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Mathematics Quarter I – Module 8: Quadratic Functions Department of Education • Republic of the Philippines 9
Mathematics Quarter I – Module 8: Quadratic Functions
Department of Education • Republic of the Philippines
9
Mathematics - Grade 9 Alternative Delivery Mode
Quarter 1 – Module 8: Quadratic Functions First Edition, 2020
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Development Team of the Module
Author/s: Julius Gregory B. Hechanova, Evaluators/Editor: Grace D. Batausa Illustrator/Layout Artist: Management Team Chairperson: Dr. Arturo B. Bayocot, CESO III Regional Director
Co-Chairpersons: Dr. Victor G. De Gracia Jr. CESO V Asst. Regional Director Roy Angelo E. Gazo, PhD, CESO V Schools Division Superintendent
Nimfa R. Lago,PhD, CESE AssistantSchools Division Superintendent Mala Epra B. Magnaong, Chief ES, CLMD Members: Neil A. Improgo, EPS-LRMS Bienvenido U. Tagolimot, Jr., EPS-ADM Henry B. Abueva OIC-CID Chief Exquil Bryan P. Aron, EPS-Math Sherlita L. Daguisonan, LRMS Manager Meriam S. Otarra, PDO II Charlotte D. Quidlat, Librarian II
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Mathematics
Quarter 1 – Module 8 Quadratic Functions
Department of Education ● Republic of the Philippines
9
This instructional material was collaboratively developed and
reviewed by educators from public and private schools, colleges,
and or/universities. We encourage teachers and other education
stakeholders to email their feedback, comments, and
recommendations to the Department of Education – Region 10 at
We value your feedback and recommendations.
Table of Contents
What This Module is About ................................................................................................................... i
What I Need to Know .............................................................................................................................. i
How to Learn from this Module ........................................................................................................... .i
Icons of this Module ............................................................................................................................... .i
Lesson 1: Graphs of Quadratic Functions ........................................................................................ 1
What I Need to Know.................................................................................................. 1
What I Know.................................................................................................................. 1
What’s In ........................................................................................................................ 2
What’s New ................................................................................................................... 3
What Is It ........................................................................................................................ 4
What’s More .................................................................................................................. 6
What Is It ........................................................................................................................ 7
What’s More .................................................................................................................. 8
What Is It …. ................................................................................................................. 9
What’s More .................................................................................................................. 10
What I Have Learned.................................................................................................. 11
What I Can Do .............................................................................................................. 12
Summary............................................................................................................................................... 13
Assessment: (Post-Test) ............................................................................................................... 14
Key to Answers .................................................................................................................................. 17
References ........................................................................................................................................... 22
What This Module is About
This module focuses on the graphs of quadratic functions. You will be learning how
to draw the graph of a quadratic function and investigate the properties of the graph through
guided questions.
What I Need to Know
In this module, you are expected to:
Determine the domain, range, intercepts, axis of symmetry, opening of the parabola and the minimum or maximum value of a given quadratic function
Invistage and analyze h=the effects of changes in the variables a, h, and k in
the graph of quadratic functions y = a(x – h)2 + k and Make generalizations; and apply the concepts learned in solving real-life
problems.
How to Learn from this Module
To achieve the objectives cited above, you are to do the following:
• Take your time reading the lessons carefully. • Follow the directions and/or instructions in the activities and exercises diligently. • Answer all the given tests and exercises.
Icons of this module
This module has the following parts and corresponding icons:
What I Need to Know
This will give you an idea of the skills or
competencies you are expected to learn in the
module.
What I Know
This part includes an activity that aims to check
what you already know about the lesson to take.
If you get all the answers correct (100%), you
may decide to skip this module.
What’s In
This is a brief drill or review to help you link the
current lesson with the previous one.
What’s New
In this portion, the new lesson will be introduced
to you in various ways such as a story, a song, a
poem, a problem opener, an activity or a
situation.
What is It
This section provides a brief discussion of the
lesson. This aims to help you discover and
understand new concepts and skills.
i
What’s More
This comprises activities for independent practice
to solidify your understanding and skills of the
topic. You may check the answers to the
exercises using the Answer Key at the end of the
module
What I Have Learned
This includes questions or blank
sentence/paragraph to be filled in to process
what you learned from the lesson.
What I Can Do
This section provides an activity which will help
you transfer your new knowledge or skill into real
life situations or concerns.
Assessment
This is a task which aims to evaluate your level of
mastery in achieving the learning competency.
Additional Activities
In this portion, another activity will be given to
you to enrich your knowledge or skill of the
lesson learned. This also tends retention of
learned concepts.
Answer Key
This contains answers to all activities in the
module.
References This is a list of all sources used in developing this
module.
At the end of this module you will also find:
The following are some reminders in using this module:
1. Use the module with care. Do not put unnecessary mark/s on any part of the
module. Use a separate sheet of paper in answering the exercises.
2. Do not forget to answer What I Know before moving on to the other activities
included in the module.
3. Read the instruction carefully before doing each task.
4. Observe honesty and integrity in doing the tasks and checking your answers.
5. Finish the task at hand before proceeding to the next.
6. Return this module to your teacher/facilitator once you are through with it.
If you encounter any difficulty in answering the tasks in this module, do not
hesitate to consult your teacher or facilitator. Always bear in mind that you are not
alone.
We hope that through this material, you will experience meaningful learning and
gain deep understanding of the relevant competencies. You can do it.
ii
Lesson
1 Graphs of Quadratic Functions
What I Need to Know
As you go over the exercises, you will develop the skills in graphing quadratic functions; determining the domain, range, intercepts, axis of symmetry, and the opening of the parabola of a given quadratic function; make generalizations; and apply the concepts learned in solving real-life problems.
Wr
Answer the following items.
1. Given the quadratic functions and , transform them into the form
2. Complete the table of values for x and y.
x -3 -2 -1 0 1 2 3 4 5
y
x -4 -3 -2 -1 0 1 2 3 4
y
What I Know
1
What’s In
Going back to your answers in “What I know”:
a. Given the quadratic functions and ,
transform them into the form
Solution:
, Solution:
b. Complete the table of values for x and y.
x -3 -2 -1 0 1 2 3 4 5
y 17 10 5 2 1 2 5 10 17
x -4 -3 -2 -1 0 1 2 3 4
y -7 -2 1 2 1 -2 -7 -14 -23
𝒚 𝒙𝟐 𝟐𝒙 𝟐
𝒚 𝒙𝟐 𝟐𝒙 𝟏 𝟐 𝟏
𝒚 𝒙𝟐 𝟐𝒙 𝟏 𝟏
𝒚 𝒙 𝟏 𝟐 𝟏
𝒚 𝒙𝟐 𝟐𝒙 𝟏 𝒚 𝒙𝟐 𝟐𝒙 𝟏 𝟏 𝟏 𝒚 𝒙𝟐 𝟐𝒙 𝟏 𝟐 𝒚 𝟏 𝒙 𝟏 𝟐 𝟐
2
What’s New
Answer the following: 1. Sketch the graph on the Cartesian plane.
2. Describe the graphs above, what do they look like?
_______________________________
__________________________________
3. Which of the 2 quadratic functions has a minimum point? Maximum point?
Quadratic Function Turning point
(At what point did the graph
changed its direction?)
Maximum or Minimum
Point
4. Observe each graph. Can you draw a line that divides the graph in such a
way that one part is a reflection of the other part? If there is any, determine
the equation of the line?
5. Take a closer look at the minimum point or the maximum point and try to
relate it to the values of h and k in the equation of the
function. Write your observations. ______________________________________________
6. Can you identify the domain and range of the functions?
Domain: __________ Range: ___________
Domain: __________ Range: ___________
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What Is It
The graph of a quadratic function is called a parabola. It is the set of all points on
the Cartesian Coordinate Plane that satisfies the function defined by or
the vertex form f(x) = a(x – h)2 + k where (h, k) is the vertex.
Consider the graph of
Example 1: Graph the function and determine its domain and range.
The arrowheads of 𝑓 𝑥 𝑥 imply that the graph extends indefinitely to the left and right. This is because the domain of f is the set of all real numbers ( ). In fact, unless it is restricted, the domain of a quadratic function is always the set of all real numbers. The range depends on whether the parabola opens upward or downward. If it opens upward, the range is the set 𝑦: 𝑦 ≥ 𝑘 ; if it opens downward,
then the range is the set 𝑦: 𝑦 ≤ 𝑘 .
Solution:
Step 1: Determine the vertex (-2,-1)
Step 2: Construct a table of values and place the vertex
in the center.
x -2
y -1
Step 3: Assign the values of x before and after -2.
x -4 -3 -2 -1 0
y -1
Step 4: Solve for y using the values of x.
If x = -4 y =
y = 𝑥 y = 4
y = 4 y = 3
follow this procedure when the vales of x are -3, -1 and 0
and complete the table of values.
Step 5: Fill in the table with the computed values of y in
terms of x.
x -4 -3 -2 -1 0
y 3 0 -1 0 3
Step 6: Plot the points in the graph as shown
below..
Domain: ℝ (Set of all real numbers)
*Since 𝑘 , and the parabola opens upward
Range: set 𝑦: 𝑦 ≥
(-4, 3)
(-3, 0)
(-2,-1)
(-1,0)
(0,3)
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Example 2: Consider the graph of the quadratic function g, given by .
Determine its domain and range.
Solution:
Step 1: Determine the vertex (0,1)
Step 2: Construct a table of values and place the
vertex in the center.
x 0
y 1
Step 3: Assign the values of x before and after -2.
x -2 -1 0 1 2
y 1
Step 4: Solve for y using the values of x.
If x = -2
y = 𝑥
y =
y = 4
y = 4
y = 3
follow this procedure when the vales of x are -1, 1
and 2 and complete the table of values.
Step 5: Fill in the table with the computed values of y
in terms of x.
x -2 -1 0 1 2
y -3 0 1 0 -3
Step 6: Plot the points in the graph as
shown below.
Domain: ℝ (Set of all real numbers)
*Since 𝑘 , and the parabola opens downward
Range: set 𝑦: 𝑦 ≤
(0,1)
(0,-2) (0,2)
(-2, -3) (2, -3)
Given the two examples above, I hope that you have learned some ideas on how to
graph the given function. In the next page you are going to try some exercises. I know
you can do it. Enjoy!
5
What’s More
Graph the given quadratic functions and determine their domain and range:
1. 2.
x x
y y
Domain: __________________ Domain: __________________ Range: __________________ Range: __________________ 3.
x
y
Domain: __________________ Range: __________________
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What is It
The x-intercept of a graph is the value of x at which the graph intersects the x-axis.
That is, if the graph passes through the point (a,0), then the graph has an x-intercept
of a. (In other words, the value/s of x when y=0).
The y-intercept of a graph is the value of y at which the graph intersects the y-axis.
Thus, if the graph passes through the point (0,b), then the graph has a y-
intercept of b. (In other words, the value of y when x=0).
Example 1: Consider the graph of the quadratic function f, given by
. Determine its x and y-intercepts.
Example 2: Consider the graph of the quadratic function g, given by .
Determine its x and y-intercepts.
Since the graph intersects the x-axis at (-3, 0) and (-1, 0), thus: x-intercepts: -1 and -3 Since the graph intersects the y-axis at (0, 3), thus:
y-intercept: 3
Since the graph intersects the x-axis at (-1, 0) and (1, 0), thus: x-intercepts: -1 and 1 Since the graph intersects the y-axis at (0, 1), thus:
y-intercept: 1
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What’s More
Graph the given quadratic functions and determine their x and y -intercepts:
1. 3 2.
x x
y y
x-intercept/s: __________________ x-intercept/s: __________________ y-intercept: __________________ y-intercept: __________________
3.
x
y
x-intercept/s: __________________
y-intercept: __________________
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What is It
You have noticed that the parabola opens upward or downward. It has a turning point
called vertex which is either the lowest point or the highest point of the graph. If the value of
a > 0, the parabola opens upward and has a minimum point. If a < 0, the parabola opens
downward and has a maximum point. There is a line called the axis of symmetry which
divides the graph into two parts such that one-half of the graph is a reflection of the other
half. If the quadratic function is expressed in the form y = a(x – h)2 + k, the vertex is the point
(h, k). The line x = h is the axis of symmetry and k is the minimum or maximum value of the
function.
Example 1: Consider the graph of the quadratic function f, given by
.
Determine the direction of the opening of the parabola, its vertex and axis of symmetry and
the minimum or maximum value.
Example 2: Consider the graph of the quadratic function g, given by .
Determine the direction of the opening of the parabola, its vertex, axis of symmetry
and the minimum or maximum value.
Axis of
symmetry
Vertex
Minimum
Point
𝒇 𝒙 𝒙 𝟐 𝟐 𝟏.
a = 1, h= -2 and k = -1 Since a=1, then the parabola opens upwards. Vertex: (-2, -1) Axis of symmetry: x = -2
Minimum Value: -1
Axis of
symmetry
Vertex
Maximum
Point 𝒈 𝒙 𝒙𝟐 𝟏.
a = -1, h= 0 and k = 1 Since a= -1, then the parabola opens downwards.
Vertex: (0, 1) Axis of symmetry: x = 0 Maximum Value: 1
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What’s More
Determine the direction of the opening of the parabola, the vertex, axis of symmetry and the
minimum or maximum value of the given quadratic functions:
1. 5 5. 5
Direction of the opening: Direction of the opening:
Vertex: Vertex:
Axis of symmetry: Axis of symmetry:
Minimum Value: Maximum Value:
2. 5 6. 5
Direction of the opening: Direction of the opening:
Vertex: Vertex:
Axis of symmetry Axis of symmetry:
Minimum Value: Minimum Value:
3. 5 7. 5
Direction of the opening: Direction of the opening:
Vertex: Vertex:
Axis of symmetry: Axis of symmetry:
Minimum Value: Minimum Value:
4. 5 8. 5
Direction of the opening: Direction of the opening:
Vertex: Vertex:
Axis of symmetry: Axis of symmetry:
Maximum Value: Maximum Value:
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What I Have Learned
Graph the given quadratic functions. Determine the domain, range, x and y-intercepts, its
vertex, axis of symmetry and minimum and maximum value.
1.
Domain: _____________
Range: _____________
x-intercept: _____________
y-intercept _____________
Vertex: _____________
Axis of Symmetry:__________
Minimum Value: ___________
2. 4
Domain: _____________
Range: _____________
x-intercept: _____________
y-intercept _____________
Vertex: _____________
Axis of Symmetry:__________
Maximum Value: __________
3.
Domain: _____________
Range: _____________
x-intercept: _____________
y-intercept _____________
Vertex: _____________
Axis of Symmetry:__________
Minimum Value: ___________
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4.
Domain: _____________
Range: _____________
x-intercept: _____________
y-intercept _____________
Vertex: _____________
Axis of Symmetry:__________
Maximum Value: ___________
5. 3
Domain: _____________
Range: _____________
x-intercept: _____________
y-intercept _____________
Vertex: _____________
Axis of Symmetry:__________
Minimum Value: ___________
Make a simple presentation of world famous parabolic arches.
Task:
1. Surf the internet for world famous parabolic arches. As you search, keep a record of where you go, and what you find on the site.
2. Organize the data you collected, including the name of the architect and the purpose of creating the design.
3. Once you completed the data make a presentation in a creative manner. You can use any of the following but not limited to: a. Multimedia presentation b. Webpages c. Poster
What I Can Do
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In Summary The graph of the quadratic function is called parabola. The
parabola opens upward when a is positive otherwise it opens downward.
The vertex is the highest or lowest point of the parabola. The axis of symmetry is
an imaginary line passing through the vertex, which divides the parabola into two identical
parts.
The x-intercept of a graph is the value of x at which the graph intersects the x-axis.
That is, if the graph passes through the point (a,0), then the graph has an x-intercept of a.
(In other words, the value/s of x when y=0).
The y-intercept of a graph is the value of y at which the graph intersects the y-axis.
Thus, if the graph passes through the point (0,b), then the graph has a y-intercept of b. (In
other words, the value of y when x=0).
If the graph passes through the origin, we can see that the graph has an x-intercept
of 0 and a y-intercept of 0.
The domain of a quadratic function is the set of all real numbers. The range
depends on whether the parabola opens upward or downward. If it opens upward, the range
is the set {y : y ≥ k}; if it opens downward, then the range is the set {y : y ≤ k}.
The minimum value is the y coordinate of the vertex of the parabola when it opens
upward or when a > 0.
The maximum value is the y coordinate of the vertex of the parabola when it
opens downward or when a < 0.
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Assessment
Find out how much you have learned in this module. Write the letter that you think is the best
answer to each question on a sheet of paper. Answer all items.
1. What is the graph of the quadratic function called?
a. Parabola
b. Cartesian Plane
c. Axis of Symmetry
d. Domain
2. What is the vertex of this quadratic function: 3 5?
a. (-2, 3) b. (3, -5) c. (-3, -5) d. (-2, -5)
3. What is the axis of symmetry of this quadratic function: 3 5?
a. X = -5 b. x = -3 c. x = -2 d. x = 3
4. What do you call the value of x at which the parabola intersects the x-axis?
a. x – axis
b. y – axis
c. y – intercept
d. x – intercept
5. What is the domain of this quadratic function: 3 5?
a. The set of all Natural Numbers c. The set of all Whole Numbers
b. The set of all integers d. The set of all Real Numbers
6. What do you call the maximum or the minimum point of the parabola?
a. Domain
b. Range
c. Vertex
d. Intercept
7. Which direction does the parabola open in 3 5?
a. Downward c. Left
b. Upward d. Right
For numbers 8 – 14, refer to the graph given below.
14
8. What is the vertex of the parabola?
a. (-1,
2)
b. (1, -
2)
c. (2, -
1)
d. (-2,
1)
9. What are the x-intercepts of the graph?
a. 0 and 1
b. 3 and 0
c. 3 and 2
d. 1 and 3
10. What is the range of the given graph?
a. : ≤
b. : ≥
c. : ≤
d. : ≥
11. What is the axis of symmetry of the given graph?
a. x = -2 b. x = -1 c. x = 1 d. x = 2
12. What is the y-intercept of the parabola?
a. -1 b. 1 c. 2 d. 3
13. What is the domain of the given graph?
a. Set of all Real nos.
b. Set of all Rational nos.
c. Set of all Irrational nos.
d. Set of all integers
14. What is the equation of the parabola?
a.
b.
c.
d.
15. What is the range of this quadratic function: 3 5?
a. : ≤ c. : ≤ 3
b. : ≤ 5 d. : ≤ 3
16. What do you call the imaginary line which divides the graph into two parts
such that one-half of the graph is a reflection of the other half?
a. x – axis
b. y – axis
c. Axis of symmetry
d. Cartesian Plane
17. What is the vertex of this quadratic function: ?
a. (1, 1)
b. (0, 0)
c. (-1, -1)
d. (1, 0)
18. What is the range of ?
a. : ≤
b. : ≤
c. : ≥
d. : ≥
15
19. What is the Axis of symmetry of ?
a. x = -1 b. x = 0 c. x = 1 d. x = 2
20. Which direction does the parabola open in ?
a. Upward
b. Downward
c. Left
d. Right
16
1
Answer Key
17
2
18
3
19
4
20
5
21
6
References Janice F. Antonio et. al., Mathe Connections in the Digital Age Grade 9, Sibs Publishing house, Inc., Quezon City, 2015. Merden L. Bryant et al. Mathematics 9 Learner’s Manual, Department of Education-Instructional Materials Council Secretariat (DepEd-IMCS), Pasig City, Philippines. https://owl.purdue.edu/owl/research_and_citation/chicago_manual_17th_edition/cmo
s_formatting_and_style_guide/chicago_manual_of_style_17th_edition.html
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