Mathematics

Quarter 1 - Module 7

Transforms the quadratic function defined by

into the form and vice versa

Department of Education ● Republic of the Philippines

9

Math- Grade 9

Alternative Delivery Mode Quarter 1 - Module 7: Transforms the quadratic function

First Edition, 2020

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Mathematics

Quarter 1 - Module 7

Transforms the quadratic function defined by

into the form and vice versa

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

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We value your feedback and recommendations.

Department of Education ● Republic of the Philippines

9

Table of Contents

What This Module is About ................................................................................................................... i

How to Learn from this Module ........................................................................................................... ii

Icons of this Module ............................................................................................................................... ii

Lesson 1:

Transforms the quadratic function defined by

into the form and vice versa ....................................... 1

What I Need to Know.................................................................................................. 1

What I Know ................................................................................................................ 2

What’s In ........................................................................................................................ 3

What’s New .................................................................................................................. 4

What Is It ...................................................................................................................... 5

What’s More ................................................................................................................. 8

What I Have Learned.................................................................................................. 10

What I Can Do .............................................................................................................. 11

Summary

Assessment: (Post-Test)

Key to Answers ..................................................................................................................................

References ...........................................................................................................................................

What This Module is About

Have you ever asked yourself why some basketball players are great shooters? How

do expert players of Angry Birds hit their targets? Do you know the secret key or techniques

in playing such games? What is the maximum height an object can reach when thrown

vertically upward given a particular condition?

One of the most interesting topics in mathematics and extremely useful in

engineering, statistics, physics, and any other math- or science-related field is quadratic

functions. The concept of quadratic functions has played a fundamental role in providing

solutions to many problems related to human life. In this module, you will be able to learn the

basic concepts of quadratic functions, which will enable you to answer the questions above.

i

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

What I Need to This part contains learning objectives that

Know are set for you to learn as you go along the

module.

What I know This is an assessment as to your level of

knowledge to the subject matter at hand,

meant specifically to gauge prior related

knowledge

What’s In This part connects previous lesson with that

of the current one.

What’s New An introduction of the new lesson through

various activities, before it will be presented

to you

What is It These are discussions of the activities as a

way to deepen your discovery and under-

standing of the concept.

What’s More These are follow-up activities that are in-

tended for you to practice further in order to

master the competencies.

What I Have Activities designed to process what you

Learned have learned from the lesson

What I can do These are tasks that are designed to show-

case your skills and knowledge gained, and

applied into real-life concerns and situations.

ii

Lesson

1

Transforms the Quadratic Function

Defined by into the form

and vice versa

What I Need to Know Let us start this lesson by recalling the concepts of completing the square,

perfect square trinomial, and square of a binomial to solve quadratic equations. The

knowledge and mathematical skill mentioned will help you transform quadratic functions

written in the form into its vertex form and vice

versa. This new skill of identifying and transforming a quadratic function in its vertex form is essential because quadratic function written in vertex form is more convenient to use when solving problems involving their graphs and vertices.

What I Know

1. The quadratic function is expressed in vertex form as

a. c.

b. d.

2. What is when written in the form ?

a. c.

b. d. 3. The graph of a quadratic function is called a a. straight line c. hyperbola b. parabola d. ellipses 4. Which of the following is NOT a quadratic function?

a. c.

b. d.

5. What constant do you add to complete the expression ?

a. 3 c. 9

b. -3 d. -9

6. In the quadratic function of , what are the values of ?

a. c.

b. d.

7. The expression can be expressed as which of the following?

a. c.

b. d.

1

For #8-10.

8. What is the value of a? a. 0 b. 2 c. 4 d. 5 9. What is the value of h?

a. b. c. 4 d. -4

10. What is the value of k?

a. b. c. 4 d. -4

What’s In

Activity 1 #LearnAsOne

Express each of the following perfect square trinomial as a square of a binomial.

1. 5. 9.

2. 6. 10.

3. 7.

4. 8.

Activity 2 The Secret During Pandemic

Match the given square of a binomial to its equivalent perfect square trinomial. Write the corresponding letter of your answer to find out the secret during pandemic.

1. t

2. O

3. Y

4. D

5. O

6. R

7. A

8. G

9. P

2

Activity 3 Completing the Square to Square of a Binomial

Perform completing the square to the following equations. Then simplify the perfect square trinomial to obtain a square of a binomial.

1.

2.

3.

4.

5.

What’s New

Fill in the boxes by following the steps found on the right side.

1. Group together the terms containing . = ⎕ Factor out the coefficient of .

= 3 Make the expression enclosed in

parentheses a perfect square trinomial by completing the squares and SUBTRACT the constant to the constant term outside the parentheses in order to maintain equality to the quadratic function. Note that the constants added are on the same side of the equation. That is why we have to subtract, NOT add, the second constant.

= 3 You should arrive at this expression.

Express the perfect square trinomial as

the square of the binomial and simplify.

2.

3

What Is It

What you did in the previous page was that you have transformed a quadratic

function written in the form into the form . It is

important to note that the form is called the standard form by some

references; others recognized it as the general form and that their standard form is the

vector form which is . To avoid confusion, in this module, the form

is NOT labeled as the standard form of quadratic function but is

referenced by its equation instead. The form is labeled as the vertex

form of the quadratic function. It is essential to know the vertex form because this is suitable to use when working with problems involving vertex and graph of a quadratic function, which will be discussed in the next module. Below are more examples of transforming

into vertex form and vice versa.

Example 1:

Express in the vertex form and give the values of

and .

Solution:

Hence, can be expressed as . The values

of and are +2 and +4, respectively. Take note that h in this case is NOT because

the vertex form is always and NOT . Example 2:

Transform into its vertex form and give the values of and .

Solution:

Hence, can be expressed as . The values of

and are 6 and 4, respectively.

4

There is another way though to rewrite a quadratic function into

its vertex form. Let us transform it.

Group together the terms containing

Factor out . Here, a=1.

Complete the enclosed expression to make it into a perfect square trinomial by

adding a constant Subtract the

same value from the constant term outside the parentheses to maintain equality.

Express the perfect square trinomial as the square of the binomial and simplify.

The vertex form is . Thus, and . With

this formula, we can convert a quadratic function in the form into the

vertex form without performing completing the square. Let us answer the first example in the previous page using this solution. Example 3:

Express in the vertex form using the values of and .

Solution:

.

and

The values of and are 2 and 4, respectively. Substituting them to the vertex form

, we obtain , which is the same with our answer in

Example 1.

5

The value of is also equal to or the function of . In other words, is equal to

. Here is a quick hack to obtain the value of in the example above:

Substituting the value of which is 2 to the variable ,

Now try answering Example 2 using and and find

out if your answer is the same with when using completing the square. Example 4:

Express into the form .

Solution:

Expand the square of a binomial.

Simpify.

Hence can be written as . Example 5:

Rewrite into the form .

Solution:

Hence can be written as .

6

What’s More

Activity 4 A Colorful Time

Identify each of the following quadratic functions below if they have the correct values of

and . If the quadratic function has the correct values of BOTH and , color the box that

contains the corresponding number of the quadratic function. If NOT, leave it uncolored. You can choose any color you want.

1.

2.

3.

4.

5.

6. 7. 8.

9.

10.

1 4 9 5 10 7 9 4 9 10 5 4 1 9 7 10 7

5 2 1 4 8 5 2 3 8 6 1 8 10 5 4 6 1

7 6 3 7 8 1 2 5 5 7 9 6 7 10 1 2 4

9 8 7 6 3 4 3 8 3 2 7 8 9 7 9 2 9

10 3 1 5 2 9 8 1 9 9 9 3 9 6 9 3 5

5 2 9 4 6 7 8 2 3 6 10 1 2 5 8 7 7

10 4 7 1 9 10 5 7 7 5 10 1 5 10 1 5 4

7 8 5 10 6 4 6 3 8 6 9 8 3 2 8 10 9

1 3 8 9 6 7 2 4 5 2 5 6 1 10 9 6 5

10 2 4 2 8 1 3 1 4 3 4 3 10 5 4 3 10

5 6 1 7 3 5 8 1 4 6 4 2 3 8 6 4 1

7 3 5 5 2 9 6 10 10 8 9 2 4 6 9 1 7

9 10 1 7 5 4 2 3 8 2 7 8 4 5 2 1 5

10 4 4 10 9 1 7 9 10 9 5 2 4 1 10 3 9

7 1 5 10 1 9 7 4 5 10 7 1 7 4 5 9 1

8 10 5 4 6 10 5 8 2 3 1 9 8 10 7 4 7

3 3 9 6 8 7 6 9 5 9 3 7 6 5 4 1 9

2 1 8 4 2 5 8 1 4 7 8 9 3 10 10 5 4

6 9 7 10 3 1 3 2 2 8 6 4 2 1 7 9 10

8 5 4 7 6 4 3 7 4 1 2 9 2 9 5 7 4

8 4 9 10 6 1 2 5 9 10 2 1 6 1 10 9 7

2 4 5 9 2 7 8 10 5 4 8 5 8 6 2 3 5

10 5 7 1 9 10 1 1 7 7 9 4 4 5 9 10 1

7

Activity 5 Reversing the Process

Rewrite the quadratic functions written in vertex form into the form by

following the steps.

Steps Quadratic Function

1. Expand

2. Multiply the perfect square trinomial by 2

3. Add 4

4. Simplify

5. Result

Steps Quadratic Function

1. Expand

2. Multiply the perfect square trinomial by 3

3. Add 5

4. Simplify

5. Result

Steps Quadratic Function

1. Expand

2. Add 11

3. Simplify

4. Result

Steps Quadratic Function

2. Expand

2. Multiply the perfect square trinomial by 2

3. Add 2

4. Simplify

5. Result

8

What I Have Learned

Activity 6 Cloud Data Storage

In the cloud storage below, write your answer in paragraph for the following questions: (1) When do you think is it necessary to express a quadratic function in the form of

? (2) When is it necessary to express it in vertex form? (3) Which solution

do you find easier in transforming a quadratic function (completing the square or using and

values) and state the reason(s) why?

__________________________________________

_________________________________________________________________

______________________________________________________________________

_________________________________________________________________________

__________________________________________________________________________

____________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

_____________________________________________________________________________

________________________________________________________________________________

_____________________________________________________________________________________

______________________________________________________________________________________

_____________________________________________________________________________________

_________________________________________________________________

______________________________________________________________

(4) Give one quadratic function written in the form of and

transform it to its vertex form. Then, transform the vertex form back to its form

. Show your solution for the whole process.

_____________________________________________________________________

____________________________________________________________________

__________________________________________________________________

__________________________________________________________

______________________________________________________

_______________________________________

________ ___________________

9

What I Can Do

Solve the following problem and write your solution on the space provided below:

While playing basketball, you are attempting to make a 3-point shot. The height in meters of the ball thrown at an angle of is given by the quadratic function

, where is the time in seconds after throwing. The ball’s horizontal

distance in meters from you is modeled by . Assuming the ball went inside the ring,

what is the horizontal distance from you to the ring?

10

Summary

In this lesson, you were guided on how to transform quadratic functions into either of the two forms, or the vertex form . As mentioned,

different authors disagree on the standard form of the quadratic function. In this module, we

label by its equation and not as the standard form.

There are two ways to transform from to the vertex form. One

method is to perform completing the square to obtain a perfect square trinomial and then

contract it into a square of a binomial. The second one is to use the formula

and

or , which is derived from performing completing the square to the

function . The values of and are then substituted to the vector form

. To write the quadratic function in vector form into , one just expand the square of a binomial and simplify.

This mathematical skill is essential especially when you encounter or solve problems involving graphs of quadratic functions, which will be discussed in the next module. We should also not undervalue the basic concepts of quadratic functions as they have many applications in engineering, statistics, physics, any other math- or science-related field, and our everyday life.

11

Assessment: (Post-Test)

1. A parabola is a shape that is formed by the graph of what polynomial function? a. linear c. exponential b. quadratic d. cubic

2. A quadratic function is equivalent to which of the following?

a. c.

b. d.

3. The vertex form of is which of the following?

a. c.

b. d.

4. What is the value of for number 3?

a. c.

b. d.

5. What is of the quadratic function ?

a. c.

b. d.

6. The vector form is transformed to which of the following function?

a. c.

b. d.

7. In the equation , what constant do you add to the trinomial

in order to become a perfect square?

a. c.

b. d.

8. What is for quadratic function ?

a. c.

b. d.

9. The quadratic function in the previous item can be transformed into which of the following form?

a. c.

b. d.

10. What is of the quadratic function if ?

a. b. c. d.

12

Key to Answers

What I Know

1. a 2. c 3. b 4. c 5. d 6. c 7. a 8. b 9. a 10. c Activity 1 #LearnAsOne

1. 7. 10.

2. 5. 8.

3. 6. 9.

Activity 2 The Secret During Pandemic

1-9. PRAY TO GOD

Activity 3 Completing the Square to Square of a Binomial

1. 2. 3. 4. 5.

What’s New

2. Activity 4 A Colorful Time

1.

2. True 3. True

4.

5.

6. True 7.

8. True

9.

10.

Activity 5 Reversing the Process

1. 3.

2. 4.

Assessment: (Post-Test) 1. b 2. c 3. d 4. c 5. a 6. b 7. c 8. b 9. d 10. d

What I Can Do

Nine meters. 13

1 4 9 5 10 7 9 4 9 10 5 4 1 9 7 10 7

5 1 4 5 1 10 5 4 1

7 7 1 5 5 7 9 7 10 1 4

9 7 4 7 9 7 9 9

10 1 5 9 1 9 9 9 9 9 5

5 9 4 7 10 1 5 7 7

10 7 1 9 10 5 7 7 5 10 1 5 10 1 5 4

7 5 10 4 9 10 9

1 9 7 4 5 5 1 10 9 5

10 4 1 1 4 4 10 5 4 10

5 1 7 5 1 4 4 4 1

7 5 5 9 10 10 9 4 9 1 7

9 10 1 7 5 4 7 4 5 1 5

10 4 4 10 9 1 7 9 10 9 5 4 1 10 9

7 1 5 10 1 9 7 4 5 10 7 1 7 4 5 9 1

10 5 4 10 5 1 9 10 7 4 7

9 7 9 5 9 7 5 4 1 9

1 4 5 1 4 7 9 10 10 5 4

9 7 10 1 4 1 7 9 10

5 4 7 4 7 4 1 9 9 5 7 4

4 9 10 1 5 9 10 1 1 10 9 7

4 5 9 7 10 5 4 5 5

10 5 7 1 9 10 1 1 7 7 9 4 4 5 9 10 1

References

Bryant, M. et al. (2014). Quadratic Functions. In Debbie Marie B. Versoza (Ed.),

Mathematics Grade 9 Learner’s Material (1st ed., pp. 120-139). Pasig City,

Philippines: DepEd-BLR.

“Vertex Form of Quadratic Functions,” MathBitsNotebook,

https://mathbitsnotebook.com/Algebra1/Quadratics/QDVertexForm.html.

“The Square Of A Binomial/Perfect Square Trinomials.” Lawrence Spector, 2020,

http://www.themathpage.com/Alg/perfect-square-trinomial.htm

“Perfect-Square Trinomials,” Elizabeth Stapel, PurpleMath,

https://www.purplemath.com/modules/specfact3.htm.

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