematics|Grade 9 | First Quarter
Page | 3

Equations in Quadratic Form

Solving Quadratic Inequalities

Application of Quadratic Inequalities

Domain and Range of Relations and
Functions

The Graph of 𝑓(𝑥) = 𝑎(𝑥 − ℎ)2 + 𝑘.

More on Graphing Quadratic Functions
Problems Involving Quadratic
Equations
Quadratic Inequalities
Quadratic Functions and Their Graphs
MY TIMELINE
August 17, 2021
Distribution of Module
August 26-27, 2021
First Monthly
Examination
August 25, 2021
Distribution of Test
Papers/Retrieval of
Worksheets (Lesson 1-2)
Page | 4
August 27, 2021
Retrieval of Test
Questionnaire for First
Monthly Exam
September 28, 2020
Submission of Worksheets
(Lesson 3 – 5)
Distribution of First Quarter
Exam
September 29 - 30, 2020
First Quarter Exam
SY 2021-2022 |Mathematics |Grade 9 | First Quarter
September 30 , 2020
Submission of
Performance Task
All other requirements
Retrieval of Test
Questionnaire for First
Quarter Exam
How to Use the Module
In this module, you will undergo through a series of learning activities to accomplish
requirements as projected in each lesson and subtopics. Each lesson contains Pre and Post-
Assessment Sheet, Vocabulary Section, Lesson or topic exercise sheet, and
Performance Task Exercises Sheet.
Summative Assessments such as Monthly and Quarterly Exams will be separated from
the module. The accomplishment of each task is on your comfort, however following the
scheduled submission of every module.
THINGS TO REMEMBER!
1.)
2.)
3.)
4.)
5.)
6.)
7.)
8.)
Carefully read all the information sheets,
Follow the directions in answering all the tasks, or exercises,
Answer all the exercises, and accomplish your performance task,
Submit the module based on the scheduled date,
Do not write unnecessary markings inside the module,
All questions should only be answered on the given worksheet after every lesson.
Lastly, inform your parents/guardians to affix their signatures on the sheet that will
be provided on the scheduled day of module distribution.
Use only the given space where you can write your solution. Use graphing paper for
every problem which require graphing.
Should you have any questions about this module, please do not hesitate to reach us via email,
group chat, or mobile number as projected on the teacher’s information above.
For the schedule of module distribution/submission and date of examination, refer to the
information box below. Please take note that the distribution of modules and
examination papers for Kinder to Grade 12 will be done inside the SMC campus.
SY 2021-2022 |Mathematics|Grade 9 | First Quarter
Page | 5
A number of physical quantities cannot be
represented by a first degree equation, function, or
inequality. There are real-world situations that are best
modeled by functions of degree greater than 1.
The word quadratic is derived from the Latin word
quadratus that means squared. Thus, quadratic pertains to
the operation of squaring or raising a number or an
expression to the second power.
Quadratic equation, a second-degree polynomial
equation, and solution to problems leading to quadratic
equations are known as early as 2000 B.C.
Quadratic functions are often used in twodimensional problems and in relationship between two
physical variables. Here, relationships are observed,
patterns are noted, and generalizations are drawn.
Some applications of quadratic relations are
manifested in solving for extreme function values maxima
or minima problems, and motions problems such as
trajectory and acceleration (Diaz et.al., 2018)
Let’s find out how much you already know about this module. Direction: Read the following
questions below and encircle the letter of the correct answer. Do not leave unanswered item.
1. Which of the following quadratic equations is in standard form?
a. 𝑥 2 = −1
b. 3𝑥 2 − 4 + 5𝑥 = 0
c. 𝑥 2 − 2 = 0
d. 2𝑥 − 1 = 𝑥 2
2. What is the standard form of 𝑥 2 + 3 = −2𝑥?
a. 𝑥 2 − 2𝑥 + 3 = 0
b. 𝑥 2 + 2𝑥 + 3 = 0
c. 𝑥 2 + 2𝑥 − 3 = 0
d. 𝑥 −2 + 2𝑥 + 3 = 0
3. Which of the following is the set of roots of 7𝑥 2 + 18𝑥 = 10𝑥 2 + 12𝑥?
a. 𝑥 = 0, 𝑥 = 1
b. 𝑥 = 1, 𝑥 = 2
c. 𝑥 = 0, 𝑥 = 2
d. 𝑥 = 0, 𝑥 = 3
Page | 6
SY 2021-2022 |Mathematics |Grade 9 | First Quarter
4. Why should 𝑎 ≠ 0 to make 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 a quadratic equation?
a. It is because if 𝑎 = 0, the equation will become a special type of quadratic
equation.
b. It is because if 𝑎 = 0, the equation will become linear equation.
c. It is because whenever the leading coefficient is equal to zero, the
equation contains complex root/s.
d. It is because the leading coefficient should be real numbers to have a
standard form of quadratic equation.
5. Which of the following is a root of 3𝑥 2 + 𝑥 = 0?
a. 0
b. 1
c. 2
d. 3
6. Which of the following quadratic equations has a root 𝑥 = 3?
a. 𝑥 2 + 9 = 0
b. 𝑥 2 + 5𝑥 + 6 = 0
c. 2𝑥 2 − 6𝑥 + 17 = 0
d. 3𝑥 2 − 𝑥 + 3 = 0
7. Which statement is true?
a. All quadratic equations can be solved by factoring.
b. All quadratic equations can be solved by the square root method.
c. All quadratic equations can be solved by quadratic formula.
d. If a quadratic equation can be solved by completing the square, then it can
be solved by factoring.
8. Which of the following quadratic equations has two complex but not real
solutions?
a. 𝑥 2 = 9
b. 𝑥 2 + 9 = 0
c. 𝑥 2 − 3𝑥 − 4 = 0
d. 𝑥 2 − 25 = 0
9. Which of the following equations whose discriminant is equal to 5?
a. 𝑥 2 + 5𝑥 + 5 = 0
b. 2𝑥 2 − 𝑥 = 9
c. 3𝑥 2 = 8
d. 𝑥 2 + 25𝑥 − 25 = 0
10. Which of the following quadratic functions is represented by the table values
below?
𝑥
𝑓(𝑥)
−3
18
−2
8
−1
2
0
0
1
2
2
8
3
18
SY 2021-2022 |Mathematics|Grade 9 | First Quarter
Page | 7
a. 𝑓(𝑥) = 𝑥 2
b. 𝑓(𝑥) = 2𝑥 2
c. 𝑓(𝑥) = 𝑥 2 + 1
d. 𝑓(𝑥) = 𝑥 2 − 1
11. What are the roots/zeroes described by the given graph?
a.
b.
c.
d.
1, 3
−1, 3
1, −3
−1, −3
12. Which of the following shows the graph of 𝑓(𝑥) = 2(𝑥 − 1)2 − 3?
a.
b.
c.
D.
13. What makes the equation −4𝑥 2 = 0 NOT a quadratic equation in standard form?
A. The leading coefficient is negative.
B. The real numbers 𝑎 and 𝑏 are missing.
C. The entire equation is equal to zero.
D. The quadratic equation is not in the form 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0.
Learning Target
:
Recall that a linear equation is an equation
containing a first degree polynomial. In solving problem
situations, we sometimes obtain an equation containing a
second-degree polynomial. Such an equation is called
quadratic.
At the end of the lesson,
1. The learners will be able to know:
a. Quadratic Equation
Page | 8
SY 2021-2022 |Mathematics |Grade 9 | First Quarter
b. Illustration of Quadratic Equations
c. Solving Quadratic Equations.
2. The learners will be able to:
a. illustrate quadratic equations
b. solve quadratic equations by: (a) extracting square roots; (b) factoring; (c)
completing the square; and (d) using the quadratic formula.
Below are some words that will help you understand the lesson quickly. Check them out!
Words
Meaning
Quadratic
Equation
It is an equation of the form Discriminant
𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0, where 𝑎, 𝑏,
and 𝑐 represent real numbers
and 𝑎 ≠ 0.
It is the radicand 𝑏 2 −
4𝑎𝑐 in the quadratic
formula, 𝑥 =
The solutions of the quadratic
Roots
equation 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 =
0 correspond to the roots of the
function 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐,
since they are the values of 𝑥 for
which 𝑓(𝑥) = 0.
The solutions to a
quadratic equation.
Solutions
Words
Meaning
−𝑏±√𝑏2 −4𝑎𝑐
.
2𝑎
Find each indicated product then answer the question that follow.
1. 3(𝑥 2 − 2)
2. (2𝑥 + 1)(𝑥)
3. (𝑤 − 2)(2𝑤)
4. (2 − 𝑠)(1 + 2𝑠)
Questions:
1. How did you find each product?
2. In finding each product, what mathematics concepts or principles did you apply? Explain
how you applied these mathematics concepts or principles.
3. How would you describe the products obtained?
SY 2021-2022 |Mathematics|Grade 9 | First Quarter
Page | 9
Quadratic Equations in One Variable
A second-degree equation in one variable is an
equation that can be expressed in the form 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 =
0, where 𝑎, 𝑏, and 𝑐 represent real numbers and 𝑎 ≠ 0. This
form of the quadratic equation is said to be in standard
form since all the nonzero terms are on the left side of the
equation and the powers of the variables are in descending
order.
Example 1:
Write each of these quadratic equations in standard form and identify the real numbers 𝑎, 𝑏,
and 𝑐.
1. 2𝑥 2 − 2𝑥 + 2 = 0
2. 𝑥 − 3𝑥 2 = 1
1
3. 7𝑥 2 = 3 𝑥
Solutions:
1. 2𝑥 2 − 2𝑥 + 2 = 0
𝑎 = 2, 𝑏 = −2, 𝑐 = 2
This equation is already in standard form.
2. 𝑥 − 3𝑥 2 = 1
𝑥 − 3𝑥 2 − 1 = 1 − 1
𝑥 − 3𝑥 2 − 1 = 0
−3𝑥 2 + 𝑥 − 1 = 0
𝑎 = −3, 𝑏 = 1, 𝑐 = 0
Add −1 to both sides (Addition Property of Equality)
so that the right hand side equals zero.
Arrange the power of the variables in descending order
1
3. 7𝑥 2 = 3 𝑥
1
1
1
7𝑥 2 − 𝑥 = 𝑥 − 𝑥
3
3
3
1
7𝑥 2 − 3 = 0
1
𝑎 = 7, 𝑏 = − , 𝑐 = 0
3
Page | 10
1
Add − 𝑥 to both sides (Addition Property of Equality)
3
so that the right hand side equals zero.
Since the constant term is missing, 𝑐 = 0.
SY 2021-2022 |Mathematics |Grade 9 | First Quarter
Solving Quadratic Equations by Factoring
When a quadratic equation is in standard
form, 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0, it may be possible to
solve it by factoring.
By setting each factor equal to 0 and solving the resulting first-degree equations, we find the
roots or the solutions to the quadratic equations. This process is based on the Principle of
Zero Products.
The Principle of Zero Products
For any real numbers 𝑥 and 𝑦, if 𝑥𝑦 = 0,
then 𝑥 = 0 or 𝑦 = 0; and if either 𝑥 = 0 or
𝑦 = 0, then 𝑥𝑦 = 0
Recall:
𝑥 is a factor of 𝑥 2 .
𝑥2 = 𝑥 ∙ 𝑥
5 and −5 are factors of −25.
−25 = 5 ∙ −5
Example 2:
Solve 3𝑥 2 = 75.
Solution:
3𝑥 2 = 75
1
1
( ) 3𝑥 2 = ( ) 75
3
3
𝑥 2 = 25
𝑥 2 − 25 = 0
(𝑥 + 5)(𝑥 − 5) = 0
Multiply both sides by
1
3
(Multiplication Property of Equality [MPE])
Rewrite in standard form.
Factor.
Applying the principle of zero products, we have:
𝑥+5=0
𝑥 − 5 = 0 Set each factor to 0.
𝑥 = −5
𝑥=5
Solve for 𝑥.
Thus, the roots are −5 and 5.
When the constant is 0, the quadratic equations will be of the form 𝑎𝑥 2 + 𝑏𝑥 = 0.
Example 3:
Solve. 3𝑥 2 + 18𝑥 = 0
Solution:
3𝑥 2 + 18𝑥 = 0
Factor. The Greatest Common Factor (GCF) of 3 and 18 is 3, and the GCF of
3𝑥(𝑥 + 6) = 0
𝑥 2 and 𝑥 is 𝑥. Thus, you can factor out 3𝑥 from the equation.
3𝑥 = 0
𝑥+6=0
𝑥=0
𝑥 = −6
The roots are 0 and −6.
Set each factor to 0.
Solve for 𝑥.
SY 2021-2022 |Mathematics|Grade 9 | First Quarter
Page | 11
The equation of this form, one root will always be equal to 0 while the other root will be
nonzero number.
Factoring may also be used to solve a quadratic equation when none of the constants 𝑎, 𝑏, or 𝑐
is 0.
Review: Multiplying Binomials
Example 4:
Solve. 𝑥 2 + 5𝑥 + 6 = 0.
Solution:
𝑥 2 + 5𝑥 + 6 = 0
To factor 𝑥 2 + 5𝑥 + 6 = 0, find first the numbers that multiply to 6
(the constant number), and add up to 5 (the 𝑥-coefficient).
(𝑥 + 2)(𝑥 + 3) = 0
These two numbers are 2 and 3 since 2 ∙ 3 = 6 and 2 + 3 = 5. Then,
add these numbers to 𝑥 to form the binomial factors: (𝑥 + 2) and
(𝑥 + 3).
𝑥+2=0
𝑥+3=0
Set each factor to 0.
𝑥 = −2
𝑥 = −3
Solve for 𝑥.
Quadratic equations,
inequalities, and functions?!
Wow, how will I learn these? I
am not a math person! I do
not have a math brain.
Is anyone born with a math brain?
Neuroscience research suggests that there is no such
thing as a math brain. Everyone can learn math and the
brain has the ability to grow and shrink. When you
struggle and make mistake, it is a most important time
for your brain. It is the time when the brain grows.
Page | 12
SY 2021-2022 |Mathematics |Grade 9 | First Quarter
Solving Quadratic Equations by Extracting
Square Root
Note that the incomplete quadratic equation 𝑥 2 − 4 = 0
can be written as 𝑥 2 = 4 by applying the Addition Property of
Equality. By substitution, two values satisfy this equation. These
are 2 and −2.
Any quadratic equation of the form 𝑥 2 = 𝑐 has two
possible solutions: 𝑥 = √𝑐 or 𝑥 = −√𝑐. This is referred to as the
square root property.
Example 5:
Solve 𝑥 2 = 49
Solution:
𝑥 2 = 49
2 =square
Get
√𝑥the
±√49root of both sides.
𝑥 = ±7
The roots of the equation are 𝑥 = 7 and 𝑥 = −7.
Square Root Property involves taking the square
roots of both sides of a quadratic equation. This is
applied when the term containing the second degree
term with 1 as its numerical coefficient is isolated.
Square Root Property:
If 𝑥 2 = 𝑐, then 𝑥 = √𝑐 or 𝑥 = −√𝑐
Speak Like a Mathematician:
The notation ±7 is read as “plus or minus
7.” It is a shorthand notation for the pair
of numbers +7 and −7.
Some perfect squares:
1
36
√1 = 1
4
49
√4 = 2
9
64
√9 = 3
16
81
√16 = 4
25
100
√25 = 5
Example 6:
Solve the equation 𝑥 2 − 25 = 0.
Solution:
𝑥 2 − 25 = 0
25 is added to both sides (APE).
𝑥 2 − 25 + 25 = 25 + 0
𝑥 2 = 25
Square root both sides of the equation (Square Root Property).
√𝑥 2 = ±√25
𝑥 = ±5
√36 = 6
√49 = 7
√64 = 8
√81 = 9
√100 = 10
The square root is simplified.
Therefore, the solutions are 5 and −5.
SY 2021-2022 |Mathematics|Grade 9 | First Quarter
Page | 13
Example 7:
Solve the equation 𝑥 2 − 8 = 0.
Solution:
𝑥2 − 8 = 0
Add 8 to both sides of the equation.
𝑥2 = 8
2
Square
root both sides of the equation (Square Root Property).
√𝑥 = ±√8
Factor the perfect square. 8 has factors of 1, 2, 4 and 8. Choose the two
𝑥 = ±√(4)(2)
factors with biggest perfect square, which are 4 and 2.
𝑥 = ±√4√2
𝑥 = ±2√2
The solutions are 2√2 and −2√2.
Example 8:
Solve. 2(𝑥 − 5)2 = 32.
Solution:
2(𝑥 − 5)2 = 32
(𝑥 − 5)2 = 16
√(𝑥 − 5)2 = ±√16
𝑥 − 5 = ±4
Divide both side of the equation by 2.
Square root both sides of the equation.
The square roots are simplified.
Since 4 may be + or −, there are two equations. Solve for each equation.
𝑥−5=4
𝑥 − 5 = −4
𝑥 =4+5
𝑥 = −4 + 5
𝑥=9
𝑥=1
The roots of the equation are 9 and 1.
We know that the quadratic equation of the form (𝑥 + 𝑦)2 = 𝑐 can be solved by finding
the square roots of both sides. Thus, if we can write a quadratic equation in this form, we can
solve it.
Completing the Square
To complete the square of the expression 𝑥 2 + 𝑏𝑥, add the square of half the coefficient of 𝑥
𝑏 2
to make 𝑥 2 + 𝑏𝑥 + (2)
Example 9:
Complete the square. 𝑥 2 − 8𝑥
Solution:
Find 𝑏 in 𝑥 2 + 𝑏𝑥. Here our 𝑏 = −8.
𝑥 2 − 8𝑥
8
− 2 = −4
(−4)2 = 16
Next, divide 𝑏 by 2 or
𝑏
2
𝑏 2
Then, square the quotient of 𝑏 and 2 or ( ) .
2
2
𝑥 − 8𝑥 + 16
The trinomial 𝑥 2 − 8𝑥 + 16 can be written as (𝑥 − 4)2 since
(𝑥 − 4)2 = (𝑥 − 4)(𝑥 − 4) = 𝑥 2 − 8𝑥 + 16.
Page | 14
SY 2021-2022 |Mathematics |Grade 9 | First Quarter
Example 10:
Solve by completing the square. 𝑥 2 + 5𝑥 + 4 = 0
Solution:
𝑥 2 + 5𝑥 + 4 = 0
𝑥 2 + 5𝑥 = −4
Rearrange by writing the constant on the right side. Use APE.
5 2
5 2
2
Since the coefficient of 𝑥 2 is 1, complete the square of 𝑥 2 +
𝑥 + 5𝑥 + ( ) = −4 + ( )
2
2
25
4
𝑥 2 + 5𝑥 + ( ) =
5 2
5𝑥 by adding (2) to both sides of the equation.
9
4
5 2
2
5 2 9
(𝑥 + ) =
2
4
Factor the perfect square trinomial. Remember that
5 2
25
(𝑥 + ) = 𝑥 2 + 5𝑥 + ( )
2
4
2
Get the square roots. Always use the principle of positive and
negative roots.
√(𝑥 + 5) = ±√9
2
4
5
25
4
Evaluate. Take note that ( ) = ( )
3
𝑥 + 2 = ±2
Solve both equations.
𝑥+2=2
𝑥 + 2 = −2
Think About This:
𝑥= −
𝑥=− −
When the solutions or roots are integers, does it
mean that you could have solved the equation by the
factoring method? Explain your answer.
5
3
3
5
2
2
−2
2
5
3
3
2
8
−2
5
2
𝑥=
𝑥=
𝑥 = −1
𝑥 = −4
The roots of the equation are −1 and −4.
When the coefficient of 𝑥 2 is not 1, the first step is to divide all terms of the equation by
the coefficient of 𝑥 2 so that it will then be equal to 1.
Example 11:
Solve by completing the square. 2𝑥 2 − 4𝑥 − 4 = 0
Solution:
2𝑥 2 − 4𝑥 − 2 = 0
Rearrange by writing the constant on the right side. Use APE.
2𝑥 2 − 4𝑥 = 2
2
2𝑥
4𝑥 2
Divide the terms by the coefficient of 𝑥 2 , which is 2.
−
=
2
2
2
𝑥 2 − 2𝑥 = 1
−2 2
−2 2
𝑥 2 − 2𝑥 + ( 2 ) = 1 + ( 2 )
𝑥 2 − 2𝑥 + (−1)2 = 1 + (−1)2
−2 2
Add ( ) to both sides of the equation.
2
Simplify.
(𝑥 − 1)2 = 2
√(𝑥 − 1)2 = ±√2
Factor the perfect square trinomial.
Get the square roots. Always use the principle of positive and
negative roots.
𝑥 − 1 = ±√2
𝑥 − 1 = √2
𝑥 − 1 = −√2
𝑥 = √2 + 1
𝑥 = −√2 + 1
Solve both equations.
SY 2021-2022 |Mathematics|Grade 9 | First Quarter
Page | 15
The famous line “You complete me” was popularized
by Tom Cruise in the movie Jerry Maguire. What do
you think makes a person feel complete?
Steps in Solving a Quadratic Equation in 𝒙
by Completing the Square
1. Write the terms with variables on one
side of the equation.
2. Arrange the terms in descending
orders.
3. If the coefficient of 𝑥 2 is not 1, divide
both sides by the coefficient of 𝑥 2 .
4. Take half of the coefficient of 𝑥 and its
square to both sides of the equation to
complete the square.
5. Factor the perfect square trinomial on
one side. Simplify the expression on
the other side.
6. Use the principle of positive and
negative roots.
7. Solve for 𝑥 in each case.
Page | 16
SY 2021-2022 |Mathematics |Grade 9 | First Quarter
Solving Quadratic Equations by Quadratic
Formula
Often, the method of completing the square is tedious especially when the values of
𝑎, 𝑏, and 𝑐 are large quantities. However, a formula for solving any quadratic equation involving
𝑎, 𝑏, and 𝑐 can be derived using completing the square.
Quadratic Formula
If 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 and 𝑎 ≠ 0, then 𝑥 =
−𝑏±√𝑏2 −4𝑎𝑐
.
2𝑎
Example 12:
Solve. 8𝑥 2 + 2𝑥 − 55 = 0
Solution:
First, find the value of 𝑎, 𝑏, and 𝑐. In the equation, 𝑎 = 8 (the coefficient of 𝑥 2 ), 𝑏 = 2 (the
coefficient of 𝑥), and 𝑐 = −55 (the constant term). Then, substitute the values of 𝑎, 𝑏, and 𝑐 to
find the value of 𝑥 using quadratic formula, we have:
SY 2021-2022 |Mathematics|Grade 9 | First Quarter
Page | 17
𝑥=
𝑥=
−𝑏±√𝑏2 −4𝑎𝑐
2𝑎
−(2)±√(2)2 −4(8)(−55)
2(8)
Substitute the values.
Simplify. Follow the PEMDAS rule.
−2 ± √4 + 1760
16
−2 ± √1764
𝑥=
16
−2±42
𝑥=
𝑥=
Solve both equations.
16
𝑥=
𝑥=
−2+42
16
40
5
=2
16
𝑥=
𝑥=
−2−42
16
−44
=
16
−
5
11
4
Therefore, the roots are 2 and −
11
.
4
Helpful Hint:
To replace 𝑎, 𝑏, and 𝑐 correctly in the quadratic
formula, write the quadratic equation in standard
form 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0.
Example 13:
Solve for 𝑥 using quadratic formula. 2𝑥 2 = 1 − 2𝑥
Solution:
The equation is not in standard form. Rewriting the equation in standard form, we have:
This is the equation in standard form.
2𝑥 2 + 2𝑥 − 1 = 0
𝑎 = 2, 𝑏 = 2, 𝑐 = −1
𝑥=
𝑥=
𝑥=
Substitute these values in the formula.
The quadratic formula.
−𝑏±√𝑏2 −4𝑎𝑐
2𝑎
−(2)±√(2)2 −4(2)(−1)
2(2)
−2±√4+8
4
Substitute the values of 𝑎, 𝑏, and 𝑐.
Simplify. Follow the PEMDAS rule.
𝑥=
−2 ± √12
4
𝑥=
−2±2√3
4
𝑥=
2(−1 ± √3)
4
Factor out 2 from the terms in numerator, then simplify.
𝑥=
−1±√3
2
Solve each equation.
𝑥=
−1+√3
2
The roots are
Simplify the radical. Using the rule of simplifying radical and
choosing the highest perfect square factor of 12 which is 4,
we have √12 = √4 ∙ 3 = √4√3 = 2√3.
𝑥=
Think About This:
−1−√3
2
−1+√3
2
and
−1−√3
.
2
What happens if 𝑏 2 − 4𝑎𝑐 in the quadratic equation
gives a negative value? That would mean that the
quadratic equation has no real solution. Can you
explain why?
Math FYI:
About 400 B.C., mathematicians of ancient Babylon discovered the method of completing the
square that led to the quadratic formula for an exact solution of any quadratic equation in the form
𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0
Page | 18
SY 2021-2022 |Mathematics |Grade 9 | First Quarter
We can say that the quadratic formula is a
very dependable partner in solving quadratic
equations. When the factoring and square root
methods do not work and when the completing
the square method is tedious to do, the quadratic
formula can be counted on to give the roots. Are
you anything like the quadratic formula to your
family and friends? How?
Activity 1
Problem: The side of a square is 2 cm. When its side is increased by 5 cm, its area
becomes 49 cm2 . How long is the side of the original square?
1. Draw a figure that will illustrate the details of the problem.
2. Represent the unknown with a variable.
3. If 𝑥 is the side of the original square, how do you represent the side of the new
square?
4. Form the equation needed to solve for the unknown.
5. Solve the equation by applying the Square Root Property.
6. Which of the two solutions is the appropriate answer? Why?
Four-Pronged
Integration
ICV/RV
Behavioral Indicators
Points to Ponder
Excellence/Competence
Like solving problems in quadratic
equations, we really need to acquire the
basic knowledge and skills so that we
can do efficiently our work or task. The
knowledge
of
solving
quadratic
equations are needed in many aspects
in our lives especially for those who
work in engineering and architecture.
Those architect or engineers must
possess great competence in their job
so as not to compromise the project
Pursues high achievement
standards in everything one
does
SY 2021-2022 |Mathematics|Grade 9 | First Quarter
Page | 19
they are working on.
Social
Orientation
Lesson Across
Discipline
Biblical Text
Page | 20
Government’s way of handling their
structural projects, like #BuildBuildBuild
(BBB) by the Duterte Administration can
make or break the nation. If they do
their job with full of accountability and
honesty, then the quality of that project
is almost guaranteed. But, if the project
is full of corruption, then the people,
whom they serve, will suffer.
Architecture
Engineers and architects have applied
the concepts of quadratic equation to
create magnificent buildings and other
structures.
Genesis 6:14-16
God Himself is a great engineer and
architect of a legendary structure that
“Make for yourself an ark of saved the human and other species
gopher wood; you shall make from extinction.
the ark with rooms, and shall
cover it inside and outside
with pitch. This is how you
shall make it: the length of the
ark three hundred cubits, its
breadth fifty cubits, and its
height thirty cubits.”
Government Structural
Projects

A quadratic equation is an equation of the form 𝑎𝑥 2 +
𝑏𝑥 + 𝑐 = 0, where 𝑎, 𝑏, and 𝑐 represent real numbers
and 𝑎 ≠ 0.

The solutions to a quadratic equation are called its
roots.

For any real number 𝑥 and 𝑦, if 𝑥𝑦 = 0, then 𝑥 = 0 or
𝑦 = 0; and if either 𝑥 = 0 or 𝑦 = 0, then 𝑥𝑦 = 0.

If 𝑥 2 = 𝑘 and 𝑘 ≥ 0, then 𝑥 = ±√𝑘.

If 𝑥 2 + 𝑘𝑥 = 𝑑, then add (2 ) to each side.

If 𝑥 2 + 𝑏𝑥 + 𝑐 = 0 and 𝑥 ≠ 0, then 𝑥 =
𝑘 2
SY 2021-2022 |Mathematics |Grade 9 | First Quarter
−𝑏±√𝑏2 −4𝑎𝑐
.
2𝑎
Quiz 1
General Instruction: Give what is asked in the given problems.
Test I: Make each expression a perfect square trinomial. Then, factor the resulting perfect
square trinomial.
1. 𝑥 2 + 6𝑥
3
2. 𝑥 2 + 4 𝑥
Test II: Transform each equation in standard form. Then, identify the values of 𝑎, 𝑏, and 𝑐.
3. 6𝑥 2 = −𝑥 + 1
4. −12𝑥 = 3 − 5𝑥 2
Test III: Apply the Square Root Property in each quadratic equation.
5. 2𝑥 2 − 19 = −1
6. 2(𝑥 − 3)2 = 18
Test IV: Solve the following quadratic equations by factoring.
7. 𝑥 2 − 16 = 0
8. 2𝑥 2 − 6 = 4𝑥
9. 𝑥(𝑥 − 4) − 2 = 43
Test V: Solve the following quadratic equations by completing the square.
10. 𝑥 2 − 2𝑥 − 8 = 0
11. 𝑥 2 + 2𝑥 − 15 = 0
12. 3𝑥 2 = 12𝑥
Test VI: Solve the following quadratic equations using the quadratic formula.
13. 𝑥 2 − 3𝑥 + 2 = 0
14. 6𝑥 2 = −𝑥 + 1
15. 2𝑥 2 = −7𝑥 − 3
SY 2021-2022 |Mathematics|Grade 9 | First Quarter
Page | 21
WORKSHEET
Name
:
Lesson No. 1
Grade Level/Section
:
Quadratic Equations
Teacher:
: Rene B. Tinquilan, LPT, MAED
Judy-Ann B. Morales, LPT
Reminder: Please write your final answer/s on the space provided. Write your
solution/s on the pages provided for the solution. If the given pages for writing your
solution is not enough, you may use intermediate pad and don’t forget to staple it
together with this worksheets.
Warm Up:
1. ___________ 2. ___________ 3. ___________ 4. ___________
Questions:
1. ______________________________________________________________
______________________________________________________________
______________________________________________________________
2. ______________________________________________________________
______________________________________________________________
______________________________________________________________
3. ______________________________________________________________
______________________________________________________________
______________________________________________________________
Think About This: (Lesson 1.1)
Reflect:
1. ___________________________________________________________
___________________________________________________________
___________________________________________________________
2. ___________________________________________________________
___________________________________________________________
___________________________________________________________
3. ___________________________________________________________
___________________________________________________________
___________________________________________________________
Lesson 1.2
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
Page | 22
SY 2021-2022 |Mathematics |Grade 9 | First Quarter
Lesson 1.3
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
Check Your Progress:
1. ______________________________________________________________
______________________________________________________________
______________________________________________________________
2. ______________________________________________________________
______________________________________________________________
3. ______________________________________________________________
______________________________________________________________
4. ______________________________________________________________
______________________________________________________________
5. ______________________________________________________________
______________________________________________________________
6. ______________________________________________________________
______________________________________________________________
7. ______________________________________________________________
______________________________________________________________
8. ______________________________________________________________
______________________________________________________________
9. ______________________________________________________________
______________________________________________________________
10. ______________________________________________________________
______________________________________________________________
11. ______________________________________________________________
______________________________________________________________
12. ______________________________________________________________
______________________________________________________________
13. ______________________________________________________________
______________________________________________________________
14. ______________________________________________________________
______________________________________________________________
15. ______________________________________________________________
______________________________________________________________
SY 2021-2022 |Mathematics|Grade 9 | First Quarter
Page | 23
Activity 1
1. Draw your figure here:
2. _________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
3. _________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
4. _________________________________________________________
_________________________________________________________
_________________________________________________________
5. _________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
Parent’s /Guardian’s Full Name and Signature
Page | 24
SY 2021-2022 |Mathematics |Grade 9 | First Quarter
Write your solutions here.
SY 2021-2022 |Mathematics|Grade 9 | First Quarter
Page | 25
Write your solutions here.
Page | 26
SY 2021-2022 |Mathematics |Grade 9 | First Quarter
Write your solutions here.
SY 2021-2022 |Mathematics|Grade 9 | First Quarter
Page | 27
Write your solutions here.
Page | 28
SY 2021-2022 |Mathematics |Grade 9 | First Quarter
Write your solutions here.
SY 2021-2022 |Mathematics|Grade 9 | First Quarter
Page | 29
Write your solutions here.
Page | 30
SY 2021-2022 |Mathematics |Grade 9 | First Quarter
The expression 𝑏 2 − 4𝑎𝑐 which occurs under the
radical sign of the quadratic formula plays important role
in describing and differentiating the characteristics of a
given quadratic function. This expression is called
discriminant.
Learning Target
:
At the end of the lesson,
1. The learners will be able to know:
a. Nature of Roots of Quadratic Equation
b. Writing Equations from Solutions
c. Solving Equations in Quadratic Form
2. The learners will be able to:
a. characterize the roots of a quadratic equation using the discriminant
b. describe the relationship between the coefficients and the roots of a quadratic
equation
c. solve equations transformable to quadratic equations (including rational algebraic
equations)
Below are some words that will help you understand the lesson quickly. Check them out!
Words
Meaning
Words
Meaning
Real
Number
It is any positive or negative
number. This includes all integers
and all rational and irrational
numbers.
Complex Numbers
It is a number that can
be expressed in the
form 𝑎 + 𝑏𝑖 ,
where 𝑎 and 𝑏 are real
numbers,
and 𝑖 is
a
solution
of
the
equation 𝑥 2 = −1.
Discriminant It is the radicand 𝑏 2 − 4𝑎𝑐 in the
quadratic formula, 𝑥 =
−𝑏±√𝑏 2 −4𝑎𝑐
2𝑎
.
SY 2021-2022 |Mathematics|Grade 9 | First Quarter
Page | 31
Using your calculator, give the value of the following radicals. Round off your answers to the
nearest hundredths. Then, answer the questions that follow.
1.
2.
3.
4.
√12
√−1
√10
√−5
Questions:
1. Did your calculator give you definite answers to all radicals? Why do you think so?
2. Do you think the value of the square root of a number depends on the sign of the
radicand (the number inside the square root sign)? Explain your answer.
3. What do you think is the meaning of the answer given by the calculator for numbers 2
and 4?
Roots of Quadratic Equations
In the quadratic formula, 𝑥 =
−𝑏±√𝑏2 −4𝑎𝑐
,
2𝑎
2
the radicand 𝑏 2 − 4𝑎𝑐 is called the
discriminant. By knowing the value of 𝑏 − 4𝑎𝑐, we can discriminate among the possible
number and types of solutions of a quadratic equation.
The table below shows the possible values of the discriminant and the number and
types of solutions of the equation.
Discriminant
𝟐
𝒃 − 𝟒𝒂𝒄
Positive
Zero
Negative
Number and Types of Solutions
Two real solutions
One real solution
Two complex but not real solution.
Example 1:
Use the discriminant to determine the number and type of solutions of each quadratic equation.
1. 𝑥 2 − 8𝑥 + 16 = 0
2. 𝑥 2 − 4𝑥 − 1 = 0
3. 8𝑥 2 + 5 = 0
Page | 32
SY 2021-2022 |Mathematics |Grade 9 | First Quarter
Solutions:
1. 𝑥 2 − 8𝑥 + 16 = 0
Find the value of 𝑎, 𝑏, and 𝑐.
𝑎 = 1, 𝑏 = −8, 𝑐 = 16
2
2
𝑏 − 4𝑎𝑐 = (−8) − 4(1)(16)
Substitute the value of 𝑎, 𝑏, and 𝑐 to 𝑏 2 − 4𝑎𝑐.
2
𝑏 − 4𝑎𝑐 = 64 − 64
Evaluate.
𝑏 2 − 4𝑎𝑐 = 0
Since 𝑏 2 − 4𝑎𝑐 = 0, this quadratic equation has one real solution.
2. 𝑥 2 − 4𝑥 − 1 = 0
Find the value of 𝑎, 𝑏, and 𝑐.
𝑎 = 1, 𝑏 = −4, 𝑐 = −1
2
2
𝑏 − 4𝑎𝑐 = (−4) − 4(1)(−1)
Substitute the value of 𝑎, 𝑏, and 𝑐 to 𝑏 2 − 4𝑎𝑐.
2
𝑏 − 4𝑎𝑐 = 16 + 4
Evaluate.
𝑏 2 − 4𝑎𝑐 = 20
Since 𝑏 2 − 4𝑎𝑐 = 20, this quadratic equation has two real solutions.
3. 8𝑥 2 + 5 = 0
Find the value of 𝑎, 𝑏, and 𝑐.
𝑎 = 8, 𝑏 = 0, 𝑐 = 5
2
2
𝑏 − 4𝑎𝑐 = (0) − 4(8)(5)
Substitute the value of 𝑎, 𝑏, and 𝑐 to 𝑏 2 − 4𝑎𝑐.
2
𝑏 − 4𝑎𝑐 = −160
Evaluate
.
Since 𝑏 2 − 4𝑎𝑐 = −160, this quadratic
equation has two complex but no real
solutions.
Math Note
Writing Equations from Solutions
By the principle of zero products, we know that (𝑥 − 1)(𝑥 + 4) = 0 has solutions 1 and
4. Suppose we want two numbers to be solutions of an equation, we can apply the principle of
zero products in reverse to find an equation.
Example 2:
Find an equation for which the given numbers are solutions.
1. −5 and 1
4
2. 2 and − 5
SY 2021-2022 |Mathematics|Grade 9 | First Quarter
Page | 33
Solutions:
1. −5 and 1
𝑥 = −5 or 𝑥 = 1
𝑥 + 5 = 0 or 𝑥 − 1 = 0
(𝑥 + 5)(𝑥 − 1) = 0
𝑥 2 + 5𝑥 − 𝑥 − 5 = 0
𝑥 2 + 4𝑥 − 5 = 0
4
2. 2 and − 5
4
𝑥 = 2 or 𝑥 = − 5
(𝑥 − 2) (𝑥 +
2
4
+ 𝑥
5
2
Equate both equations to zero.
Apply the principle of zero products. Then, multiply.
Combine like terms.
Equate the given solutions to 𝑥.
4
5
𝑥 − 2 = 0 or 𝑥 + = 0
4
)
5
Equate the given solutions to 𝑥.
Equate both equations to zero.
=0
Apply the principle of zero products. Then, multiply.
8
5
Multiply both sides by 5 to clear the fractions.
𝑥
− 2𝑥 − = 0
5𝑥 + 4𝑥 − 10𝑥 − 8 = 0
5𝑥 2 − 6𝑥 − 8 = 0
Combine like terms.
Alternate Solution: Another way of clearing the equation of fractions is by multiplying
4
5
𝑥 + = 0 by 5 before using the principle of zero product.
4
2 and − 5
4
𝑥 = 2 or 𝑥 = − 5
Equate the given solutions to 𝑥.
4
5
𝑥 − 2 = 0 or 𝑥 + = 0
𝑥 − 2 = 0 or 5𝑥 + 4 = 0
5𝑥 2 + 4𝑥 − 10𝑥 − 8 = 0
5𝑥 2 − 6𝑥 − 8 = 0
1.
2.
4
Equate both equations to zero. Multiply 𝑥 + = 0 by 5.
5
Equate both equations to zero.
Combine like terms.
Equations in Quadratic Form
Some equations may appear complicated but when transformed into a quadratic form, their
solution is facilitated.
Any equation of the form 𝑎𝑓 2 + 𝑏𝑓 + 𝑐 = 0, where 𝑓is any algebraic expression, may be
described as an equation in quadratic form.
Example 3:
Solve the following equations for 𝑥 by factoring.
1. 9𝑥 4 − 52𝑥 2 + 64 = 0
2. (2𝑥 − 1)2 − 6(2𝑥 − 1) = −8
Page | 34
SY 2021-2022 |Mathematics |Grade 9 | First Quarter
Solutions:
These equations can be solved by using simple substitution. The expressions to be substituted
are generally found by looking at the first-term degree.
1. 9𝑥 4 − 52𝑥 2 + 64 = 0
Rewrite the original equation. Transform the first-term degree
9(𝑥 2 )2 − 52(𝑥 2 ) + 64 = 0
𝑥 4 to (𝑥 2 )2 . Then, let 𝑓 = 𝑥 2 .
9𝑓 2 − 52𝑓 + 64 = 0
(𝑓 − 4)(9𝑓 − 16) = 0
𝑓 − 4 = 0 or 9𝑓 − 16 = 0
16
𝑓=4
or 𝑓 = 9
Then, solve for 𝑥.
16
𝑥 2 = 𝑓 = 4 or 𝑥 2 = 𝑓 = 9
or 𝑥 =
𝑥 = ±√4
or 𝑥 = ±√ 9
𝑥 = ±2
or 𝑥 = ± 3
2
Factor the equation.
Use the principle of zero products.
Since 𝑓 = 𝑥 2 , then, 𝑥 2 = 4 or 𝑥 2 =
16
9
𝑥 =4
2
Substitute 𝑓 to all 𝑥 2 .
16
9
. Then, solve for 𝑥.
16
4
4 4
Hence, the roots are −2, − 3 , 3 , 2.
2. (2𝑥 − 1)2 − 6(2𝑥 − 1) = −8
𝑔2 − 6𝑔 = −8
Rewrite the original equation. Let, 𝑔 = 2𝑥 − 1. Then,
substitute 𝑔 for all 2𝑥 − 1.
𝑔2 − 6𝑔 + 8 = 0
Write the quadratic equation in standard form.
(𝑔 − 4)(𝑔 − 2) = 0
Factor the equation
𝑔 − 4 = 0 or 𝑔 − 2 = 0
Use the principle of zero products.
𝑔=4
or 𝑔 = 2
Then, solve for 𝑥.
Since 𝑔 = 2𝑥 − 1, then, 2𝑥 − 1 = 4 or 2𝑥 − 1 = 2. Then, solve
2𝑥 − 1 = 𝑔 = 4 or 2𝑥 − 1 = 𝑔 = 2
for 𝑥.
2𝑥 − 1 = 4
or 2𝑥 − 1 = 2
2𝑥 = 5
or 2𝑥 = 3
5
3
𝑥=
or 𝑥 =
2
Hence, the roots are
5
2
2
3
and 2.
Math FYI:
In 300 B.C., Euclid used geometric
construction in solving quadratic equations. In
the year 1000, Arab mathematicians had
developed their way of writing quadratic
equation.
SY 2021-2022 |Mathematics|Grade 9 | First Quarter
Page | 35
Unlike the discriminant we used in this lesson, to discriminate
against individuals or groups of people defiles human dignity.
Discrimination is still rampant today, and it is a lifetime goal to
dismantle it. One of the prominent names in history who fight against
racial discrimination is Nelson Mandela. Read his life at
https://www.biography.com/political-figure/nelson-mandela.
What are the similarities and differences between discriminant
and discrimination?
In your simples ways, how can you end discrimination in
society?
Activity 2
Use the discriminant to determine the number and types of solutions of each quadratic
equation.
1. 𝑥 2 + 4𝑥 + 5 = 0
2. 4𝑥 2 + 20𝑥 + 25 = 0
3. 2𝑥 2 + 7𝑥 − 15 = 0
Four-Pronged
Integration
ICV/RV
Page | 36
Behavioral Indicators
Points to Ponder
Service/ Stewardship
In the topic on discriminant you were
able to determine the nature of roots
Respects and nurtures the without
solving
the
equation.
giftedness in others to Discriminant helps you to classify the
promote the growth of roots, and you may use this to check if
persons and communities
the solution is correct. However, the
word discriminate, which the word
discriminant
is
derived
from,
oftentimes used to racially oppress
individuals or group of people. As
Ignatian-Marian student, we have to
always remember that despite our
differences in race, color, or social
status, we need to respect one another.
SY 2021-2022 |Mathematics |Grade 9 | First Quarter
Social
Orientation
Lesson Across
Discipline
Biblical Text
In this way, we nurture ourselves into
becoming productive, peaceful, and
God-loving citizens.
Discrimination on
Racial discrimination is still rampant in
Indigenous People
our society. In our local context, we
oftentimes use the term Mandaya to
refer to ignorant persons. In doing so,
we degrade our tribe and teach other
Sociology (Indigenous
people to treat our tribe as such. This
People’s Protection)
practice should end. We should be
proud of our tribe and we should try our
best to protect, preserve and promote
its tradition and cultures.
Galatians 3:28
In this Bible text, we are one in the
name of Jesus Christ. The Christian
“There is neither Jew nor community does not discriminate. Thus
Greek, there is neither slave it is just and right if we treat each other
nor free, there is no male and brothers and sisters in Christ.
female, for you are all one in
Christ Jesus.”
The number of real solutions to 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 with 𝑎 ≠ 0 can be
found by evaluating the discriminant , 𝑏 2 − 4𝑎𝑐.
1. If 𝑏 2 − 4𝑎𝑐 > 0, there are two real solutions.
2. If 𝑏 2 − 4𝑎𝑐 = 0, there is one real solution.
3. If 𝑏 2 − 4𝑎𝑐 < 0, there are no real solutions. Instead, there are two
complex solutions.
SY 2021-2022 |Mathematics|Grade 9 | First Quarter
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Quiz 2
General Instruction: Give what is asked in the given problems.
Test I: Without solving the equation, determine the nature of its roots.
1. 𝑥 2 − 𝑥 − 12 = 0
2. 6𝑥 2 − 3 = 8𝑥
3. 7 + 5𝑥 − 2𝑥 2 = 0
Test II: Write in standard form a quadratic equation whose solution set is given.
4. {−6,1}
5. {4,4}
1 1
6. {− 3 , 2}
Test III: Solve for 𝑥 in each equation.
7. 𝑥 4 − 17𝑥 2 + 16 = 0
8. 𝑥 4 + 33 = 14𝑥 2
9. (𝑥 2 − 7)2 + (𝑥 2 − 7) = 6
10. (𝑥 2 − 4𝑥)2 + 7(𝑥 2 − 4𝑥) = −12
Page | 38
SY 2021-2022 |Mathematics |Grade 9 | First Quarter
WORKSHEET
Name
:
Lesson No. 2
Grade Level/Section
:
Nature of Roots of
Quadratic Equations
Teacher:
: Rene B. Tinquilan, LPT, MAED
Judy-Ann B. Morales, LPT
Reminder: Please write your final answer/s on the space provided. Write your
solution/s on the pages provided for the solution. If the given pages for writing your
solution is not enough, you may use intermediate pad and don’t forget to staple it
together with this worksheets.
Warm Up
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Check Your Progress:
1. ______________________________________________________________
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2. ______________________________________________________________
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3. ______________________________________________________________
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Activity 2
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SY 2021-2022 |Mathematics |Grade 9 | First Quarter
Write your solutions here.
SY 2021-2022 |Mathematics|Grade 9 | First Quarter
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Write your solutions here.
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SY 2021-2022 |Mathematics |Grade 9 | First Quarter
Write your solutions here.
SY 2021-2022 |Mathematics|Grade 9 | First Quarter
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Write your solutions here.
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SY 2021-2022 |Mathematics |Grade 9 | First Quarter
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Write your solutions here.
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SY 2021-2022 |Mathematics |Grade 9 | First Quarter
Recall that a word problem describes a situation
that involves both known and unknown quantities, and
certain relations between the quantities. A number of
stated problems, which deal with products or quotients
involving physical quantities, lead to quadratic
equations.
Learning Target
:
At the end of the lesson,
1. The learners will be able to know:
a. Application of Quadratic Equation
2. The learners will be able to:
a. solve problems involving quadratic equations and rational algebraic equations.
Solving real-life problems, like
personal or family problems, requires much
effort. To know, however, if your effort is
leading to positive and productive results
depend on several factors. One of the
factors is being systematic. Unplanned,
hasty solutions may make the problem at
hand worse, thus making the situation much
difficult, and the problem remains unsolved.
In the same way, solving word
problems requires careful analysis and a
systematic approach. Do you still recall how
you solve the last word problem given to
you by your math teacher? Did you follow a
particular set of steps in solving? Is it
effective? What is the importance of being
systematic in solving a word problem?
SY 2021-2022 |Mathematics|Grade 9 | First Quarter
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In this lesson, we will consider Polya’s 4step problem-solving process. We, however, will
make use of different wordings here, though they
are essentially the same with the original Polya’s
4 steps. These steps are:
1. Understand the Problem. In this step,
you will give what is asked in the
problem and enumerate the given data.
2. Write the Equation. In this step, you
will write the mathematical equation to
be used to solve the problem.
3. Solve the Equation. After writing the
mathematical equation, you can now
solve the equation based on the given
data.
4. Check. Look back at the problem if
the solution arrived is logical and sound.
Example 1:
Two numbers differ by 9. The sum of their squares is 653. What are the numbers?
Solution:
1. Understand the Problem
What is asked in the problem?
The numbers (both smaller and larger numbers).
What are the given data?
Let 𝑥 be the smaller number.
Let 𝑥 + 9 be the larger number.
2. Write the Equation
𝑥 2 + (𝑥 + 9)2 = 653
Assign a variable for a given quantity. In this case let us use 𝑥.
Since the smaller and larger number differ by 9, then the larger
number must be 9 more than the smaller number.
After assigning variables, follow the conditions set in the
problem. According to the problem, each number must be
squared before added. Then, equate their sum to 653.
3. Solve the Equation
𝑥 2 + (𝑥 + 9)2 = 653
𝑥 2 + 𝑥 2 + 18𝑥 + 81 = 653
𝑥 2 + 𝑥 2 + 18𝑥 + 81 − 653 = 0
2𝑥 2 + 18𝑥 − 572 = 0
𝑥 2 + 9𝑥 − 286 = 0
(𝑥 + 22)(𝑥 − 13) = 0
Expand the equation
Equate the equation to 0.
Combine like terms.
Divide all terms by 2.
Factor the equation. You may use another methods if you want.
Applying the principle of zero products, solve for 𝑥.
𝑥 + 22 = 0
or
𝑥 − 13 = 0
𝑥 = −22
or
𝑥 = 13
If 𝑥 = −22, then 𝑥 + 9 = −13
If 𝑥 = 13, then 𝑥 + 9 = 22.
The numbers are −22 and 13, or 13 and 22.
4. Check
−13 − (−22) = 9 and 22 − 13 = 9
(−22)𝟐 + (−13)𝟐 = 653 and (13)2 + (22)2 = 653
Page | 48
The difference of two numbers is 9.
The sum of their squares is 653
SY 2021-2022 |Mathematics |Grade 9 | First Quarter
Example 2:
A 13-meter ladder is leaning against the side of a
building and is positioned such that the base of the ladder is 5
meters from the base of the building. How far above the ground
is the point where the ladder touches the building?
Solution:
1. Understand the Problem
What is asked in the problem?
The length of the part of the building
from the ground to the point where the
ladder touches the building.
𝑏
Math Note
Pythagorean Theorem
What are the given data?
Since the ladder formed a right triangle
when lean against the building, we can
use the Pythagorean Theorem.
Let 𝑐 be the hypothenuse or the length
of the ladder. 𝑐 = 13m
Let 𝑏 be the base of the right triangle
formed. 𝑏 = 5
Let 𝑎 be the altitude or the height.
Here, 𝑎 is unknown.
2. Write the Equation
𝑐 2 = 𝑎2 + 𝑏 2
Since the problem is about a right triangle, consider
using the Pythagorean Theorem.
𝑎2 = 𝑐 2 − 𝑏 2
Derive 𝑎2 from the Pythagorean Theorem since 𝑎 is the
unknown of the problem. This will serve as the
mathematical equation of the problem.
3. Solve the Equation
𝑎2 = 𝑐 2 − 𝑏 2
Substitute the values of 𝑐 and 𝑎.
𝑎2 = (13)2 − (5)2
2
Evaluate.
𝑎 = 169 − 25
2
𝑎 = 144
Apply the square root methof of finding the roots.
√𝑎2 = ±√144
𝑎 = ±12
Therefore, the length of the part of the building from the ground to the point where the
ladder touches the building is 12 meters.
4. Check
𝑐 2 = 𝑎2 + 𝑏 2
(13)2 = (12)2 + (5)2
169 = 144 + 25
169 = 169
Think About This:
In this problem, the roots are 12 and −12. Why did
we discard −12 as a solution? Why did we only
consider 12 as the solution to the problem?
SY 2021-2022 |Mathematics|Grade 9 | First Quarter
Page | 49
Math Note
Math FYI:
Pythagoras, more accurately known as
Pythagoras of Samos, was best known as a
Greek mathematician. His famous theorem,
the Pythagorean theorem, is believed to
have been used by the ancient Babylonians
and the Indians, although their use of this is
based on evidence of the understanding of
the principle, and not on recorded writings.
Activity 3
Suppose the picture of Archimedes measuring 6 cm by 8 cm, has a frame of uniform width
a total area equal to the area of the picture, how wide is the frame?
1. Draw a sketch.
2. Represent the problem with a variable.
3. Find the relation that exists between the area of the picture together with the
frame, and the area of the picture.
4. Solve the working equation.
Four-Pronged
Integration
Ignacian Core
Values/ Related
Values (ICV/RV)
Behavioral Indicators
Points to Ponder
Excellence/
Resourcefulness
Problem solving in mathematics,
particularly
involving
quadratic
equation,
entails
efforts,
critical
thinking, and responsibilities. The skill
in solving problem helps us to deal with
problems we encounter, whether it is
Seeks continual improvement
in performing tasks and
responsibilities
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SY 2021-2022 |Mathematics |Grade 9 | First Quarter
Social
Orientation
Dealing with problems
Biblical Text
Philippians 4:13
“I can do all things through
him who strengthens me.”
small or big. It is important to note that
solving problems is fun and it helps us
continuously improve in performing our
tasks and responsibilities.
You can deal your problems easily if
you have skills in problem solving. Real
word problems in mathematics, like in
this lesson on quadratic equation, can
be solved easily using the Polya’s
method. In life, we have to follow the
concept of Polya’s. In any problems in
life that we need to solve, we need to
identify the problem, devise ways to
solve it, and execute the plan. And if the
proposed solution does not work, repeat
the process. In the end, our character is
determined not only on getting the
solutions to our problem, but also on the
way we solve the problem.
Though we need our skills in problem
solving to find solutions to our
problems, as Ignatian-Marian student,
we need the guidance of God in
everything we do. Thus, when dealing
with problems in life, we must always
ask God for enlightenment and
strengths.
Polya’s 4 steps. These steps are:
1. Understand the Problem. In this step,
you will give what is asked in the
problem and enumerate the given data.
2. Write the Equation. In this step, you
will write the mathematical equation to
be used to solve the problem.
3. Solve the Equation. After writing the
mathematical equation, you can now
solve the equation based on the given
data.
4. Check. Look back at the problem if
the solution arrived is logical and sound.
SY 2021-2022 |Mathematics|Grade 9 | First Quarter
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Quiz 3
General Instruction: Solve each problem involving quadratic equations.
1. The sum of two numbers is 16, and the sum of their squares is 146. Find the two
numbers.
2. The base of a right triangle is 1 dm longer than its altitude. Its area is 6 dm2. Find the
1
lengths of the base and the altitude. [Note: The 𝐴𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 = 2 𝑏ℎ, where 𝐴 is the area, 𝑏
is the base, and ℎ is the height or altitude.]
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SY 2021-2022 |Mathematics |Grade 9 | First Quarter
WORKSHEET
Name :
Grade Level/Section :
Teacher
Lesson No. 3
Problems Involving
Quadratic Equations
: Rene B. Tinquilan, LPT, MAED
Judy-Ann B. Morales, LPT
Warm Up
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SY 2021-2022 |Mathematics |Grade 9 | First Quarter
Write your solutions here.
SY 2021-2022 |Mathematics|Grade 9 | First Quarter
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Write your solutions here.
Parent’s /Guardian’s Full Name and Signature
Page | 56
SY 2021-2022 |Mathematics |Grade 9 | First Quarter
Inequalities are used to represent various realworld situations in which a quantity must fall within a
range of possible values. For example, figure skaters and
gymnasts frequently want to know what they need to
score to win a competition. That score can be represented
by an inequality.
Learning Target
:
At the end of the lesson,
1. The learners will be able to know:
a. Solving Quadratic Inequalities
b. Application of Quadratic Inequalities
2. The learners will be able to:
a. illustrates quadratic inequalities
b. solves quadratic inequalities
c. solves problems involving quadratic inequalities
Match the linear inequality with its graph.
A
B.
1. 𝑥 > 3
∗
∗
2. 𝑥 < 2
∗
∗
3. 𝑥 ≥ −1
∗
∗
4. 𝑥 ≤ 8
∗
∗
SY 2021-2022 |Mathematics|Grade 9 | First Quarter
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1.
2.
Equations in Quadratic Form
A quadratic equation can be written as 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0, with 𝑎 ≠ 0. It becomes a
quadratic inequality when the equality symbol is replaced by <, >, ≤ or ≥. A solution of a
quadratic inequlity in one variable is a value of the variable that makes the inequality a true
statement.
Example 1:
Solve 𝑥 2 − 2𝑥 − 3 < 0.
Solution:
To solve the inequality 𝑥 2 − 2𝑥 − 3 < 0, we are looking for all values that make the expression
𝑥 2 − 2𝑥 − 3 less than 𝟎 or negative.
Write the related equation.
𝑥 2 − 2𝑥 − 3 = 0
Factor the equation.
(𝑥 + 1)(𝑥 − 3) = 0
𝑥+1=0
𝑥−3=0
Apply the Principle of Zero Product
𝑥 = −1
𝑥=3
So, (𝑥 + 1)(𝑥 − 3) is 0 when 𝑥 = −1 or 𝑥 = 3.
Divide the number line into two regions with these two numbers.
A
B
𝑥 < −1
−1 < 𝑥 < 3
C
𝑥>3
To see whether the inequality 𝑥 2 − 2𝑥 − 3 < 0 is true or false in each region, choose a test
point from each region and substitute its value for 𝑥 in the inequality 𝑥 2 − 2𝑥 − 3 < 0. If the
resulting inequality is true, then the region containing the test point is a solution region.
Region
Page | 58
A
Test Point Value
−2
B
0
C
4
(𝑥 + 1)(𝑥 − 3) < 0
(−2 + 1)(−2 − 3) < 0
(−1)(−5) < 0
5<0
(0 + 1)(0 − 3) < 0
(1)(−3) < 0
−3 < 0
(4 + 1)(4 − 3) < 0
(5)(1) < 0
5<0
Result
SY 2021-2022 |Mathematics |Grade 9 | First Quarter
False
True
False
The values in Region B satisfy the inequality. Since the inequality symbol is <, the number −1
and 3 are not included in the solution set.
Thus, the solution set is −1 < 𝑥 < 3.
Example 2
Solve 𝑥 2 + 𝑥 − 2 ≥ 0.
Solution:
Write the related equation.
𝑥2 + 𝑥 − 2 = 0
(𝑥 + 2)(𝑥 − 1 = 0)
𝑥+2=0
𝑥 = −2
Factor the equation.
𝑥−1=0
𝑥=1
Apply the Principle of Zero Products.
Then, divide the number line into 3 regions.
A
B
C
𝑥 ≤ −2
−2 < 𝑥 < 1
𝑥≥1
Choose test points for each region.
Region
A
Test Point Value
−3
B
0
C
2
(𝑥 + 2)(𝑥 − 1) ≥ 0
(−3 + 2)(−3 − 1) ≥ 0
(−1)(−4) ≥ 0
−4 ≥ 0
(0 + 2)(0 − 1) ≥ 0
(2)(−1) ≥ 0
−2 ≥ 0
(2 + 2)(2 − 1) ≥ 0
(4)(1) ≥ 0
4≥0
Result
True
False
True
The values in Regions A and C satisfy the inequality. Since the inequality symbol is ≥, the
numbers −2 and 1 are included in the solution set.
Thus, the solution set is 𝑥 ≤ −2 and 𝑥 ≥ 1.
SY 2021-2022 |Mathematics|Grade 9 | First Quarter
Page | 59
Application of Quadratic Inequalities
Example 3:
The profit 𝑃 that the company earns for selling 𝑥 number of toy cars can be modeled by
𝑃(𝑥) = −25𝑥 2 + 1000𝑥 − 3000.
How many toy cars must be sold for a profit of at
least Php 5000?
Solution:
1. Understand the Problem
What is asked in the problem?
The number of toys to be sold to earn
Php 5000.
What are the given data?
Target profit is Php 5000.
2. Write the Equation
Since the profit 𝑃(𝑥) has to be at least
Php 5000, then write the quadratic inequality
as
−25𝑥 2 + 1000𝑥 − 3000 ≥ 5000.
3. Solve the Equation
−25𝑥 2 + 1000𝑥 − 3000 = 5000
−25[𝑥 2 − 40𝑥 + 120] = 5000
𝑥 2 − 40𝑥 + 120 = −200
𝑥 2 − 40𝑥 + 120 + 200 = 0
𝑥 2 − 40𝑥 + 320 = 0
Write the related equation.
Factor out −25.
Divide both sides by −25.
Add both sides by 200 (APE). Then,
simplify.
Solving 𝑥 2 − 40𝑥 + 320 = 0 using the quadratic formula, approximately, we get
𝑥 = 28.94 or 𝑥 = 11.06.
Thus,
A
B
𝑥 ≤ 11.06
𝑥 ≥ 28.94
11.06 < 𝑥 < 28.94
11.06
Region
Page | 60
C
Test Point
28.94
−25𝑥 2 + 1000𝑥 − 3000 ≥ 5000
SY 2021-2022 |Mathematics |Grade 9 | First Quarter
Result
A
B
C
Value
5
15
30
1375 ≥ 5000
6375 ≥ 5000
4500 ≥ 5000
False
True
False
The range of possible values of 𝑥 is 11.06 < 𝑥 < 28.94. Since we are talking of number of toy
cars (which must be a whole number), the range of possible values of 𝑥 is 12 ≤ 𝑥 ≤ 28. Thus,
to make a profit of at least Php 5000, at least 12 and at most 28 toy cars must be sold by the
company.
Activity 4
Solve the following problem:
The total profit function 𝑃(𝑥) for a company producing 𝑥 thousand of pens is given by
𝑃(𝑥) = 2𝑥 2 + 26𝑥 + 1320.
Find the values of 𝑥 for which the company makes a profit.
[Hint: The company makes a profit when 𝑃(𝑥) > 0.]
Four-Pronged
Integration
ICV/RV
Behavioral Indicators
Points to Ponder
Excellence/Competence
Like solving problems in quadratic
inequalities, we really need to acquire
the basic knowledge and skills so that
we can do efficiently our work or task.
The knowledge of solving quadratic
inequalities are needed in many
aspects in our lives especially for those
who
work
in
engineering
and
architecture.
Those
architect
or
engineers
must
possess
great
competence in their job so as not to
compromise the project they are
working on.
Pursues high achievement
standards in everything one
does
SY 2021-2022 |Mathematics|Grade 9 | First Quarter
Page | 61
Social
Orientation
Lesson Across
Discipline
Biblical Text
Government’s way of handling their
structural projects, like #BuildBuildBuild
(BBB) by the Duterte Administration can
make or break the nation. If they do
their job with full of accountability and
honesty, then the quality of that project
is almost guaranteed. But, if the project
is full of corruption, then the people,
whom they serve, will suffer.
Architecture
Engineers and architects have applied
the concepts of quadratic equation to
create magnificent buildings and other
structures.
Genesis 6:14-16
God Himself is a great engineer and
architect of a legendary structure that
“Make for yourself an ark of saved the human and other species
gopher wood; you shall make from extinction.
the ark with rooms, and shall
cover it inside and outside
with pitch. This is how you
shall make it: the length of the
ark three hundred cubits, its
breadth fifty cubits, and its
height thirty cubits.”
Government Structural
Projects
Quiz 4
Solve these Problems.
1. An object tossed downward with an initial
m
speed (𝑉0 ) of 9.81 s will travel a distance of
of 𝑑 meters, where 𝑑 = 4.9𝑡 2 + 𝑉0 𝑡 and 𝑡 is
measured in seconds. Suppose an object is
dropped from a helicopter at an altitude of
75 m. Approximately how long does it take
the coin to reach the ground?
2. A projectile is fired straight up from the
ground with an initial velocity of 80 feet per
second. Its height 𝑑(𝑡) in feet at anytime 𝑡 by
the function 𝑑(𝑡) = −16𝑡 2 + 80𝑡. Find the
interval of time for which the height of the
projectile is greater than 96 feet.
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SY 2021-2022 |Mathematics |Grade 9 | First Quarter
WORKSHEET
Name :
Lesson No. 4
Grade Level/Section :
Teacher
Quadratic Inequalities
: Rene B. Tinquilan, LPT, MAED
Judy-Ann B. Morales, LPT
Warm Up
1. ______
2. ______
3. ______
4. ______
Check Your Progress
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Write your solutions here.
Page | 64
SY 2021-2022 |Mathematics |Grade 9 | First Quarter
Write your solutions here.
Parent’s /Guardian’s Full Name and Signature
SY 2021-2022 |Mathematics|Grade 9 | First Quarter
Page | 65
Learning Target
:
At the end of the lesson,
1. The learners will be able to know:
a. Domain and Range of Relations and Functions
b. The Graph of 𝑓(𝑥) = 𝑎(𝑥 − ℎ)2 + 𝑘.
c. More on Graphing Quadratic Functions
The learners will be able to:
a. model real-life situations using quadratic functions
b. represent a quadratic function using: (a) table of values; (b) graph; and (c)
equation
c. transform the quadratic function defined by 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 into the form 𝑦 =
𝑎(𝑥 − ℎ)2 + 𝑘
d. graph a quadratic function: (a) domain; (b) range; (c) intercepts; (d) axis of
symmetry; (e) vertex; (f) direction of the opening of the parabola
e. analyze the effects of changing the values of 𝑎, ℎ and 𝑘 in the equation 𝑦 =
𝑎(𝑥 − ℎ)2 + 𝑘 of a quadratic function on its graph
f. determine the equation of a quadratic function given: (a) a table of values; (b)
graph; (c) zeros
g. solve problems involving quadratic functions
Domain and Range of Relations and
Functions
A function is a correspondence between a first set called domain and a second set,
called range such that each member of the domain corresponds to exactly one member of
the range.
Domain
Page | 66
Correspondence
SY 2021-2022 |Mathematics |Grade 9 | First Quarter
Range
Example 1:
For each relation, (a) write the domain; (b) write the range, and (c) determine whether the
correspondence is a function.
1. {(2,3), (5,4), (6,4), (7,5)}
2.
𝑥
4
−3
−2
−1
𝑥
−2
4
5
Solution:
1. Domain:
Range:
{2,5,6,7}
{3,4,5}
It is a function since each 𝑥-coordinate or abscissa, has only one 𝑦-coordinate or
ordinate.
2. Domain:
Range:
{−2,4,5}
{4, −3, −2, −1}
It is NOT a function since −2 has two ordinates, 4 and −3.
Graphing 𝒇(𝒙) = 𝒂(𝒙 − 𝒉)𝟐 + 𝒌
You learned that a real number is related to a square of that number and this
relationship can be expressed as 𝑦 = 𝑥 2 . You also learned that this is a function since each to
each real number, there corresponds a square of the number. Hence, you can write 𝑦 = 𝑥 2 as
𝑓(𝑥) = 𝑥 2 . This function is not linear because the 𝑥 2 term is raised to a second degree. It is a
quadratic function.
SY 2021-2022 |Mathematics|Grade 9 | First Quarter
Page | 67
The Graph of 𝒇(𝒙) = 𝒙𝟐
To graph 𝑓(𝑥) = 𝑥 2 , start first with finding the ordered pairs solutions. A simple way to find the
ordered pair is by making a table of values. Let the domain 𝑥 be a set of integers such that
−3 ≤ 𝑥 ≤ −3. We have:
𝑥
−3
−2
−1
0
1
2
3
If 𝑥 = −3,
If 𝑥 = −2,
If 𝑥 = −1,
If 𝑥 = 0,
If 𝑥 = 1,
If 𝑥 = 2,
If 𝑥 = 3,
𝑦 = 𝑓(𝑥) = 𝑥 2
then 𝑦 = 𝑓(𝑥) = (−3)2 or 9
then 𝑦 = 𝑓(𝑥) = (−2)2 or 4
then 𝑦 = 𝑓(𝑥) = (−1)2 or 1
then 𝑦 = 𝑓(𝑥) = (0)2 or 0
then 𝑦 = 𝑓(𝑥) = (1)2 or 1
then 𝑦 = 𝑓(𝑥) = (2)2 or 4
then 𝑦 = 𝑓(𝑥) = (3)2 or 9
𝑦 = 𝑓(𝑥)
9
4
1
0
1
4
9
Quadratic Function
A quadratic function is a function that can be written in the standard form
𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 where 𝑎 ≠ 0. The domain of a quadratic function is all real
numbers.
Thus, the set of ordered pairs is {(−3,9), (−2,4), (−1,1), (0,0), (1,1), (2,4), (3,9)}.
Then, plot the ordered pairs.
Page | 68
Lastly, connect them with a smooth curve
SY 2021-2022 |Mathematics |Grade 9 | First Quarter
All quadratic functions have graphs similar to this graph. This U-shaped curve is called
parabola. It can go upward or downward. The turning point of the graph is called the vertex
of the parabola. In the figure above, the vertex is (0,0). It is the lowest on the graph.
Notice that with the exception of 0, all the other 𝑦-values correspond to two different 𝑥values. For example, 22 = 4 and (−2)2 = 4. As a result, the graph is a mirror image of itself
along the 𝑦-axis. Thus, we can say that the parabola is symmetric with respect to a line that
goes through the center of the parabola and the vertex. This line is called the parabola’s axis
of symmetry.
In a quadratic function, if 𝑎 > 0, the
parabola opens upward. If 𝑎 < 0, the
parabola opens downward. If the parabola
opens downward, the vertex of the parabola is
the highest point. The axis of the parabola is
the vertical line that passes through the vertex.
The Graph of 𝒇(𝒙) = 𝒂𝒙𝟐
The graph of any function of the form 𝑦 = 𝑎𝑥 2 has a vertex of (0,0) and an axis of
symmetry of 𝑥 = 0.
Example 1:
1
Graph 𝑓(𝑥) = 𝑥 2 , 𝑔(𝑥) = 2𝑥 2 and ℎ(𝑥) = 2 𝑥 2 .
Solution:
Each function has different value of 𝑎. To determine the effect of different values of 𝑎 for each
function, let us graph the functions.
1
First, generate the table of values of 𝑔(𝑥) = 2𝑥 2 and ℎ(𝑥) = 2 𝑥 2 . Let the domain 𝑥 be a set of
integers such that −2 ≤ 𝑥 ≤ −2. We will not include anymore the table of values for 𝑓(𝑥) = 𝑥 2
since it was already presented in the preceding discussion.
𝑥
𝑔(𝑥) = 2𝑥 2
𝑥
−2
8
−2
−1
2
−1
0
0
0
1
2
1
2
8
2
SY 2021-2022 |Mathematics|Grade 9 | First Quarter
1
ℎ(𝑥) = 𝑥 2
2
2
1
or 0.5
2
0
1
or 0.5
2
2
Page | 69
The ordered pairs for 𝑔(𝑥) = 2𝑥 2 are (−2,8), (−1,2), (0,0), (1,2), (2,8).
1
1
1
The ordered pairs for ℎ(𝑥) = 𝑥 2 are (−2,2), (−1, ) , (0,0), (1, ) , (2,2).
2
2
𝑔(𝑥) = 2𝑥 2
2
Then, plot the ordered pairs and connect them
with a smooth curve. What can you say about
the three graphs?
𝑓(𝑥) = 𝑥 2
1
ℎ(𝑥) = 𝑥 2
2
Notice that the graph of 𝑔(𝑥) = 2𝑥 2 is
narrower than the graph of 𝑓(𝑥) = 𝑥 2 . Also,
1
the graph of ℎ(𝑥) = 2 𝑥 2 is wider than the
graph of 𝑓(𝑥) = 𝑥 2 .
Example 2:
1
Graph 𝑓(𝑥) = −𝑥 2 , 𝑔(𝑥) = −2𝑥 2 , and ℎ(𝑥) = − 4 𝑥 2 on the same set of axes. Let the domain 𝑥
be a set of integers such that −2 ≤ 𝑥 ≤ −2.
Solution:
First generate table of values for each function.
Page | 70
𝑥
𝑔(𝑥) = −2𝑥 2
(𝑥) = −𝑥 2
−2
−8
−4
−1
−2
−1
0
0
0
1
−2
−1
2
−8
−4
1
ℎ(𝑥) = − 𝑥 2
4
−1
−
1
or 0.25
4
0
−
1
or 0.25
4
−1
SY 2021-2022 |Mathematics |Grade 9 | First Quarter
Next, plot the ordered pairs and connect them with a smooth curve.
Compare the graphs of 𝑓(𝑥) =
𝑥 2 and 𝑓(𝑥) = −𝑥 2. The
parabola formed by 𝑓(𝑥) =
−𝑥 2 is the same shape as
𝑓(𝑥) = 𝑥 2 , but opens
downward.
𝑔(𝑥) = −2𝑥 2
𝑓(𝑥) = −𝑥 2
1
ℎ(𝑥) = 𝑥 2
4
Also, notice that the graph of
𝑓(𝑥) = −𝑥 2 is wider than the
graph of 𝑔(𝑥) = −2𝑥 2 but
narrower than the graph of
1
ℎ(𝑥) = − 𝑥 2 . All graphs open
4
downwards because 𝑎 < 0.
The Graph of 𝒇(𝒙) = 𝒙𝟐 + 𝒌
Example 3:
Graph 𝑓(𝑥) = 𝑥 2 and 𝑔(𝑥) = 𝑥 2 + 3 on the same set of axes. For table of values, let the domain
𝑥 be a set of integers such that −2 ≤ 𝑥 ≤ −2.
Solution:
Generate the table of values first.
𝑥
−2
−1
0
1
2
𝑓(𝑥) = 𝑥 2
4
1
0
1
4
SY 2021-2022 |Mathematics|Grade 9 | First Quarter
𝑔(𝑥) = 𝑥 2 + 3
7
4
3
4
7
Page | 71
𝑔(𝑥) = 𝑥 2 + 4
Next, identify the ordered pairs. Then, plot the
ordered pairs and connect them with a
smooth curve.
Consider the table of values and notice that
for each 𝑥-value, the 𝑦-value of 𝑔(𝑥) = 𝑥 2 + 3
is the same as the graph of 𝑓(𝑥) = 𝑥 2 shifted
3 units upward. Likewise, the two graphs
share the same axis of symmetry and that is
the 𝑦-axis.
𝑓(𝑥) = 𝑥 2
The Graph of 𝒇(𝒙) = (𝒙 − 𝒉)𝟐
Example 4:
Graph 𝑓(𝑥) = 𝑥 2 and 𝑔(𝑥) = (𝑥 − 4)2 on the same set of axes.
Solution:
As usual, generate first the table of values. But, in this case, the domain we will use for each
function differs. Let us find ℎ first in 𝑔(𝑥) = (𝑥 − 4)2 .
𝑥−ℎ = 𝑥−4
Both 𝑥 − ℎ and 𝑥 − 4 are equal since 𝑔(𝑥) = (𝑥 − 4)2 is in the
form 𝑓(𝑥) = (𝑥 − ℎ)2 .
𝑥−𝑥−ℎ=𝑥−𝑥−4
Add −𝑥 to both sides of the equation (APE).
−ℎ = −4
Simplify.
ℎ=4
Since our ℎ = 4, the domain we will consider for the function 𝑔(𝑥) = (𝑥 − 4)2 is the domain of
(𝑥) = 𝑥 2 increased by 4. Thus, out table of values will be:
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SY 2021-2022 |Mathematics |Grade 9 | First Quarter
𝑓(𝑥) = 𝑥 2
4
1
0
1
4
𝑥
−2
−1
0
1
2
𝑥
2
3
4
5
6
(𝑥) = (𝑥 − 4)2
4
1
0
1
4
Increase by 4 so that 𝑓(𝑥) = 𝑔(𝑥).
Next, identify the ordered pairs. Then,
plot the ordered pairs and connect
them with a smooth curve.
Note that the axis of symmetry for the
graph of 𝑔(𝑥) = (𝑥 − 4)2 is also shifted
4 units to the right.
Exercise 4:
Graph 𝑓(𝑥) = 𝑥 2 and 𝑔(𝑥) = (𝑥 + 3)2
on the same set of axes.
𝑓(𝑥) = 𝑥 2
𝑔(𝑥) = (𝑥 − 4)2
The Graph of 𝒇(𝒙) = (𝒙 − 𝒉)𝟐 + 𝒌
For the following example, you will combine vertical and
horizontal shifts.
Example 5:
Graph 𝑔(𝑥) = (𝑥 − 2)2 + 3 and ℎ(𝑥) = (𝑥 + 3)2 − 4
Solution:
For these functions, you only consider the graph of
𝑓(𝑥) = 𝑥 2 . Then, identify the ℎ and the 𝑘. The value of ℎ
will determine the number of units the graph of 𝑓(𝑥) = 𝑥 2
will shift horizontally while the value of 𝑘 will determine
the number of units the graph of 𝑓(𝑥) = 𝑥 2 will shift
vertically.
SY 2021-2022 |Mathematics|Grade 9 | First Quarter
Page | 73
For 𝑔(𝑥) = (𝑥 − 2)2 + 3, ℎ = 2. Thus, the graph of 𝑓(𝑥) = 𝑥 2 shifts 2 units horizontally to the
right (since ℎ is positive). Meanwhile, 𝑘 = 3, which makes the graph of (𝑥) = 𝑥 2 shifts 3 units
upward (since 𝑘 is positive).
The vertex of the graph is (2,3) and the axis of symmetry is 𝑥 = 2.
For ℎ(𝑥) = (𝑥 + 3)2 − 4, however, ℎ = −3 and 𝑘 =
−4. Thus, the graph of (𝑥) = 𝑥 2 shifts 3 units
horizontally to the left (since ℎ is negative).
Meanwhile, 𝑘 = −4, which makes the graph of
𝑓(𝑥) = 𝑥 2 shifts 4 units downward (since 𝑘 is
negative).
The vertex of the graph is (−3, −4) and the axis of
symmetry is 𝑥 = −3.
The Graph of 𝒇(𝒙) = 𝒂(𝒙 − 𝒉)𝟐 + 𝒌.
Example 6:
Graph 𝑔(𝑥) = 2(𝑥 − 4)2 − 3.
Solution:
In 𝑔(𝑥) = 2(𝑥 − 4)2 − 3, ℎ = 4 and 𝑘 = −3. Thus,
the graph of 𝑔(𝑥) = 2(𝑥 − 4)2 − 3 looks like the
graph of 𝑓(𝑥) = 2𝑥 2 but moved 4 units to the right
and 3 units down. Since 2 is positive, the graph
opens upward.
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SY 2021-2022 |Mathematics |Grade 9 | First Quarter
Example 7:
Graph ℎ(𝑥) = −3(𝑥 + 1)2 + 2.
Solution:
In ℎ(𝑥) = −3(𝑥 + 1)2 + 2, ℎ = −1 and 𝑘 = 2. Thus,
the graph of 𝑔(𝑥) = 2(𝑥 − 4)2 − 3 looks like the
graph of 𝑓(𝑥) = 2𝑥 2 but moved 4 units to the right
and 3 units down. Since 2 is positive, the graph
opens upward.
Exercise 6:
Graph 𝑔(𝑥) = −4(𝑥 − 1)2 + 3 and
1
ℎ(𝑥) = (𝑥 − 2)2 + 1.
2
More on Graphing Quadratic
Functions
It is easy to graph a quadratic function when we know the vertex, line of symmetry, and
any intercept of the parabola. Rewriting the quadratic function in the form 𝑦 = (𝑥 − ℎ)2 + 𝑘
gives us (ℎ, 𝑘) which is the vertex, and 𝑥 = ℎ, which is the line of symmetry.
Completing the square is useful in rewriting a quadratic equation in the form
(𝑥
𝑦 = − ℎ)2 + 𝑘.
Example 8:
Graph 𝑓(𝑥) = 𝑥 2 − 2𝑥 − 3. Find the vertex, line of symmetry, and any intercepts.
SY 2021-2022 |Mathematics|Grade 9 | First Quarter
Page | 75
Solution:
We know that the graph of a quadratic function is a parabola. Let us rewrite the function in the
form 𝑦 = (𝑥 − ℎ)2 + 𝑘. To do this, apply the completing the square.
The original function.
𝑓(𝑥) = 𝑥 2 − 2𝑥 − 3
𝑦 = 𝑥 2 − 2𝑥 − 3
𝑦 + 3 = 𝑥 2 − 2𝑥 − 3 + 3
𝑦+3+
−2 2
(2)
2
= 𝑥 − 2𝑥 +
Let 𝑓(𝑥) = 𝑦
−2 2
(2)
Add 3 to both sides of the equation (APE)
−2 2
Add ( ) to both sides of the equation.
2
𝑦 + 3 + 1 = 𝑥 2 − 2𝑥 + 1
𝑦 + 4 = (𝑥 − 1)2
𝑦 = (𝑥 − 1)2 − 4
Simplify.
𝑓(𝑥) = (𝑥 − 1)2 − 4
Let 𝑦 = 𝑓(𝑥).
Factor 𝑥 2 − 2𝑥 + 1.
Add −4 to both sides of the equation (APE)
From the above quadratic function in the form 𝑓(𝑥) = (𝑥 − ℎ)2 + 𝑘, ℎ = 1 and 𝑘 = −4. Since
(ℎ, 𝑘) is the vertex, then the vertex of the parabola is (1,4), and the axis of symmetry is 𝑥 = 1.
Since 𝑎 > 0 (which in this function 𝑎 = 1), the parabola opens upward and will have two 𝑥intercepts and one 𝑦-intercept.
To find the 𝑥-intercepts:
Let 𝑓(𝑥) = 0
𝑓(𝑥) = 𝑥 2 − 2𝑥 − 3
0 = 𝑥 2 − 2𝑥 − 3
𝑥 2 − 2𝑥 − 3 = 0
(𝑥 + 1)(𝑥 − 3) = 0
Applying the property of zero product,
𝑥+1=0
𝑥−3=0
𝑥 = −1
𝑥=3
To find the 𝑦-intercept:
Let 𝑥 = 0
𝑓(0) = 𝑥 2 − 2𝑥 − 3
𝑓(0) = (0)2 − 2(0) − 3
𝑓(0) = −3
The two 𝑥-intercepts are −1 and 3 and 𝑦intercept is −3. The graph of 𝑓(𝑥) = 𝑥 2 − 2𝑥 − 3
is shown.
There is another way of finding the vertex without transforming 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 to 𝑓(𝑥) =
(𝑥 − ℎ)2 + 𝑘.
Page | 76
SY 2021-2022 |Mathematics |Grade 9 | First Quarter
Example 9:
Find the vertex of the graph of 𝑓(𝑥) = 𝑥 2 − 2𝑥 − 3.
Solution:
Let 𝑎 = 1, 𝑏 = −2, and 𝑐 = −3. So,
−𝑏
−(−2)
2
= 2(1) = 2 = 1
2𝑎
The 𝑥-coordinate of the vertex is 1. To find the corresponding 𝑦-value, find 𝑓(1).
𝑓(𝑥) = 𝑥 2 − 2𝑥 − 3
𝑓(1) = (1)2 − 2(1) − 3
𝑓(1) = 1 − 2 − 3
𝑓(1) = −4
The 𝑦-coordinate of the vertex is −4.
Thus, the vertex is (1, −4).
Alternate Solution
−𝑏 4𝑎𝑐−𝑏2
)
4𝑎
In this solution set, we will use ( 2𝑎 ,
to find the vertex.
Let 𝑎 = 1, 𝑏 = −2, and 𝑐 = −3. So,
−𝑏 4𝑎𝑐−𝑏2
)
4𝑎
( 2𝑎 ,
−(−2) 4(1)(−3)−(−2)2
,
)
2(1)
4(1)
=(
The Vertex of a Parabola
The graph of 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐, when 𝑎 ≠ 0, is a parabola with vertex
−𝑏 4𝑎𝑐−𝑏2
,
) or
2𝑎
4𝑎
−𝑏
vertex is 2𝑎 .
(

The 𝑥-coordinate of the

The 𝑦-coordinate of the vertex is found by computing 𝑓 ( 2𝑎 ).

The axis of symmetry is 𝑥 =
−(−2) 4(1)(−3)
, 4(1) )
2(1)
(
−2 −12−4
)
4
(2 ,
−𝑏
−𝑏
, 𝑓 ( ))
2𝑎
2𝑎
(
−𝑏
−𝑏
.
2𝑎
−2 −12−4
)
4
=(2 ,
= (−1,
−16
)
4
= (−1, −4)
Thus, the vertex is (1, −4).
SY 2021-2022 |Mathematics|Grade 9 | First Quarter
Page | 77
Activity 5
Solve the following problems:
1
1. Graph 𝑔(𝑥) = −4(𝑥 − 1)2 + 3 and ℎ(𝑥) = 2 (𝑥 − 2)2 + 1.
2. Graph 𝑓(𝑥) = 𝑥 2 + 4𝑥 − 5. Find the vertex, line of symmetry, and any intercepts.
3. Find the vertex of the graph of 𝑓(𝑥) = 𝑥 2 + 4𝑥 − 5.
Four-Pronged
Integration
ICV/RV
Behavioral Indicators
Points to Ponder
Faith/ Strong faith in God
Our life is like the openings of the
graphs of quadratic equations – it can
be upward or downward. In our ups and
downs, God is always with us.
Manifests a strong sense and
experience of God’s loving
presence developed through
personal prayer and
reflection.
Social
Orientation
Lesson Across
Discipline
Biblical Text
Page | 78
“The Wheel of Life”
The openings of the graphs of quadratic
equations reminds the “wheel of life” –
that sometimes we’re up and down.
There are instances that we are happy,
fulfilled, and successful. There are also
times where we are sad, dissatisfied
and failed. These are parts of our life. If
we are now in downs, always remember
hope for better days are coming, and if
you are up, always be grateful and
humble
Architecture
Engineers and architects have applied
the concepts of the graphs of quadratic
equation to create magnificent buildings
and other structures.
Genesis 6:14-16
God Himself is a great engineer and
architect of a legendary structure that
“Make for yourself an ark of saved the human and other species
gopher wood; you shall make from extinction.
the ark with rooms, and shall
cover it inside and outside
SY 2021-2022 |Mathematics |Grade 9 | First Quarter
with pitch. This is how you
shall make it: the length of the
ark three hundred cubits, its
breadth fifty cubits, and its
height thirty cubits.”
The general form of quadratic function is 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐,
where 𝑎, 𝑏, and 𝑐 are real numbers and 𝑎 ≠ 0.
4. The graph of a quadratic equation is a parabola.
5. The vertex of the graph is its turning point.
6. The axis of symmetry of the graph is the vertical line that
goes through the center of the parabola and the vertex.
The general form of quadratic function can be transformed into
𝑓(𝑥) = 𝑎(𝑥 − ℎ)2 + 𝑘.
1.)
𝑎 is the leading coefficient.
2.)
(ℎ, 𝑘) is the vertex.
3.)
𝑥 = ℎ is the axis of symmetry.
4.)
If 𝑎 > 0, the parabola opens upward.
5.)
If 𝑎 < 0, the parabola opens downward.
The coefficient 𝑎 in 𝑓(𝑥) = 𝑎𝑥 2 makes the graph narrower or wider.

If |𝑎| > 1, the graph is narrower than the graph of 𝑦 = 𝑥 2 .

If |𝑎| < 1, the graph is wider than the graph of 𝑦 = 𝑥 2.
The graph of 𝑓(𝑥) = 𝑥 2 + 𝑘 has the same shape as the graph of 𝑦 =
𝑥 2 . The graph of 𝑦 = 𝑥 2 is
 shifted upwards when 𝑘 is positive.
 shifted downwards when 𝑘 is negative.
The graph of 𝑓(𝑥) = (𝑥 − ℎ)2 has the same shape as the graph of
𝑦 = 𝑥 2 . The graph of 𝑦 = 𝑥 2 is

shifted ℎ units to the right if ℎ is positive.

shifted ℎ units to the left if ℎ is negative.
SY 2021-2022 |Mathematics|Grade 9 | First Quarter
Page | 79
Quiz 5
Solve these Problems.
1. For each relation, (a) write the domain; (b) write the range, and (c) determine whether
the correspondence is a function.
a. {(1,1), (2,3), (4, −2), (4,2)}
b.
𝑥
𝑥
−2
4
4
−3
5
2. Graph 𝑓(𝑥) = 𝑥 2 , 𝑔(𝑥) = 𝑥 2 + 4, and ℎ(𝑥) = 𝑥 2 − 5 on the same set of axes. For table of
values, let the domain 𝑥 be a set of integers such that −2 ≤ 𝑥 ≤ −2.
3. Graph 𝑓(𝑥) = 𝑥 2 and 𝑔(𝑥) = (𝑥 + 3)2 on the same set of axes.
4. Graph 𝑔(𝑥) = (𝑥 + 1)2 − 2 and ℎ(𝑥) = (𝑥 − 4)2 + 5.
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SY 2021-2022 |Mathematics |Grade 9 | First Quarter
WORKSHEET
Name :
Grade Level/Section :
Teacher
Lesson No. 5
Quadratic Functions
and Their Graphs
: Rene B. Tinquilan, LPT, MAED
Judy-Ann B. Morales, LPT
Reminder: Please write your final answer/s on the space provided. Write your
solution/s on the pages provided for the solution. If the given pages for writing your
solution is not enough, you may use intermediate pad and/or graphing paper, and
don’t forget to staple it together with this worksheets.
Check Your Progress
1.
a. ____________________________________________________________
____________________________________________________________
b. ____________________________________________________________
____________________________________________________________
2.
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
Generate your table of values here
𝑥
−2
−1
0
1
2
SY 2021-2022 |Mathematics|Grade 9 | First Quarter
Page | 81
Sketch your graph here.
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SY 2021-2022 |Mathematics |Grade 9 | First Quarter
3.
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
Generate your table of values here
𝑥
−2
−1
0
1
2
Sketch your graph here.
SY 2021-2022 |Mathematics|Grade 9 | First Quarter
Page | 83
4.
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
Generate your table of values here
𝑥
−2
−1
0
1
2
Sketch your graph here.
Page | 84
SY 2021-2022 |Mathematics |Grade 9 | First Quarter
Write your solutions here.
SY 2021-2022 |Mathematics|Grade 9 | First Quarter
Page | 85
Write your solutions here.
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SY 2021-2022 |Mathematics |Grade 9 | First Quarter
Write your solutions here.
SY 2021-2022 |Mathematics|Grade 9 | First Quarter
Page | 87
Write your solutions here.
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SY 2021-2022 |Mathematics |Grade 9 | First Quarter
Final Output
Now that you are almost at the final phase of this module, let us check how you would
apply what had learned in your own context. Still not ready? You can always go back on the
previous pages for some missing key points. Ready? Let’s do this in action!
Mathematics, as a discipline, is known for its complexities. According to Dossey (1992)
and Freudenthal (1973), the perception on the cognitive complexity of mathematics becomes
the hardest challenge by educators. This perceptive bias makes it one of the contributing factors
for mathematics achievement decline in the past years. It is being reflected, in general sense,
in the Philippine ranking. The Philippines is ranked 67th of 140 countries for 2015 – 2016 and
79th in 138th in the 2016 – 2017 data in quality of math and science education of the Global
Competitive Report of the World Economic Forum.
Performance Task/s G.R.A.S.P.S
Quarter 1: Quadratic Equations and Inequalities
Goal: Your goal is apply the concept of quadratic equations and inequalities in real life situation.
Role: You can either be a :
 Writer
 Architect
 Businessperson
Audience: Audience for this activity shall be composed of high school students and teachers.
Situation:
Particularly here in St. Mary’s College, Baganga, Inc., it is alarming that we got low mean
score in mathematics in two annual exams. We got 18.72 mean score and 37.45 mean
percentage score in 2015 – 2016 National Achievement Tests (NAT), and 19.40 mean score
and 40.42 mean percentage score in 2016 – 2017 RVM Test.
To address this concern, you will be tasked to show the interrelatedness of mathematics to
other discipline, showing the lighter side of mathematics. You are also tasked to show the
application of mathematical concept, particularly the topic quadratic equations and inequalities.
Product/Performance:
Your task is to show the interrelatedness of mathematics to other discipline, showing the
lighter side of mathematics. You are also tasked to show the application of mathematical
concept, particularly the topic quadratic equations and inequalities. It is a must that you include
in your material the steps to be undertaken to ensure the success of your activities.
You may create any of these products/performance (Choose only one):
SY 2021-2022 |Mathematics|Grade 9 | First Quarter
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
Product 1: Feature Article: You will be writing a feature article on how to determine
quadratic equation given the roots, and the sum and product of roots.
 Product 2: Sketch Plans or Design: You will be making sketch plans and design that
illustrate quadratic equations, quadratic inequalities, and rational algebraic equations.
 Product 3: Making and Interpreting Business Graph: You will be researching a business
problem that can be solved using quadratic equations, make use of actual or imaginative
data, and then graph. Interpret the data and write its interpretation in narrative form.
Standards:
Your product/performance will be evaluated based on criteria/categories:



Feature Writing: word choice, lead, focused topic, conclusion, and grammar and
spelling
Sketch Plans or Design: drawing, creativity, craftsmanship and efforts
Making and Interpreting Business Graph
Category
SCORING RUBRICS
Feature Writing
Adopted from: http://www.menifee.k12.ky.us
4
3
2
WORD
CHOICE
Writer uses vivid
words and phrases
that linger or draw
pictures in the
reader's mind, and
the choice and
placement seems
accurate, natural,
and not forced.
LEAD (× 2)
The lead is inviting
and draws the
reader into the
article.
FOCUSED
TOPIC (× 4)
There is one clear,
well-focused,
defined topic. Main
idea is supported
with detailed
information.
The writer draws an
appropriate
conclusion and
writes it well.
Writer made no
errors in grammar
or spelling that
distract the reader
from the content.
CONCLUSIO
N
(× 2)
GRAMMAR
AND
SPELLING
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Writer uses vivid
words and
phrases that
linger or draw
pictures in the
reader's mind,
but occasionally
the words are
used
inaccurately or
seem overdone.
The lead states
the topic, and
makes an
attempt to draw
the reader into
the article.
Main idea is
clear, but the
supporting
information is
general.
The writer draws
a conclusion, but
it is not
satisfying.
Writer made 1-2
errors in
grammar or
spelling that
distract the
1
Writer uses
words that
communicate
clearly, but the
writing lacks
variety, punch or
flair.
Writer uses a
limited
vocabulary that
does not
communicate
strongly or
capture the
reader's interest.
The lead is not
inviting to the
reader.
No lead, or very
weak lead.
Main idea is
somewhat clear,
or the writer tries
to bring in too
much off topic
material.
The writer
rambles in the
conclusion.
The main idea is
not clear. There
is a seemingly
random
collection of
information.
There is no
conclusion.
Writer made 3-4
errors in
grammar or
spelling that
distract the
Writer made
more than 4
errors in
grammar or
spelling that
SY 2021-2022 |Mathematics |Grade 9 | First Quarter
reader from the
content.
reader from the
content.
distract the
reader from the
content.
Sketch Plans or Design
Adopted from: https://www.rcampus.com/rubricshow
Category
4
3
2
1
DRAWING
My drawings show
My drawings
My drawings
My drawings are
that I applied my
show that I
show some
not completed.
(× 2)
best effort and my
applied efforts
effort. A lack of
drawing was well
and planning.
planning is
planned.
evident.
CREATIVITY
My design
My design
My design lacks
My design shows
demonstrates a
demonstrates
sincere
little or no
unique level of
originality.
originality.
evidence of
originality.
original thought.
CRAFTSMANSHIP
My drawings are
My drawings are My drawings are My drawings are
(× 2)
very neat and
neat and show
somewhat
messy and
shows no evidence very little
messy and show shows marks
of marks, rips, tears evidence of
either marks or
and rips, tears,
or folds.
marks, rips, tear, rips, tears, or
or folds.
or folds.
folds.
I completed my
I completed my
I finished my
I did not finish
EFFORT (×
drawings and far
drawings in an
drawings but
my drawings in a
5)
exceeded the
above average
they lack
satisfactory
requirements for the manner, yet
finishing touches manner.
projects.
more could have or can be
been done.
improved with
little efforts.
Reading and Interpreting Business Graph
Adopted from: https://www.rcampus.com/rubricshow (Slightly Modified)
Category
3 – Good
2 – Fair
1 - Poor
Graph Interpretation
RELATIONSHIP
Have fully determine and
Able to determine and NOT able to
OF DATA
analyze the relationship of analyze the
determine and
the data
relationship of the
analyze the
(× 3)
data
relationship of the
data
CONCLUSIONS
Able to thoroughly explain Able to either explain
NOT able to explain
the results and deduce an results or deduce an
results or deduce an
(× 2)
outcome.
outcome but not both outcome.
Calculations and Accuracy
CALCULATIONS Able to derive accurate
Able to derive
NOT able to derive
calculations to interpret
approximate
accurate calculations
(× 3)
data. Showing work to
calculations to
to interpret the data.
solve the problem and
interpret data
Does not show work
including units.
including units. Does
to solve the problem
not show work to
or include units.
solve the problem.
CLARITY
Able to describe he clarity Able to describe the
NOT able to describe
of a graph or table, and
clarity of a clarity of a the clarity of a graph
(× 2)
design an appropriate
graph or table, or
or table, or design an
plan for improvements.
design an appropriate appropriate plan for
plan for improvements improvement.
but not both.
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SY 2021-2022 |Mathematics|Grade 9 | First Quarter
Books:
Diaz Z., Maharlika M., Suzara J., Mercado J., Esparrago M., Reyes N., & Orines F. (2014). Next
Century Mathematics: Grade 9. Phoenix Publishing House. Quezon City.
Diaz Z., Maharlika M., Suzara J., Mercado J., Esparrago M., Reyes N., & Orines F. (2014). Next
Century Mathematics: Grade 9 (Learning Guide). Phoenix Publishing House. Quezon
City.
Nivera G., & Lapinid M. R. (2018). Grade 9 Mathematics: Pattern and Practicalities (Rev. Ed.).
SalesianaBooks by Don Bosco Press Inc. Makati City.
Nivera G., & Lapinid M. R. (2018). Grade 9 Mathematics: Pattern and Practicalities (Rev. Ed.)
Teacher’s Manual with Assessment Guide. SalesianaBooks by Don Bosco Press Inc.
Makati City.
E-book:
Department of Education, (2020). K-12 Most Essential Learning Competencies with
Corresponding CG Codes. Retrieved at: https://commons.deped.gov.ph/K-to-12-MELCSwith-CG-Codes.pdf
Websites:
https://www.mathsisfun.com/algebra/inequality-quadratic-solving.html
https://www.toppr.com/content/story/amp/introduction-to-projectile-motion-406
https://www.coursehero.com/sg/college-algebra/properties-of-quadratic-functions/
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