REWRITING QUADRATIC
EQUATIONS
for Mathematics Grade 9
Quarter 1 / Week 7.a
1
FOREWORD
This Self Learning Kit will serve as a guide in transforming
the quadratic function defined by y = ax2 + bx + c into the
form y = a(x – h)2 + k. It will be your aid as you learn new
ideas and enrich your existing knowledge about
mathematical concepts.
In this learning kit you will gain knowledge in transforming
the quadratic function defined by y = ax2 + bx + c into the
form y = a(x – h)2 + k.
OBJECTIVES:
K: Enumerate the forms of quadratic functions
S: Rewrite quadratic functions y = ax2 + bx + c in the form
y = a(x – h)2 + k
A: Appreciate the utility of transforming quadratic
functions
LEARNING COMPETENCY:
Transforms the Quadratic function defined by
y = ax2 + bx + c into the form y = a(x – h)2 + k (M9AL-Ih-1)
I.
What Happened
Hello fellow Grade 9 students!
I’m Danny. Join me as we discover
the different ways in rewriting
quadratic functions in different
forms!
1
PRE-TEST
ACTIVITY 1. What’s Your Value?
Copy the table in your notebook. Identify the values of a, b and c of the quadratic
functions given below.
Quadratic Function
a
b
c
1. f(x) = x2 – x
2. f(x) = 7 - 3x + x2
3. f(x) = x2 + 4x + 10
4. f(x) = 3x2 -5x + 2
5. f(x) = 2x2 -4x
ACTIVITY 2. Make Me Perfect
Determine the number that must be added to make each of the following a
perfect square trinomial. Write your answer in your notebook.
1.
2.
3.
4.
x2 + 4x + ______
y2 + 20y + ______
t2 - 24t + ______
r2 + 2r + ______
5. x2 – 30x + ______
ACTIVITY 3. Matching Type
Match the quadratic function in the form y = ax2 + bx + c with its
y = a(x-h)2 + k form. Write only the letter in your notebook.
A
B
2
1. y = 3x + 12x + 17
a. y = (x + 2)2
2. y = -x2 + 10x -26
b. y = 3(x + 2)2 + 5
3. y = x2 – 12x + 36
c. y = 6(x + 7)2 - 120
2
4. y = 6x + 84x + 174
d. y = -(x – 5)2 - 1
5. y = x2 + 4x + 4
e. y = (x – 6)2
2
II.
What You Need to Know
You learned that f(x) = ax2 + bx + c is the standard form of a
quadratic function. This function can be written in an equivalent
form using the process of completing the square.
Study the steps as shown below.
f(x) = ax2 + bx + c
f(x) = a(x 2 +
Standard form of a quadratic function
b
x) + c
a
Factor out a from x2 and x terms
2
 2 b  b 2 
 b 
f(x) = ax + x +    + c − a 
a  2a  
 2a 


 b2
b
b2 
f(x) = a x 2 + x + 2  + c − a 2
a 4a 

 4a

b
b2
f(x) = a x 2 + x + 2
a
4a

Complete the square by adding
 b 
and subtracting a 
 2a 



2
Expand the terms added and
subtracted in the previous step
2
 b2 
b2
Simplify a 2  to
4a
 4a 

b2
 + c −
4a

Factor the trinomial inside the
bracket and simplify the last two
terms
2
b 
4ac − b 2

f(x) = a x +  +
4a
 2a 
3
b
4ac −b 2
= − h and
= k . Substituting this to the
2a
4a
equation above will result to f(x) = a(x – h)2 + k.
From the result let
Hence, f(x) = ax2 + bx + c is equivalent to f(x) = a(x – h)2 + k.
Examples:
A. Rewrite the following quadratic functions in the form f(x)
= a(x- h)2 + k.
1. f(x) = x2 - 2x - 15
2. f(x) = 2x2 – 7
Solution 1: Using completing the square.
1.
f(x) = x2 - 2x – 15
Given
f(x) = (x2 - 2x) – 15
Factor out 15 in the x terms.
f(x) = (x2 - 2x + 1) - 15 – 1 Complete the square inside the parenthesis by
adding and subtracting 1.
2.
f(x) = (x - 1)2 – 16
Factor the trinomial inside the parenthesis and
simplify the last 2 terms
f(x) = 2x2 – 7
f(x) = 2x2 - 0x – 7
Given
Write function in the form f(x) = ax2 + bx + c.
0 

f(x) = 2 x 2 − x  − 7
2 

Factor out 2 in the x terms.
0


f(x) = 2  x 2 − x + 0  − 7 − 0
2


Complete the square by adding and
subtracting 0.
f(x) = 2(x - 0)2 – 7
Factor the trinomial inside the parenthesis
and combine the last two terms.
4
Solution 2: Using the formula in solving the values of h and k:
b
, the value of h can be obtained using the
2a
−b
multiplication property of equality so that h =
.
2a
In the relation –h =
1. f(x) = x2 - 2x - 15.
Substitute the values a = 1, b = -2, and c = -15 in the formula.
h =
−b −( −2) 2
=
= =1
2a
2(1) 2
4ac −b 2 4(1)( −15) −( −2) 2 − 60 − 4 − 64
k=
=
=
=
= −16
4a
4(1)
4
4
Substituting the values of h and k to f(x) = a(x – h)2 + k.
Thus, f(x) = x2 – 2x – 15 is equivalent to f(x) = (x – 1)2 - 16.
2. f(x) = 2x2 – 7.
Substitute the values a = 2, b = 0 and c = -7 in the formula.
h=
−b −0 0
=
= =0
2a 2(2) 4
k=
4ac − b 2 4(2)(−7) − 0 2 − 56 − 0 − 56
=
=
=
= −7
4a
4(2)
8
8
Substitute the values of h and k to f(x) = a(x – h)2 + k.
Therefore, f(x) = 2x2 – 7 is equivalent to f(x) = 2(x – 0)2 – 7.
5
Observe that the two solutions resulted to the same answer.
Thus, a quadratic function in the form f(x) = ax 2 + bx + c can be transformed in
−b
the form f(x) = a(x – h)2 + k by: completing the square; or the relation h =
2a
2
4ac − b
and k =
(EASE Module, 2005)
4a
Now, how will you transform a quadratic
function in the form f(x) = a(x – h)2 + k to the
standard form f(x) = ax2 + bx + c?
To do this, simply follow the given steps.
1.
2.
3.
Expand the square of the binomial indicated in the function.
Multiply the result by the value of a.
Combine the similar terms (Bryant, et al. 2014)
Now, study the examples below.
Examples:
Transform the following equation to standard form.
1. f(x) = (x – 3)2 – 7
2. f(x) = -2[x – (-5)]2 + 50
Solutions:
1. f(x) = (x – 3)2 + 7
f(x) = x2 – 6x + 9 + 7
Square the binomial
f(x) = x2 – 6x + 16
Combine the similar terms
6
2. f(x) = -2[x – (-5)]2 + 50
f(x) = -2(x + 5)2 + 50
Simplify
the term inside the parenthesis
f(x) = -2(x2 + 10x + 25) + 50
Square the binomial
f(x) = -2x2 - 20x -50 + 50
Multiply the result by -2
f(x) = -x2 - 20x
Combine the similar terms
REMEMBER:
1. Quadratic functions can be written in two forms- the standard form f(x) = ax2
+ bx + c or its equivalent form f(x) = a(x – h)2 + k.
2. To rewrite a quadratic function from the form f(x) = ax 2 + bx + c to the form
f(x) = a(x – h)2 + k, use completing the square; or determine the values of a,
b, and c then solve for h and k. Substitute the obtained values in f(x) = a(x –
h)2 + k. To find the values of h and k, use the relationships,
−b
4ac - b2
h=
and k =
2a
4a
3. To rewrite a quadratic function from the form f(x) = a(x-h)2 + k to the form f(x)
= ax2 + bx + c, expand the square of the binomial, multiply by a and add k,
then simplify by combining similar terms (Abramson 2019)
III.
What Have I Learned
A. Match the following quadratic functions to its f(x) = a(x – h)2 +k form. Write
only the letter in your notebook.
1. f(x) = 2x2 -4x + 5
A. f(x) = (x + 3)2 – 9
2. f(x) = x2 + 6x
B. f(x) = 2 (x – 1)2 + 3
3. f(x) = 5x2 – 4
C. f(x) = 5 (x - 0)2 -4
D. f(x) = 2 (x - 4)2 + 2
7
B.
Transform the following quadratic function to
f(x) = ax2 + bx + c by
following each task below. Write your answers in your notebook.
1. f(x) = 5(x – 2)2 + 2
2. f(x) = -3(x + 1)2 – 4
Task
Answer
a. Expand the square of
the binomial
b. Multiply the result by
the value of a
c. Combine similar terms
POST TEST:
What is the mathematical name for the Division Sign?
Direction: To find the answer to the question above, write the indicated letter of
the quadratic function in the form f(x) = a (x – h)2 + k into the box that corresponds
to its f(x) = ax2 + bx + c form. Copy the puzzle in your notebook.
1 2
3
E
f(x) = (x+ 2) + 4
L
f(x) = 2(x - 3)2- 5
N
f(x) = 3(x + 4)2 + 5
O
f(x) = (x - 1)2 + 3
U
f(x) = 3(x - 1)2 +
S
f(x) = 4(x - 5)2 + 3
B
f(x) = (x - 0)2 + 25
R
f(x) = (x - 4) + 3
1
3
8
1 2
DEPARTMENT OF EDUCATION
SCHOOLS DIVISION OF NEGROS ORIENTAL
SENEN PRISCILLO P. PAULIN, CESO V
Schools Division Superintendent
FAY C. LUAREZ, TM, Ed.D., Ph.D.
OIC - Assistant Schools Division Superintendent
Acting CID Chief
ADOLF P. AGUILAR
OIC - Assistant Schools Division Superintendent
NILITA L. RAGAY, Ed.D.
OIC - Assistant Schools Division Superintendent
ROSELA R. ABIERA
Education Program Supervisor – (LRMS)
ARNOLD R. JUNGCO
Education Program Supervisor – (SCIENCE & MATH)
MARICEL S. RASID
Librarian II (LRMDS)
ELMAR L. CABRERA
PDO II (LRMDS)
CHRISTIAN EVEN D. SANTILLAN
Writer/Illustrator/Lay-out Artist
_________________________________
ALPHA QA TEAM
FLORENCIO BARTOLO JR.
TERESITA P. BUBOLE
MELBA S. TUMARONG
BETA QA TEAM
ELIZABETH A. ALAP-AP
EPIFANIA Q. CUEVAS
NIDA BARBARA S. SUASIN
VRENDIE P. SYGACO
MELBA S. TUMARONG
HANNAHLY I. UMALI
9
SYNOPSIS
ANSWER KEY
c. f(x) = 5x2 – 20X + 22
2. a. f(x) = -3(x2 + 2x + 1) – 4
b. f(x) = 5x2 – 20x + 20 + 2
WHAT HAVE I LEARNED:
A. 1. B
2. A
3. C
B. 1. a. f(x) = 5(x2 – 4x + 4) + 2
ACTIVITY 3
3. e 4. c
5. A
2. d
1. b
ACTIVIY 2
2. 100 3. 144 4. 1 5. 225
1. 4
ABOUT THE AUTHOR
b. f(x) = -3x2 – 6x - 3 – 4
c. f(x) = -3x2 – 6x – 7
Come on and join us in discovering
the different ways of transforming
quadratic functions defined by y = ax2 +
bx + c into the form y = a (x-h)2 + k.
EVALUATION/POST TEST:
The learners are expected to
develop their skills and knowledge and
use them in real-life situations.
OBELUS
This Self Learning Kit is focused on
problem
solving
in
Mathematics
especially on transforming quadratic
functions defined by y = ax2 + bx + c into
the form y = a (x-h)2 + k.
PRE-ACTIVITIES/PRE-TEST:
ACTIVITY 1
1. a=1, b=-1, c=0
2. a=1, b=-3, c=7
3. a=1, b=4, c=10
4. a=3, b=-5, c=2
5. a=2, b=-4, c=0
CHRISTIAN EVEN D.
SANTILLAN is a
Teacher I of La
Libertad TechnicalVocational School.
He finished Bachelor of Secondary Education
Major in Mathematics at Negros Oriental State
University Main Campus in 2017.
REFERENCES
Abramson, J. January 9, 2019. Quadratic Functions. Mathematics LibreTexts.
Arizona State University: OpenStax College. Retrieved from
math.libretexts.org/Box
Bryant, M., et al. 2014. Mathematics Grade 9 Learner’s Module First Edition.
Pasig City: DepEd-IMCS
Quadratic Functions. 2005. EASE Module 1, Year 4. DepEd
10