9

Introduction to Quadratic
Functions
Learner's Module in Mathematics 9
Quarter 1 ● Module 6

RENANTE F. RAGOS
CLAIRE O. VALENCIA
Developers

Department of Education • Cordillera Administrative Region

NAME:________________________ GRADE AND SECTION ____________

TEACHER: ____________________ SCORE _________________________
 Republic of the Philippines
DEPARTMENT OF EDUCATION
Cordillera Administrative Region
SCHOOLS DIVISION OF BAGUIO CITY
Military Cut-off, Baguio City

Published by:
DepEd Schools Division of Baguio City
Curriculum Implementation Division

COPYRIGHT NOTICE
2020

Section 9 of Presidential Decree No. 49 provides:

“No copyright shall subsist in any work of the Government of the Philippines.
However, prior approval of the government agency of office wherein the work is
created shall be necessary for exploitation of such work for profit.”

This material has been developed for the implementation of K-12 Curriculum
through the DepEd Schools Division of Baguio City – Curriculum Implementation
Division (CID). It can be reproduced for educational purposes and the source must be
acknowledged. Derivatives of the work including creating an edited version, an
enhancement or a supplementary work are permitted provided all original work is
acknowledged and the copyright is attributed. No work may be derived from this
material for commercial purposes and profit.

ii
 PREFACE

This module is a project of the DepEd Schools Division of Baguio City through
the Curriculum Implementation Division (CID) which is in response to the
implementation of the K to 12 Curriculum.

This Learning Material is a property of the Department of Education, Schools

Division of Baguio City. It aims to improve students’ academic performance specifically
in Mathematics.

Date of Development : June 2020

Resource Location : DepEd Schools Division of Baguio City
Learning Area : Mathematics
Grade Level :9
Learning Resource Type : Module
Language : English
Quarter/Week : Q1/W6
Learning Competency/Code : Models real-life situations using quadratic
functions. (M9AL-Ig-2)
: Represents a quadratic function using: (a) table
of values; (b) graph; and (c) equation.
(M9AL-Ig-2)
: Transforms the quadratic function defined by
𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 into the form 𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘
(M9AL-lh-1)

iii
 ACKNOWLEDGEMENT

The developers would like to express their gratitude to those who, in one way
or another, have contributed in the development of this learning material.

Appreciation for all the collaboration and cooperation given by the Grade-9
Mathematics teachers. Boundless gratitude goes to their friends for sharing their time
and talent in crafting this learning resource and to all the students of Baguio City
National High School who are hoping to learn despite this pandemic. Lastly, thanks to
their school’s supervisory office led by their school principal, Madam Brenda M. Cariño
and the DepEd Division of Baguio City for all the support.

Development Team
Authors: Renante F. Ragos and Claire O. Valencia
Illustrators: Marilyn Degay-Bugatti (Label Icons)
Renante F. Ragos, Ian Tomin

School Learning Resources Management Committee

Brenda M. Cariño School Principal
Editha L. Laop Subject/ Learning Area Specialist
Niño E. Martinez Subject/ Learning Area Specialist
Sherwin Fernando School LR Coordinator

Quality Assurance Team

Francisco C. Copsiyan EPS – Mathematics
Lourdes B. Lomas-e PSDS – BCNHS District

Learning Resource Management Section Staff

Loida C. Mangangey EPS – LRMDS
Victor A. Fernandez Education Program Specialist II - LRMDS
Christopher David G. Oliva Project Development Officer II – LRMDS
Priscilla A. Dis-iw Librarian II
Lily B. Mabalot Librarian I

CONSULTANTS

JULIET C. SANNAD, EdD

Chief Education Supervisor – CID

SORAYA T. FACULO, PhD

Asst. Schools Division Superintendent

MARIE CAROLYN B. VERANO, CESO V

Schools Division Superintendent

iv
 TABLE OF CONTENTS

Page
COPYRIGHT NOTICE…………………………………………….……...….…. ii
PREFACE ………………………………………………………………………... iii
ACKNOWLEDGEMENT………………………………………………………… iv
TABLE OF CONTENTS……………………………………………….……....... v
TITLE PAGE ……………………………………………………..………………. 1
What I Need to Know ……………………………………………………..…….. 2
What I Know………………………..……………………………….……………. 3
What’s In………………………………………………………………….………. 5
Activity 1. Describe Me in Many Ways!
What’s New ………………..……………….……………………...……..……… 6
Activity 2. Identify Me!
Activity 3. Compare Me!
What Is It…….…………………………………………………………….……… 8
What’s More.....…….………………………………………..…...……………… 10
Activity 4. Step by Step!
Activity 5. Reversing the Process!
What I Have Learned.................................................................................... 11
Activity 6. Your Own!
What I Can Do.............................................................................................. 12
Activity 7. Hit or Miss!
Post Assessment …………………...…………………………………….…… 13
Additional Activity ……………………………………………………………….. 15
Activity 8. The Hidden Message!
Answer Key…………………………………………………………………….…. 16
Reference Sheet…………………………………………………...…….…….… 18

v
Title Page

Introduction to Quadratic
Functions
Learner's Module in Mathematics 9
Quarter 1 ● Module 6

RENANTE RAGOS
CLAIRE O. VALENCIA
Developers

Department of Education • Cordillera Administrative Region

 What I Need to Know

Hello learner! This module was designed and written with you in mind. Primarily,
its scope is to develop your understanding on the definition of a quadratic function.

While going through this module, you are expected to:

1. represent a quadratic function using: (a) table of values; (b) graph; and (c)
equation;
2. transform the quadratic function defined by 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 into the form
𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘 and vice versa; and
3. model real-life situations using quadratic functions.
By the way, always remember to use the answer sheet for you to write
your answers on the different activities and assessments presented in this
learning module. DO NOT ANSWER HERE directly.
Now, here is an outline of the different parts of your learning module. The
descriptions will guide you on what to expect on each part of the module.

Icon Label Description

What I Need to Know This states the learning objectives that you need to
achieve as you study this module.
What I Know This is to check what you already know about the
lesson on this module. If you answered all the
questions here correctly, then you may skip
studying this module.
What’s In This connects the current lesson with a topic or
concept necessary to your understanding.

What’s New This introduces the lesson to be tackled through an

activity.
What’s In It This contains a brief discussion of the learning
module lesson. Think of it as the lecture section of
the lesson.
What’s More These are activities to check your understanding
and to apply what you have learned from the
lesson.
What I have Learned This generalizes the essential ideas tackled from
this module.

What I Can Do This is a real-life application of what you have

learned.
Assessment This is an evaluation of what you have learned
from this learning material.

Additional Activity This is an activity that will strengthen and fortify

your knowledge about the lesson.

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 What I Know

If you answer all the test items correctly in this pre-assessment, then you may
skip studying this learning material and proceed to the next learning module.

DIRECTION: Let us determine how much you already know about a quadratic
function. Read and understand each item, then choose the letter of your answer and
write it on your answer sheet.

1) Which of the following equations represent a quadratic function?

A. 𝑦 = 3 + 2𝑥 2 C. 𝑦 = 3𝑥 − 22
B. 2𝑦 + 3 = 𝑥 D. 𝑦 = 2𝑥 − 3
2) What is the graph of a quadratic function?
A. straight line C. parabola
B. circle D. bar graph
3) _____________is a function defined by 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐, where 𝑎, 𝑏, and 𝑐 are
real numbers and 𝑎 ≠ 0.
A. Exponential function C. Linear function
B. Quadratic function D. Inverse function
4) Given the following table of values for 𝑥 𝑎𝑛𝑑 𝑦, which table describes a quadratic
function?
A. x 1 2 3 4 C. x −3 −2 −1 0
y 2 3 4 5 y −4 −2 0 2

B. x 1 2 3 4 D. x 0 1 2 3
y 2 5 10 17 y −2 −1 6 25
5) In the quadratic function 𝑓(𝑥) = 4𝑥 2 − 2𝑥 + 1, which is the quadratic term?
A. 4𝑥 2 C. 𝑥 2
B. −2𝑥 D. 1
6) What are the values of 𝑎, 𝑏, and 𝑐 in the quadratic 𝑓(𝑥) = −2𝑥 2 − 5𝑥 − 12.
A. 𝑎 = 2, 𝑏 = 5, 𝑐 = 12 C. 𝑎 = −2, 𝑏 = −5, 𝑐 = −12
B. 𝑎 = −2, 𝑏 = −5, 𝑐 = 12 D. 𝑎 = 2, 𝑏 = −5, 𝑐 = 12
7) Which of the following is the degree of every quadratic function?
A. 1 C. 3
B. 2 D. 4
8) All equations represent a quadratic function EXCEPT:
A. 𝑓(𝑥) = (𝑥 − 3)(𝑥 + 2) C. 2𝑥(𝑥 − 30) − 𝑦 = 0
B. 22 + 𝑥 = 𝑦 D. 2𝑥 2 − 2𝑥 + 1 = 𝑦

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9) Which of the following is the graph of a quadratic function?
A. C.

B. D.

10) What is 𝑓(𝑥) = −3(𝑥 + 2)2 + 2 when written in the form 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐?
A. 𝑓(𝑥) = −3𝑥 2 + 12𝑥 − 10 C. 𝑓(𝑥) = −3𝑥 2 + 12𝑥 + 10
B. 𝑓(𝑥) = 3𝑥 2 − 12𝑥 + 10 D. 𝑓(𝑥) = −3𝑥 2 − 12𝑥 − 10
11) What are the values of h and k in the function 𝑓(𝑥) = 3𝑥(𝑥 − 2) + 1?
A. ℎ = 2, 𝑘 = 1 C. ℎ = 1, 𝑘 = 2
B. ℎ = −2, 𝑘 = 1 D. ℎ = −1, 𝑘 = −2
12) The quadratic function 𝑓(𝑥) = 𝑥 2 + 2𝑥 − 1 when expressed in vertex form
𝑓(𝑥) = 𝑎(𝑥 − ℎ)2 + 𝑘 is ____?
A. 𝑓(𝑥) = (𝑥 + 1)2 + 1 C. 𝑓(𝑥) = (𝑥 + 1)2 − 2
B. 𝑓(𝑥) = (𝑥 + 1)2 − 1 D. 𝑓(𝑥) = (𝑥 + 1)2 + 2
13) A quadratic function has how many zeros?
A. 𝑡𝑤𝑜 C. 𝑎𝑡 𝑚𝑜𝑠𝑡 𝑡𝑤𝑜
B. 𝑎𝑡 𝑙𝑒𝑎𝑠𝑡 𝑡𝑤𝑜 D. 𝑜𝑛𝑒
14) All of the following table of values for x and y describes a quadratic function
EXCEPT?
A. x −2 −1 0 1 C. x 1 2 3 4
y 1 0 1 4 y −2 5 −10 17

B. x 0 1 2 3 D. x −1 0 1 2
y −6 −10 −12 −12 y −1 −3 −1 −5

15) Which of the following equation is equivalent to 𝑓(𝑥) = 2(𝑥 − 3)2 + 5 ?

A. 𝑓(𝑥) = 2𝑥 2 − 6𝑥 + 12 C. 𝑓(𝑥) = 2𝑥 2 − 12𝑥 − 15
B. 𝑓(𝑥) = 2𝑥 2 − 12𝑥 + 23 D. 𝑓(𝑥) = 2𝑥 2 − 6𝑥 + 23

4
 What’s In

Let us start this lesson by recalling ways of representing a

linear function. The knowledge and skills in doing this activity
will help you a lot in understanding the quadratic function. In
going over this lesson, you will be able to identify a quadratic
function and represent it in different ways.

Activity 1. Describe Me in Many Ways!

Perform this activity.

a. Observe the pattern and draw the 4th and 5th figures.

? ?

1 2 3 4 5

b. Use the table to illustrate the relation of the figure number to number of blocks.

Figure Number (x) 1 2 3 4 5

Number of blocks (y) 1 4 7

c. Write the pattern observed from the table. _____________________________

d. List the following:
Set of ordered pairs _______________________________
Domain ____________________ Range ____________________
e. What equation describes the pattern? ________________________________
f. Graph the relation using the Cartesian Plane.

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 From the given activity, I hope that you remembered
what you have learned about linear functions. A linear
function is a function of degree 1 and its graph is a
straight line. In any function, the domain are all the
values of 𝑥 and the range are all the values of 𝑦.

What’s New

Now you are ready to learn about quadratic functions. A

Quadratic function is a function that can be written in the form
𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐, where 𝑎, 𝑏, and 𝑐 are real numbers and
𝑎 ≠ 0. The highest exponent of the variable is 2, hence it is
called a second degree function. Note that the degree of a
function, determines the maximum number of zeros.

Activity 2. Identify Me!

State whether each of the following equation represents a quadratic function or not.
Justify your answer.

Equations Yes or No Justification

1. 𝑦 = 𝑥 2 + 2
2. 𝑦 = 2𝑥 − 10
3. 𝑦 = 9 − 2𝑥 2
4. 𝑦 = 2𝑥 + 2
5. 𝑦 = 3𝑥 2 + 𝑥 3 + 2

Activity 3. Compare Me!

Consider the given functions 𝑔(𝑥) = 3𝑥 + 1 and 𝑓(𝑥) = 𝑥 2 − 2𝑥 − 3.

1. What kind of function is 𝑔(𝑥)? 𝑓(𝑥)?

2. Complete the following table of values using the indicated function.

𝑔(𝑥) = 3𝑥 + 1 𝑓(𝑥) = 𝑥 2 − 2𝑥 − 3
x −3 −2 −1 0 1 2 3 x −3 −2 −1 0 1 2 3
y y

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3. What are the difference between two adjacent x-values in each table?

4.Find the differences between each table, and write them on the blanks provided.
𝑔(𝑥) = 3𝑥 + 1 𝑓(𝑥) = 𝑥 2 − 2𝑥 − 3
x -3 -2 -1 0 1 2 3 x -3 -2 -1 0 1 2 3
y y

5. What do you observe from the differences in the values of y in each function?

6. How can you recognize a quadratic function when the table of values is given?

7. Using the table of values, graph the two functions and compare the results.

𝑔(𝑥) = 3𝑥 + 1 𝑓(𝑥) = 𝑥 2 − 2𝑥 − 3

8. Compare the graph of linear function and quadratic function.

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 What Is It
You have already learned in the previous activity that in a
quadratic function, equal differences in the independent
variable 𝑥, produces equal second differences in the
dependent variable 𝑦. The graph of a quadratic function is a
smooth curve which opens upward or downward called
parabola. This time you will learn how to transform a quadratic
function from the general form 𝒇(𝒙) = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 to
vertex form 𝒇(𝒙) = 𝒂(𝒙 − 𝒉)𝟐 + 𝒌 and vice versa.

TRANSFORMING FROM GENERAL FORM TO VERTEX FORM OF QUADRATIC

FUNCTION
Examples:
A. Transform each quadratic function in vertex form. Then give the value of h and k.
Example 1: 𝒇 (𝒙) = 𝒙𝟐 − 𝟔𝒙 + 𝟐
Method 1: By Completing the Square
Steps Task
1. Group the terms containing x. 𝑓(𝑥) = (𝑥 2 − 6𝑥) + 2
2. Factor out a. Here a = 1 𝑓(𝑥) = (𝑥 2 − 6𝑥) + 2
3. Complete the expression in parenthesis to make it
−6 2
a perfect square trinomial by adding( 2 ) = 9 and 𝑓(𝑥) = (𝑥 2 − 6𝑥 + 9) + 2 − 9
subtracting the same value from the constant term.
4. Express the perfect square trinomial as the square
𝑓(𝑥) = (𝑥 − 3)2 − 7
of a binomial.
5. Give the value of h. ℎ= 3
6. Give the value of k. 𝑘 = −7
Thus, the vertex form of 𝒇 (𝒙) = 𝒙𝟐 − 𝟔𝒙 + 𝟐 is 𝒇(𝒙) = (𝒙 − 𝟑)𝟐 − 𝟕.
−𝒃 𝟒𝒂𝒄−𝒃𝟐
Method 2: By applying the formula 𝒉 = and 𝒌 =
𝟐𝒂 𝟒𝒂

In the equation 𝑓 (𝑥) = 𝑥 2 − 6𝑥 + 2, 𝑎 = 1, 𝑏 = −6 𝑎𝑛𝑑 𝑐 = 2

Substitute the values in the formula.

ℎ=
−𝑏 4𝑎𝑐 − 𝑏 2
2𝑎 𝑘=
−(−6) 4𝑎
ℎ= 4(1)(2) − (−6)2
2(1) 𝑘=
6 4𝑎
ℎ= 8 − 36
2 𝑘=
ℎ=3 4(1)
−28
𝑘=
4
𝑘= −7

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 By substituting the solved values of ℎ 𝑎𝑛𝑑 𝑘 in 𝑓(𝑥) = 𝑎(𝑥 − ℎ)2 + 𝑘, we
obtain 𝑓(𝑥) = 1(𝑥 − 3)2 + (−7) or 𝑓(𝑥) = (𝑥 − 3)2 − 7.

Example 2: 𝒚 = −𝟑𝒙𝟐 − 𝟐𝟒𝒙 + 𝟏𝟏

Method 1: By completing the square.

Steps Task
1. Group the terms containing 𝑥. 𝑦 = (−3𝑥 2 − 24𝑥) + 11

2. Factor out 𝑎. Here 𝑎 = −3 𝑦 = −3(𝑥 2 + 8𝑥) + 11

3. Complete the expression in
parenthesis to make it a perfect
8 2 𝑦 = −3(𝑥 2 + 8𝑥 + 16) + 11 − (−3)(16)
square trinomial by adding(2) = 16 𝑦 = −3(𝑥 2 + 8𝑥 + 16) + 11 + 48
and subtracting the same value from
the constant term.
4. Express the perfect square trinomial
𝑦 = −3(𝑥 + 4)2 + 59
as the square of a binomial.
5. Give the value of ℎ. ℎ = −4
6. Give the value of 𝑘. 𝑘 = 59

Thus, the vertex form of 𝒚 = −𝟑𝒙𝟐 − 𝟐𝟒𝒙 + 𝟏𝟏 is 𝒚 = −𝟑(𝒙 + 𝟒)𝟐 + 𝟓𝟗.

−𝒃 𝟒𝒂𝒄−𝒃𝟐
Method 2: By using the formulas 𝒉 = and 𝒌 =
𝟐𝒂 𝟒𝒂

In the equation 𝑦 = −3𝑥 2 − 24𝑥 + 11, 𝑎 = −3, 𝑏 = −24 𝑎𝑛𝑑 𝑐 = 11

Substitute the values in the formula.

ℎ=
−𝑏 4𝑎𝑐 − 𝑏 2
2𝑎 𝑘=
−(−24) 4𝑎
ℎ= 4(−3)(11) − (−24)2
2(−3) 𝑘=
24 4𝑎
ℎ= −132 − 576
−6 𝑘=
ℎ = −4 4(−3)
−708
𝑘=
−12
𝑘= 59

By substituting the solved values of ℎ 𝑎𝑛𝑑 𝑘 in 𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘, we

obtain 𝑦 = −3(𝑥 + 4)2 + 59.

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B. Transform the given Quadratic Function in the form 𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘 into
𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐. Then, identify the values of 𝑎, 𝑏 𝑎𝑛𝑑 𝑐.

Example 1: 𝒇(𝒙) = (𝒙 − 𝟑)𝟐 − 𝟕

Steps Task
1. Expand (𝑥 − 3)2 𝑓(𝑥) = 𝑥 2 − 6𝑥 + 9 − 7
2. Simplify 𝑓(𝑥) = 𝑥 2 − 6𝑥 + 2

The function 𝑓(𝑥) = (𝑥 − 3)2 − 7 can be written as 𝑓(𝑥) = 𝑥 2 − 6𝑥 + 2.

Therefore, 𝑎 = 1, 𝑏 = −6, 𝑎𝑛𝑑 𝑐 = 2.

Example 2: 𝒚 = −𝟑(𝒙 + 𝟒)𝟐 + 𝟓𝟗

Steps Tasks
1. Expand (𝑥 + 4)2 𝑦 = −3(𝑥 2 + 8𝑥 + 16) + 59
2. Multiply the perfect square trinomial by −3 𝑦 = −3𝑥 2 − 24𝑥 − 48 + 59
3. Simplify by combining similar terms. 𝑦 = −3𝑥 2 − 24𝑥 + 11

The function 𝑦 = −3(𝑥 + 4)2 + 59 can be written as 𝑦 = −3𝑥 2 − 24𝑥 + 11.

Then 𝑎 = −3, 𝑏 = −24, 𝑎𝑛𝑑 𝑐 = 11.

What’s More

Activity 4: Step by Step!

Transform the given quadratic functions into the form 𝑦 = 𝑎( 𝑥 − ℎ)2 + k by following
the steps below.
a. 𝑦 = 𝑥 2 − 4𝑥 − 10
Steps Task
1. Group the terms containing x.
2. Factor out 𝑎.
3. Complete the expression in parenthesis
to make it a perfect square trinomial.
4. Express the perfect square trinomial as
the square of a binomial
5. Give the value of h.
6. Give the value of k.

10
 b. 𝑦 = 3𝑥 2 − 4𝑥 + 1

Steps Task
1. Group the terms containing 𝑥.
2. Factor out 𝑎.
3. Complete the expression in parenthesis
to make it a perfect square trinomial.
4. Express the perfect square trinomial as
the square of a binomial.
5. Give the value of ℎ.
6. Give the value of 𝑘.

Activity 5: Reversing the Process!

Rewrite 𝑦 = 2(𝑥 − 2)2 + 3 in the form 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 by following the given steps.
Steps Task
1. Expand (𝑥 − 2)2
2. Multiply the perfect square trinomial by 2
3. Simplify
4. Add 3
5. Result
6. Identify the values of 𝑎, 𝑏, 𝑎𝑛𝑑 𝑐.

What I Have Learned

Activity 6: Your Own!
Concepts Give your own illustrative
example.
A Quadratic function is a function that can be
written in the form 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐, where
𝑎, 𝑏, and 𝑐 are real numbers and 𝑎 ≠ 0.
Transforming a quadratic function in the form
𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 into the form
𝑦 = 𝑎(𝑥 – ℎ)2 + 𝑘.

Transforming a quadratic function in the form

𝑦 = 𝑎(𝑥 – ℎ) 2 + 𝑘 into the form
𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + c.

11
 Let me know how well you did in this module. Complete the
following statements. Please be honest with your answers.

1. I am doing well in __________________________________________________.

2. I still need help in __________________________________________________.
3. I am really confused in _____________________________________________.
I am proud of myself for _____________________________________________.

What I Can Do

Activity 7: Hit or Miss!

Solve this problem and show your solution.

An antenna is 𝟓 𝒎 high and 𝟏𝟓𝟎 𝒎

away from the firing place.
Suppose the path of the bullet shot
from the firing place is determined
𝟏 𝟐
by the equation 𝒚 = − 𝟏𝟓𝟎 𝒙𝟐 + 𝟏𝟓 𝒙,

where 𝒙 is the distance (in meters) of the bullet from the firing place and 𝒚 is its
height. Will the bullet go over the antenna? If yes/no, show your justification.

12
 Post Assessment
DIRECTION: Let us determine how much you have learned from this module. Read
and understand each item, then choose the letter of your answer and write it on your
answer sheet.

1) Which of the following equations represent a quadratic function?

A. 2𝑦 + 3 = 𝑥 C. 𝑦 = 5𝑥 2 − 5
B. 𝑦 = −22 + 3𝑥 D. 𝑦 = 2𝑥 − 3

2) What is the graph of a quadratic function?

A. straight line C. parabola
B. circle D. hyperbola

3) ______________ is a function defined by 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐, where 𝑎, 𝑏, and 𝑐

are real numbers and 𝑎 ≠ 0.
A. Linear function C. Exponential function
B. Inverse function D. Quadratic function

4) Given the following table of values for x and y, which table describes a quadratic
function?
A. x 1 2 3 4 C. x −3 −2 −1 0
y 2 5 10 17 y −4 −2 0 2

x 1 2 3 4 D. x 0 1 2 3
B. y 2 3 4 5 y −2 −1 6 25

5) In the quadratic function 𝑓(𝑥) = 3𝑥 2 − 𝑥 + 1, which is the quadratic term?

A. 3𝑥 2 C. 𝑥 2
B. −𝑥 D. 1

6) Which of the following is the degree of every quadratic function?

A. 0 C. 2
B. 1 D. 3

7) What are the values of 𝑎, 𝑏, and 𝑐 in the quadratic 𝑓(𝑥) = −2𝑥 2 − 5𝑥 − 12.
A. 𝑎 = 2, 𝑏 = 5, 𝑐 = 12 C. 𝑎 = −2, 𝑏 = −5, 𝑐 = 12
B. 𝑎 = −2, 𝑏 = −5, 𝑐 = −12 D. 𝑎 = 2, 𝑏 = −5, 𝑐 = 12

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8) Which of the following is the graph of a quadratic function?
A. C.

B. D.

9) What is 𝑓(𝑥) = −3(𝑥 + 2)2 + 2 when written in the form 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐?

A. 𝑓(𝑥) = −3𝑥 2 + 6𝑥 − 1 C. 𝑓(𝑥) = −3𝑥 2 + 6𝑥 + 1
B. 𝑓(𝑥) = 3𝑥 2 − 6𝑥 + 1 D. 𝑓(𝑥) = −3𝑥 2 − 6𝑥 − 1
10) All equations represent a quadratic function EXCEPT:
A. 𝑓(𝑥) = (𝑥 − 4)(𝑥 + 1) C. 3𝑥(𝑥 − 3) − 𝑦 = 0
B. 32 + 𝑥 = 𝑦 D. 3𝑥 2 − 6𝑥 + 1 = 𝑦
11) In the quadratic function 𝑓(𝑥) = 3(𝑥 + 2)2 − 6, what is the constant term?
A. 6 C. −6
B. 12 D. −12
12) Which of the following equation is equivalent to 𝑓(𝑥) = 2(𝑥 − 1)2 + 5 ?
A. 𝑓(𝑥) = 2𝑥 2 − 4𝑥 + 7 C. 𝑓(𝑥) = 2𝑥 2 + 4𝑥 − 7
B. 𝑓(𝑥) = 2𝑥 2 − 4𝑥 − 7 D. 𝑓(𝑥) = 2𝑥 2 + 4𝑥 + 7
13) The quadratic function 𝑓(𝑥) = 𝑥 2 + 4𝑥 − 1 when expressed in vertex form
𝑓(𝑥) = 𝑎(𝑥 − ℎ)2 + 𝑘 is ____?
A. 𝑓(𝑥) = (𝑥 + 2)2 − 3 C. 𝑓(𝑥) = (𝑥 − 2)2 − 3
2
B. 𝑓(𝑥) = (𝑥 + 2) + 3 D. 𝑓(𝑥) = (𝑥 − 2)2 + 3
14) How many zeros does a quadratic function have?
A. 𝑡𝑤𝑜 C. 𝑎𝑡 𝑚𝑜𝑠𝑡 𝑡𝑤𝑜
B. 𝑎𝑡 𝑙𝑒𝑎𝑠𝑡 𝑡𝑤𝑜 D. 𝑜𝑛𝑒
15) All of the following table of values for 𝑥 𝑎𝑛𝑑 𝑦 describes a quadratic function
EXCEPT?
A. x −2 −1 0 1 C. x 1 2 3 4
y 1 0 1 4 y −2 5 −10 17
D.

x 0 1 2 3 x −1 0 1 2
y −6 −10 −12 −12 y −1 −3 −1 −5

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 Additional Activity

Activity 8: The Hidden Message!

Write the indicated letter of the quadratic function in the form 𝒚 = 𝒂(𝒙 − 𝒉)𝟐 + 𝒌 into
the box that corresponds to its equivalent general form 𝒚 = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄.

I 𝑦 = (𝑥 − 1)2 − 4 T 𝑦 = (𝑥 − 1)2 − 16
5 2 49
S 𝑦 = 2 ൬𝑥 + ൰ − F 𝑦 = (𝑥 − 3)2 + 5
4 8
2 2 1 2 3
E 𝑦 = ൬𝑥 − ൰ + 2 M 𝑦 = ൬𝑥 − ൰ +
3 2 2
1
A 𝑦 = 3(𝑥 + 2)2 − U 𝑦 = −2(𝑥 − 3)2 + 1
2

N 𝑦 = (𝑥 − 0)2 − 36 H 𝑦 = 2(𝑥 + 1)2 − 2

DIALOG BOX

𝟕
𝒚 = 𝒙𝟐 − 𝒙 +
𝟒
𝟐𝟑
𝒚 = 𝟑𝒙𝟐 + 𝟏𝟐𝒙 +
𝟐

𝒚 = 𝒙𝟐 − 𝟐𝒙 − 𝟏𝟓

𝒚 = 𝟐𝒙𝟐 + 𝟒𝒙

𝒚 = 𝒙𝟐 − 𝟐𝒙 − 𝟑

𝒚 = 𝟐𝒙𝟐 + 𝟓𝒙 − 𝟑

𝒚 = 𝒙𝟐 − 𝟔𝒙 + 𝟏𝟒

𝒚 = −𝟐𝒙𝟐 + 𝟏𝟐𝒙 − 𝟏𝟕

𝒚 = 𝒙𝟐 − 𝟑𝟔

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 16
What I Know Activity 1. Don’t Forget Me!
a.
1. A 6. C 11. A
2. C 7. B 12. C
3. B 8. B 13. C
4. B 9. C 14. C b.
5. B 10. D 15. B x 1 2 3 4 5
y 1 4 7 10 13
c. Add 3 to the previous number of blocks.
d.(1,1), (2,4), (3,7), (4,10), (5,13)
e. y = 3x – 2
f.
Activity 2. Identify Me!
Equations Yes or No Justification
1. 𝑦 = 𝑥 2 + 2 Yes The highest exponent of the variable is 2.
2. 𝑦 = 2𝑥 − 10 No The highest exponent of the variable is 1.
3. 𝑦 = 9 − 2𝑥 2 Yes The highest exponent of the variable is 2.
4. 𝑦 = 2𝑥 + 2 No The exponent is a variable.
5. 𝑦 = 3𝑥 2 + 𝑥 3 + 2 No The highest exponent of the variable is 3.
Activity 3. Compare Me!
1. g(x) is a linear function and f(x) is a quadratic function.
2. 𝑔(𝑥) = 3𝑥 + 1 𝑓(𝑥) = 𝑥 2 − 2𝑥 − 3
x -3 -2 -1 0 1 2 3 x -3 -2 -1 0 1 2 3
y -8 -5 -2 1 4 7 10 y 12 5 0 -3 -4 -3 0
3. The difference between two adjacent x-values in each table are all equal to 1.
4. 𝑔(𝑥) = 3𝑥 + 1 𝑓(𝑥) = 𝑥 2 − 2𝑥 − 3
x -3 -2 -1 0 1 2 3 x -3 -2 -1 0 1 2 3
y -8 -5 -2 1 4 7 10 y 12 5 0 -3 -4 -3 0
5. In the linear function, the first differences in the values of y are equal, and in the quadratic
function the second differences in the values of y are equal.
ANSWER KEY
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7. Graphs 𝑓(𝑥) = 𝑥 2 − 2𝑥 − 3
Activity 7. Hit or Miss!
8. The graph of a linear function is a
straight line while the graph of a quadratic Is it yes when the bullet hits the top of the
function is a curve antenna. Find the height of the bullet when x =
150, which is the distance of the antenna from
the firing place.
Activity 4. Step by Step! 1 2
Substituting in 𝑦 = − 𝑥2 + 𝑥
1500 15
a. 𝑦 = 𝑥 2 − 4𝑥 − 10
1 2
Tasks 𝑦=− (150)2 + (150)
1500 15
1.) 𝑦 = (𝑥 2 − 4𝑥) − 10
2.) 𝑦 = 1(𝑥 2 − 4𝑥) -10 𝑦 = −15 + 20
3.) 𝑦 = 1(𝑥 2 − 4𝑥 + 4) −
𝑦=5
10 − 4
4.) 𝑦 = 1(𝑥 − 2)2 − 14 Thus, the height of the bullet is 5m, which is
5.) ℎ = 2 the same as the height of the antenna.
6.) 𝑘 = −14
b. 𝑦 = 3𝑥 2 − 4𝑥 + 1 Activity 8. The Hidden Message!
Tasks
1.) 𝑦 = (3𝑥 2 − 4𝑥) + 1
2.) 𝑦 = 3(𝑥 2 − 4𝑥) + 1
3.) 𝑦 = 3(𝑥 2 − 4𝑥 + 4) + 1 −
4
4.) 𝑦 = 3(𝑥 − 2)2 − 3
5.) ℎ = 2
6.) 𝑘 = −3
Activity 5. Reversing the Process!
𝑦 = 2(𝑥 − 2)2 + 3
Tasks
.
1.) 𝑦 = 2(𝑥 2 − 4𝑥 + 4) + 3
2.) 𝑦 = 2𝑥 2 − 8𝑥 + 8 + 3
3.) 𝑦 = 2𝑥 2 − 8𝑥 + 11
Post Assessment
4.) 𝑎 = 2, 𝑏 = −8, 𝑐 = 11
1. C 6. C 11. A
2. C 7. B 12. A
Activity 6. Let Me Summarize and Reflect! 3. D 8. C 13. B
4. A 9. D 14. C
* Answers to this activity may vary
5. A 10. B 15. C
REFERENCES
Jose-Dilao, Soledad, Ed.D., and Julieta G. Bernabe. Intermediate Algebra. Quezon
City: SD Publications, Inc., 2009.

Bryant, Merden L., et.al. Mathematics 9 Learner's Manual. Pasig City: Department of
Education, 2014.

Nivera, Gladys C., and Minie Rose C. Lapinid. Grade 9 Mathematics Patterns and
Practicalities. Makati City: SalesianaBOOKS by Don Bosco Press, Inc., 2013.

https://www.bitmoji.com

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