9
Finding the Equation and
Application of Quadratic
Functions
Learner's Module in Mathematics 9
Quarter 1 ● Module 8
JONATHAN C. DELA CRUZ
Developer
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/W8
Learning Competency/Code : Finding the equation and application of quadratic
Function (M9-lj-1/M9lla-1)
iii
ACKNOWLEDGEMENT
The developer would like to express his 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 all his 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
Author: Jonathan C. Dela Cruz
Illustrators: Ian T. Tomin (Cover page Illustration)
Marilyn D. Bugatti (Icon Illustration)
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
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TABLE OF CONTENTS
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
What’s New ................................................................................................................ 5
What Is It .................................................................................................................... 6
What’s More ............................................................................................................. 12
What I Have Learned ............................................................................................... 14
What I Can Do ……………………………………………………………………………. 14
Post Assessment ………………………………………………………………………… 15
Additional Activity ..................................................................................................... 17
KEY ANSWER ……………………………………………………………………………. 18
REFERENCES ......................................................................................................... 19
v
Finding the Equation and
Application of Quadratic
Functions
Learner's Module in Mathematics 9
Quarter 1 ● Module 8
JONATHAN C. DELA CRUZ
Developer
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 quadratic function, you are expected
to:
1. find the equation of a quadratic function given the (a) zeros (b) table of values
and (c) graph; and
2. solve worded problems using the concepts learned in quadratic function.
By the way, always remember to use the answer sheet for you to write
your answers on the different activities 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 Is 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.
Post 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.
2
What I Know
If you answer all the test items correctly in this pretest, 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 finding the equation
of the quadratic function and its application to real life problems. Read and understand
each item, then choose the letter of your answer and write it on your answer sheet.
1. Which of the following is not a quadratic function?
A. 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 C. 𝑓(𝑥) = 𝑎(𝑥 − ℎ)2 + 𝑘
B. 𝑓(𝑥) = 𝑎𝑥 2 D. 𝑓(𝑥) = 𝑎𝑥 + 𝑐
For number 2-4, use the following table of values.
𝑥 2 1 0 −1 −2
𝑓(𝑥) 4 1 0 1 4
2. Which ordered pair is the vertex?
A. (1,1) B. (0,0) C. (−1, 1) D. (−2,4)
3. What is the zero of the quadratic function?
A. −1 B. 1 C. 2 D. 0
4. What is the equation of the quadratic function?
A. 𝑓(𝑥) = 2𝑥 2 B. 𝑓(𝑥) = 3𝑥 2 C. 𝑓(𝑥) = 𝑥 2 D. 𝑓(𝑥) = 4𝑥 2
5. The table of values below shows the function 𝑓(𝑥) = 𝑥 2 − 2, what is 𝑓(𝑥) when
𝑥 = 0?
x 2 1 0 −1 −2
f(x) 2 −1 ? −1 2
A. 0 B. −3 C. −4 D. −2
6. Which quadratic function describes the graph below (figure 1)?
A. 𝑓(𝑥) = −𝑥 2 C. 𝑓(𝑥) = 𝑥 2
B. 𝑓(𝑥) = 𝑥 2 + 1 D. 𝑓(𝑥) = 𝑥 2 − 1
Figure 1
3
7. The graph below (figure 2) shows which quadratic function?
A. 𝑓(𝑥) = 𝑥 2 + 2 C. 𝑓(𝑥) = 𝑥 2 − 2
2
B. 𝑓(𝑥) = 2𝑥 + 2 D. 𝑓(𝑥) = 2𝑥 2 − 2
Figure 2
8. In figure 3, the graph is described by,
A. 𝑓(𝑥) = 2(𝑥 + 1)2 − 1 C. 𝑓(𝑥) = 2(𝑥 − 1)2 + 1
B. 𝑓(𝑥) = 2(𝑥 + 1)2 + 1 D. 𝑓(𝑥) = −2(𝑥 + 1)2 − 1
Figure 3
9. In number 8, what is the vertex of the graph?
A. (0, −3) B. (0,0) C. (−1, −1) D. (−2, −3)
10. The maximum number of zeros/roots of the quadratic function is _______.
A. 0 B. 1 C. 2 D. 3
11. What is the equation of the quadratic function when the zeros are 3 𝑎𝑛𝑑 2?
A. 𝑓(𝑥) = 𝑥 2 − 5𝑥 + 6 C. 𝑓(𝑥) = 𝑥 2 + 5𝑥 + 6
B. 𝑓(𝑥) = 𝑥 2 − 5𝑥 − 6 D. 𝑓(𝑥) = 𝑥 2 + 5𝑥 − 6
12. In the function 𝑓(𝑥) = 𝑥 2 + 5𝑥 − 6, what is the sum and product of the zeros
respectively?
A. 5 𝑎𝑛𝑑 6 B. −5 𝑎𝑛𝑑 − 6 C. 5 𝑎𝑛𝑑 − 6 D. −5 𝑎𝑛𝑑 6
For numbers 13 – 14, use the following problem.
A ball is tossed upward from the ground. Its height in 𝑓𝑒𝑒𝑡 above the
ground after 𝑡 𝑠𝑒𝑐𝑜𝑛𝑑𝑠 is given by the function. 𝑓(𝑡) = −10𝑡 2 + 20𝑡.
13. What is the time when the ball reaches its maximum height?
A. 3 𝑠𝑒𝑐 B. 0.5 𝑠𝑒𝑐 C. 2 𝑠𝑒𝑐 D. 1 𝑠𝑒𝑐
14. What is the maximum height reach by the ball?
A. 10 𝑓𝑡 B. 15 𝑓𝑡 C. 5 𝑓𝑡 D. 8 𝑓𝑡
15. How long did the ball reach the ground?
A. 3 𝑠𝑒𝑐 B. 0.5 𝑠𝑒𝑐 C. 2 𝑠𝑒𝑐 D. 1 𝑠𝑒𝑐
4
What’s In
In your previous modules you learned some characteristics of the
quadratic function, the forms and the graphs. Also, you learned some
properties of the graph like the kind of curve, the vertex and zeros. These
concepts that you learned will be very helpful in the understanding of our
lesson in this module.
Activity 1. Match Me Please! Match column A with column B.
_____1. Vertex A. The values of 𝑥 when 𝑓(𝑥) is zero
_____2. Parabola B. 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
_____3. Zeros of the function C. 𝑓(𝑥) = 𝑥 2 + 𝑘
_____4. The vertex form D. The highest or lowest point of the parabola
_____5. The general form E. The graph of the quadratic function
F. 𝑓(𝑥) = 𝑎(𝑥 − ℎ)2 + 𝑘
What’s New
Hey guys, if I will ask you to give the quadratic
equation of a quadratic function given its roots, or
say, given the table of values or maybe, given the
graph, will you be able to give?
What I am trying to say is that, if the zeros of the quadratic
function are 2 𝑎𝑛𝑑 − 3, is there a way we can determine the
equation of the function? Or say the table of values shown
below, or a given parabola, will you be able to identify the
equation that will describe such quadratic function?
𝑥 2 1 0 −1 −2
𝑓(𝑥) 4 1 0 1 4
5
What Is It
FINDING THE EQUATION OF A QUADRATIC FUNCTION
Now, we will discuss finding the equation of a quadratic function given the
zeros, table of values and the graph.
A. Given the zeros
The zeros of the quadratic function are 2 𝑎𝑛𝑑 − 3, what is its quadratic
equation?
Let us use the form 𝑓(𝑥) = 𝑎(𝑥 2 + 𝑏𝑥 + 𝑐).
Now, follow the step by step procedures.
Step 1. Make each zero as values of 𝑥 𝑥 = 2; 𝑥 = −3
Step 2. Equate each linear equation to zero 𝑥– 2 = 0; 𝑥 + 3 = 0
Step 3. Combine the two equations into a quadratic equation (𝑥– 2)(𝑥 + 3) = 0
Step 4. Multiply the left side of the equation by FOIL method
(𝑥 – 2 )( 𝑥 + 3 ) = 0
𝑥 2 + 3𝑥 – 2𝑥 – 6 = 0
𝑥2 + 𝑥 – 6 = 0
Step 5. Write the equation 𝑓(𝑥) = 𝑎(𝑥 2 + 𝑏𝑥 + 𝑐).
𝑓(𝑥) = 𝑎(𝑥 2 + 𝑥 − 6).
Step 6. Make your conclusion
Therefore, given the zeros 2 𝑎𝑛𝑑 − 3 the quadratic function is
𝑓(𝑥) = 𝑎(𝑥 2 + 𝑥 − 6).
Important!
2 𝑎𝑛𝑑 − 3 are not just the zeros of 𝑓(𝑥) = 𝑥 2 + 𝑥 − 6. They
can also be zeros of other functions like 𝑓(𝑥) = 2𝑥 2 + 2𝑥 − 12
etc., So a is included in the equation where a is any nonzero
constant.
6
B. Given the table of values
𝑥 2 1 0 −1 −2
𝑓(𝑥) 0 −5 −8 −9 −8
The table of values above describes a quadratic function. What is the equation
that describes it? Note that in this example, we will follow the form 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
Our goal to achieve the form 𝒇(𝒙) = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 is to find the values of
𝒂, 𝒃 𝒂𝒏𝒅 𝒄.
Now, let us follow the step by step procedures.
Step 1. Select 3 convenient points (points with small numbers) in the table.
Like (2,0), (1, −5) 𝑎𝑛𝑑 (0, −8)
Step 2. Substitute these points in the equation 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
Let us start with (2,0) 2 is the value of 𝑥 while 0 is the value of 𝑓(𝑥).
0 = 𝑎(2)2 + 𝑏(2) + 𝑐
0 = 4𝑎 + 2𝑏 + 𝑐 Let us label this as equation 1
Next with (1, −5) 1 is the value of 𝑥 while – 5 is the value of 𝑓(𝑥).
−5 = 𝑎(1)2 + 𝑏(1) + 𝑐
−5 = 𝑎 + 𝑏 + 𝑐 Let us label this as equation 2
Lastly with (0, −8) 0 is the value of 𝑥 while −8 is the value of 𝑓(𝑥).
−8 = 𝑎(0)2 + 𝑏(0) + 𝑐
−8 = 0 + 0 + 𝑐
−8 = 𝑐 Let us label this as equation 3
Step 3. Solve for the values of 𝑎, 𝑏 𝑎𝑛𝑑 𝑐.
Equation 1 0 = 4𝑎 + 2𝑏 + 𝑐
Equation 2 −5 = 𝑎 + 𝑏 + 𝑐
Equation 3 −8 = 𝑐
Equation 1 minus equation 2, to eliminate c.
0 = 4𝑎 + 2𝑏 + 𝑐
−(−5 = 𝑎 + 𝑏 + 𝑐 )
5 = 3𝑎 + 𝑏 Let us label this as equation 4
7
Since in equation 3, 𝑐 = −8, we substitute this in equation 1 or equation 2.
Let us choose equation 1.
0 = 4𝑎 + 2𝑏 + 𝑐
0 = 4𝑎 + 2𝑏 + (−8) Transposed −8 to the left side of the
equation
8 = 4𝑎 + 2𝑏 Let us label this as equation 5
Equation 4 5 = 3𝑎 + 𝑏
Equation 5 8 = 4𝑎 + 2𝑏
Let us eliminate 𝑏 by subtracting equation 5 from equation 4.
5 = 3𝑎 + 𝑏 Equation 4
–
8 = 4𝑎 + 2𝑏 Equation 5
– 2(5 = 3𝑎 + 𝑏) Multiply both sides of the equation by 2
8 = 4𝑎 + 2𝑏 Equation 5
10 = 6𝑎 + 2𝑏
–
8 = 4𝑎 + 2𝑏
2 = 2𝑎 + 0
2 = 2𝑎 Divide both sides of the equation by 2
1 = 𝑎 Reflexive property
𝑎 = 1
Substitute the value of 𝑎 = 1 in equation 4.
5 = 3𝑎 + 𝑏
5 = 3(1) + 𝑏
5 = 3 + 𝑏
5– 3 = 𝑏 Transpose 3 to the left side of the
equation
2 = 𝑏
𝑏 = 2 Reflexive property
Let us summarize the values of 𝑎, 𝑏 𝑎𝑛𝑑 𝑐.
𝑎 = 1 𝑏 = 2 𝑐 = −8
Step 4. Substitute the values of a, b and c in the form 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
𝑓(𝑥) = 1𝑥 2 + 2𝑥 + (−8)
𝑓(𝑥) = 𝑥 2 + 2𝑥 − 8
Step 5. Make your conclusion
Therefore, the equation is 𝒇(𝒙) = 𝒙𝟐 + 𝟐𝒙 − 𝟖
8
C. Given the graph
Example 1
The graph on the right side pictures a
quadratic function. Notice that the
graph opens upward, that will tell you
that 𝑎 > 0. Notice also that the graph
crosses the x – axis twice which
indicates that there are two zeros of
the function. These ideas will help
you determine the equation of the
function.
Let us follow the step by step procedures.
Step 1. Identify the zeros of the graph. In this case the zeros are 2 𝑎𝑛𝑑 3.
Step 2. Make each zero as values of 𝑥
𝑥 = 2; 𝑥 = 3
Step 3. Equate each linear equation to zero.
𝑥– 2 = 0; 𝑥– 3 = 0
Step 4. Combine the two equations into a quadratic equation and simplify.
(𝑥 – 2)(𝑥 – 3) = 0
𝑥 2 − 3𝑥 − 2𝑥 + 6 = 0
𝑥 2 − 5𝑥 + 6 = 0
Step 5. Make your conclusion.
Hence, the equation of the function shown by the graph is f(x) = x 2 − 5x + 6.
Example 2
Another graph describing the
quadratic function is shown at the
right. Notice that the graph opens
downward which tells us that a<0 or
negative. Also notice that the
parabola did not cross the x-axis
which indicates that there are no
zeros but if you look closely the vertex
has a point with exact coordinates.
We can use this vertex to determine
the equation of the function by using
the form 𝒇(𝒙) = 𝒂(𝒙 − 𝒉)𝟐 + 𝒃, where
vertex (ℎ, 𝑘).
9
Let us follow the step by step procedures.
Step 1 Identify the vertex and a convenient point on the graph
The vertex is (−3, −1)
The convenient point is (−2, −3)
Step 2 Solve for 𝑎 by substituting the vertex and the convenient point in the equation
f(x) = a(x − h)2 + b.
f(x) = a(x − h)2 + b
−3 = a[−2 − (−3)]2 + (−1) Substitute ℎ = −3, 𝑘 = −1, 𝑥 = −3
and 𝑓(𝑥) = −2 in f(x) = a(x − h)2 + b
−3 = a(−2 + 3)2 − 1 Simplify
−3 = a(1)2 − 1
−3 = a(1) − 1
−3 + 1 = a Transpose −1 to the left side of the
equation
−2 = a Reflexive property
𝑎 = −2
Step 3 Substitute the vertex (ℎ, 𝑘) and the value of a in the equation
𝒇(𝒙) = 𝒂(𝒙 − 𝒉)𝟐 + 𝒃.
𝑓(𝑥) = −2[𝑥 − (−3)]2 − 1 Substitute ℎ = −3, 𝑘 = −1, and
𝑎 = −2 in f(x) = a(x − h)2 + b
𝑓(𝑥) = −(𝑥 + 3)2 − 1
Step 4 Make your generalization
Hence, the equation of the function shown by the graph is f(x) = −2(x + 3)2 − 1
Remember this!
The form 𝑓(𝑥) = 𝑎(𝑥 − ℎ)2 + 𝑏 can be
simplified as 𝑓(𝑥) = 𝑥 2 + 𝑏𝑥 + 𝑐.
10
APPLICATIONS OF QUADRATIC FUNCTIONS
The quadratic function is a very important concept in our study in this module.
The concepts like the vertex and the zero plays an important role in answering some
real life problems. Let us have fun in our next discussions.
Example 1
If the perimeter of the rectangle is 100 𝑚𝑒𝑡𝑒𝑟𝑠, find its dimensions if its
Area is maximum.
𝑊𝑖𝑑𝑡ℎ (𝑤) 𝑚 5 10 15 20 25 30 35 40 45 50
𝐿𝑒𝑛𝑔𝑡ℎ (𝑙) 𝑚 45 40 35 30 25 20 15 10 5 0
𝐴𝑟𝑒𝑎 (𝐴) 𝑚2 225 400 525 600 625 600 525 400 225 0
1. What is the largest area obtained? __________
2. What are the dimensions of a rectangle with the largest area? ______
3. What is the length of the rectangle when the width is 50 𝑚𝑒𝑡𝑒𝑟𝑠? _____
4. What is the area of the rectangle when the width is 50 𝑚𝑒𝑡𝑒𝑟𝑠? ______
5. Is 0 𝑚2 possible as area of the rectangle? Explain your answer in one
sentence. _______________________________________________
Example 2
What are the dimensions of the largest rectangular field that can be
enclosed by 60 𝑚𝑒𝑡𝑒𝑟𝑠 of fencing wire?
Step 1 Sketch the figure of the situation if possible. Label the figure to better
understand the situation.
𝐿𝑒𝑛𝑔𝑡ℎ (𝑙)
𝑊𝑖𝑑𝑡ℎ (𝑤)
𝐴𝑟𝑒𝑎 (𝐴)
𝑃𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟 (𝑃) = 60 𝑚𝑒𝑡𝑒𝑟𝑠
Step 2 Use the formula of perimeter and area of a rectangle to establish a
working equation)
𝐴𝑟𝑒𝑎 = (𝑙𝑒𝑛𝑔𝑡ℎ)(𝑤𝑖𝑑𝑡ℎ)
𝐴 = 𝑙𝑤 Equation 1
𝑃𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟 = 2𝑙𝑒𝑛𝑔𝑡ℎ + 2𝑤𝑖𝑑𝑡ℎ
𝑃 = 2𝑙 + 2𝑤 Substitute the 𝑃 = 60
60 = 2𝑙 + 2𝑤 Divide both side of the equation by 2
30 = 𝑙 + 𝑤 Solve for 𝑙 in terms of 𝑤
𝑙 = 30 – 𝑤 Equation 2
11
Substitute equation 2 in equation 1.
𝐴 = 𝑙𝑤
𝐴(𝑤) = (30 – 𝑤 ) 𝑤
Step 3 Solve for the unknown variables 𝑤 𝑎𝑛𝑑 𝑙.
𝐴(𝑤) = 30𝑤 – 𝑤 2 Use the distributive property of
multiplication
𝐴(𝑤) = −𝑤 2 + 30𝑤
𝐴(𝑤) = − (𝑤 2 – 30𝑤 + 225) + 225 Use completing the square
𝐴(𝑤) = − (𝑤 – 15 )2 + 225
From the equation, the vertex is (15,225), this means that the
𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑎𝑟𝑒𝑎 is 225 𝑚2 while the 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑤𝑖𝑑𝑡ℎ is 15 𝑚. Since the
𝑤𝑖𝑑𝑡ℎ is 15 𝑚𝑒𝑡𝑒𝑟𝑠, we can solve for the 𝑙𝑒𝑛𝑔𝑡ℎ using equation 2.
𝑙 = 30 – 𝑤
𝑙 = 30 – 15
𝑙 = 15
The length is 15 𝑚𝑒𝑡𝑒𝑟𝑠.
Step 4 Make your conclusion
The dimensions of the largest rectangular field that can be
enclosed by 60 𝑚𝑒𝑡𝑒𝑟𝑠 of fencing wire is 15𝑚 𝑥 15𝑚 or the 𝑙𝑒𝑛𝑔𝑡ℎ 𝑖𝑠 15
𝑚𝑒𝑡𝑒𝑟𝑠 and the 𝑤𝑖𝑑𝑡ℎ is also 15 𝑚𝑒𝑡𝑒𝑟𝑠.
What’s More
Activity 2. What is My Quadratic Function Given My Zeros?
Given the zeros of the quadratic functions, find the equation that
describes each. Show your solution on a separate sheet.
1. 0 𝑎𝑛𝑑 2 _______________________
2. 3 𝑎𝑛𝑑 0 _______________________
3. −1 𝑎𝑛𝑑 1 _______________________
4. 2 𝑎𝑛𝑑 – 1 _______________________
5. 4 𝑎𝑛𝑑 2 _______________________
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Activity 3. What is My Quadratic Function Given My Table of Values?
Each table of values show a quadratic function. What equation can we
derive from each? Show your solution on a separate sheet.
1.
𝑥 2 1 0 −1 −2
𝑓(𝑥) 6 3 2 3 6
2.
𝑥 0 1 2 3 4
𝑓(𝑥) 0 3 12 27 48
3.
𝑥 −2 −1 0 1 2
(𝑥) 0 1 4 9 16
Activity 4. What is My Quadratic Function Given My Graph?
Using the graphs below, determine the equation of each quadratic function.
1. ________________________ 2. ________________________
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What I Have Learned
Activity 5. Fill in the blanks. Write the word or words or symbols to complete the
each statement.
1. Given the zeros of the quadratic function, _________________ is the form
of equation to be used.
2. Given the table of values of the quadratic function choose any
__________________convenient points to start with in finding the equation.
3. When the graph of the quadratic function is given, you can make use of
the _______________ and the _____________in finding the equation.
4. If in the quadratic function, the value of a is less than zero, then the parabola
opens ______________ .
What I Can Do
Activity 6: Application. Answer each question based on the problem.
From a 100 𝑓𝑒𝑒𝑡 building, a ball is thrown straight up into the air then follows a
trajectory. The height ℎ(𝑡) of the ball above the building after 𝑡 𝑠𝑒𝑐𝑜𝑛𝑑𝑠 is given by the
function ℎ(𝑡) = −20(𝑡 − 3)2 + 180.
1. What is the vertex of the graph shown by the function?
2. How far does the ball reach at its maximum point?
3. How long does the ball reach the highest point?
4. After how many seconds does the ball reach the ground?
5. What is the value of 𝑎 in the equation?
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Post Assessment
DIRECTION: Let us determine how much you already know about finding the equation
of the quadratic function and its application to real life problems. Read and understand
each item, then choose the letter of your answer and write it on your answer sheet.
1. Which of the following is not a quadratic function?
A 𝑓(𝑥) = 𝑎𝑥 + 𝑐 C. 𝑓(𝑥) = 𝑎(𝑥 − ℎ)2 + 𝑘
B. 𝑓(𝑥) = 𝑎𝑥 2 D. 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
2. The table of values below shows the function 𝑓(𝑥) = 𝑥 2 − 2, what is 𝑓(𝑥) when
𝑥 = 0? 𝑥 2 1 0 −1 −2
𝑓(𝑥) 2 −1 ? −1 2
A. 0 B. −3 C. −4 D. −2
For numbers 𝟑 − 𝟓, use the following table of values.
𝑥 2 1 0 −1 −2
𝑓(𝑥) 4 1 0 1 4
3. Which ordered pair is the vertex?
A. (1,1) B. (0,0) C. ( −1, 1) D. (−2,4)
4. What is the zero of the quadratic function?
A. −1 B. 1 C. 2 D. 0
5. What is the equation of the quadratic function?
A. 𝑓(𝑥) = 2𝑥 2 B. 𝑓(𝑥) = 3𝑥 2 C. 𝑓(𝑥) = 𝑥 2 D. 𝑓(𝑥) = 4𝑥 2
6. The graph below (figure 1) shows which quadratic function?
A. 𝑓(𝑥) = 𝑥 2 + 2 C. 𝑓(𝑥) = 2𝑥 2 + 2
2
B. 𝑓(𝑥) = 𝑥 − 2 D. 𝑓(𝑥) = 2𝑥 2 − 2
𝐹𝑖𝑔𝑢𝑟𝑒 1
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7. Which quadratic function describes the graph below (figure 2)?
A. 𝑓(𝑥) = 𝑥 2 − 1 C. 𝑓(𝑥) = 𝑥 2
2
B. 𝑓(𝑥) = 𝑥 + 1 D. 𝑓(𝑥) = −𝑥 2
𝐹𝑖𝑔𝑢𝑟𝑒 2
8. In figure 3, the graph is described by______________.
A. 𝑓(𝑥) = 2(𝑥 + 1)2 − 1 C. 𝑓(𝑥) = 2(𝑥 − 1)2 − 1
B. 𝑓(𝑥) = −2(𝑥 + 1)2 − 1 D. 𝑓(𝑥) = −2(𝑥 − 1)2 − 1
𝐹𝑖𝑔𝑢𝑟𝑒 3
9. In figure 3, what is the vertex of the graph?
A. (0, −1) B. (0,1) C. (1,1) D. (−1, −1)
10. The maximum number of zeros/roots of the quadratic function is ___.
A. 0 B. 1 C. 2 D. 3
11. What is the equation of the quadratic function when the zeros are 3 𝑎𝑛𝑑 1?
A. 𝑓(𝑥) = 𝑥 2 − 4𝑥 + 3 C. 𝑓(𝑥) = 𝑥 2 + 4𝑥 + 3
B. 𝑓(𝑥) = 𝑥 2 − 4𝑥 − 3 D. 𝑓(𝑥) = 𝑥 2 + 4𝑥 − 3
12. In the function 𝑓(𝑥) = 𝑥 2 + 𝑥 − 6, what is the sum and product of the zeros
respectively?
A. 2 𝑎𝑛𝑑 3 B. −2 𝑎𝑛𝑑 − 3 C. 2 𝑎𝑛𝑑 − 3 D. −2 𝑎𝑛𝑑 3
For numbers 13 – 14, use the following problem.
A ball is tossed upward from the ground. Its height in feet above the
ground after 𝑡 𝑠𝑒𝑐𝑜𝑛𝑑𝑠 is given by the function 𝑓(𝑡) = −10𝑡 2 + 20𝑡.
13. What is the time when the ball reaches its height?
A. 3 𝑠𝑒𝑐 B. 0.5 𝑠𝑒𝑐 C. 2 𝑠𝑒𝑐 D. 1 𝑠𝑒𝑐
14. What is the maximum height reach by the ball?
A. 10 𝑓𝑡 B. 15 𝑓𝑡 C. 5 𝑓𝑡 D. 8 𝑓𝑡
15. How long did the ball reach the ground?
A. 3 𝑠𝑒𝑐 B. 0.5 𝑠𝑒𝑐 C. 2 𝑠𝑒𝑐 D. 1 𝑠𝑒𝑐
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Additional Activity
Activity 7. Graph Me
Plot the table of values in the coordinate plane and sketch the graph that
describes the function and answer the following questions.
𝑥 −2 −1 0 1 2
𝑓(𝑥) 8 2 0 2 8
f(x)
1. Vertex ________________
2. Zeros ________________
3. Opening ________________
4. Equation ________________
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I. What I Know II. What’s In III. Application of QF
1. D 9. C 1. D 1. 625 𝑚2
2. B 10. C 2. E 2. 𝑙 = 2 5𝑚 ; 𝑤 = 25𝑚
3. D 11. A 3. A 3. 0 𝑚
4. C 12. B 4. F 4. 0 𝑚2
5. D 13. D 5. B 5. No, the figure will be one
dimensional
6. C 14. A
7.A 15. C
8. D
IV. What’s More: Activity 2 V. What’s More: Activity 3
1. 𝑓(𝑥) = 𝑎(𝑥 2 − 2𝑥) 1. 𝑓(𝑥) = (𝑥 − 1)2 − 3
2. 𝑓(𝑥) = 𝑎(𝑥 2 − 3𝑥) 2. 𝑓(𝑥) = 3𝑥 2
3. 𝑓(𝑥) = 𝑎(𝑥 2 − 1) 3. 𝑓(𝑥) = 𝑥 2 + 4𝑥 + 4
4. 𝑓(𝑥) = 𝑎(𝑥 2 − 𝑥 − 2)
5..𝑓(𝑥) = 𝑎(𝑥 2 − 6𝑥 + 8)
VI. What’s More: Activity 4 VII. What I Have Learned: Activity 5
1. 𝑓(𝑥) = 𝑥 2 − 3𝑥 − 4 1. 𝑓(𝑥) = 𝑎(𝑥 2 + 𝑏𝑥 + 𝑐)
2. 𝑓(𝑥) = (𝑥 − 1)2 − 3 2. Three
3. vertex, zeros (in any order)
4. downward
VIII. What I Can Do: Activity 6
Post Assessment:
1. (3, 180) 1. A 6. A 11. A
2. 180 𝑓𝑒𝑒𝑡 2. D 7. C 12. C
3. 3 𝑠𝑒𝑐𝑜𝑛𝑑𝑠 3. B 8. B 13. D
4. 6 𝑠𝑒𝑐𝑜𝑛𝑑𝑠 4. D 9. D 14. A
5. −20 5. C 10. C 15. C
IX. Activity 7
1. 0,0
2. 0
3. upward
4. 𝑓(𝑥) = 2𝑥 2
KEY ANSWER
REFERENCES
Nivera, Gladys (2013), Grade 9 Mathematics, Quadratic Function.
n.a. (2014). Mathematics 9 Learner’s Material, Quadratic Function.
Orines,Fernando B.(2013), Next Century Mathematics,
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