Mathematics 9 Quarter 1 Module 15: Representing Quadratic Function Through Table of Values and Graphs

Mathematics Quarter 1-Module 15 Representing Quadratic Function Through Table of Values and Graphs

Mathematics

Quarter 1-Module 15

Representing Quadratic Function Through Table of Values and Graphs

Week 6

Learning Code – M9AL-Ig-11.1

[illustration: teacher on a laptop screen with students using computers, books, and speech bubbles]

GRADE 9

Learning Module for Junior High School Mathematics

Development Team of the Module

Writers: Jhon Edward M. Valera- TI Rowena F. Reyes- TI Marvin G. Sollera – MTI Analynn M. Argel – MTII

Editor: Sally C. Caleja – Head Teacher VI Rolando A. Ibayan – Head Teacher VI Corazon T. Misa – Head Teacher VI

Validators: Remylinda T. Soriano, EPS, Math Angelita Z. Modesto, PSDS George B. Borromeo, PSDS

Illustrator: Writers

Layout Artist: Writers

Management Team: Malcolm S. Garma, Regional Director Genia V. Santos, CLMD Chief Dennis M. Mendoza, Regional EPS in Charge of LRMS and Regional ADM Coordinator Maria Magdalena M. Lim, CESO V, Schools Division Superintendent Aida H. Rondilla, Chief-CID Lucky S. Carpio, Division EPS in Charge of LRMS and Division ADM Coordinator

Quarter 1 – Module 15 – New Normal Math for G9 First Edition 2020 Copyright © 2020

Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalties.

Borrowed materials (i.e. songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this module are owned by their respective copyright holders. Every effort has been exerted to locate and seek permission to use these materials from their respective copyright owners. The publisher and authors do not represent nor claim ownership over them.

Published by the Department of Education Secretary: Leonor Magtolis Briones Undersecretary: Diosdado M. San Antonio

MODULE 15

REPRESENTING QUADRATIC FUNCTION THROUGH TABLE OF VALUES AND GRAPHS

Aside from the equation form f(x) = ax² + bx + c or y = ax² + bx + c, where a, b, c are real numbers and a ≠ 0, we can also represent quadratic functions in many form. In this module, you will learn how to represent quadratic function through table of values and graphs.

WHAT I NEED TO KNOW

LEARNING COMPETENCY

The learners will be able to:

• represent a quadratic function using: (a) table of values and (b) graph

M9AL-Ig-11.1

WHAT I KNOW

Find out how much you already know about representing quadratic function through table of values and graphs. Write the letter that you think is the best answer to each question on your answer sheet. Answer all items. After taking and checking this short test, take note of the items that you were not able to answer correctly and look for the right answer as you go through this module.

1. Complete the given table of values with equation y = 2x² − 12x + 19.

x	0	1	2	3	4
y

A.

x	0	1	2	3	4
y	19	33	51	73	97

C.

x	0	1	2	3	4
y	19	9	3	1	3

B.

x	0	1	2	3	4
y	19	33	51	33	19

D.

x	0	1	2	3	4
y	9	3	1	3	9

2. Which of the following table of values does not represent a quadratic function?

A.

x	-2	-1	0	1	2
y	2	1	0	1	2

C.

x	-2	-1	0	1	2
y	4	1	0	1	4

B.

x	-2	-1	0	1	2
y	2	1	0	-1	-2

D.

x	-2	-1	0	1	2
y	8	1	0	-1	-8

GRADE 9

Learning Module for Junior High School Mathematics

3. Which of the following graph describes a quadratic function y = ax² + bx + c with a > 0?

A. [illustration: coordinate graph of a parabola opening upward, with the vertex near the origin below the x-axis]

C. [illustration: coordinate graph of a parabola opening downward, with the vertex above the x-axis to the right]

B. [illustration: coordinate graph of a parabola opening downward, to the left of the y-axis]

D. [illustration: coordinate graph with a zigzag curve crossing the x-axis several times]

4. Use the coordinates from the table to determine which equation models the table below:

x	-3	-2	-1	0	1	2	3	4
y	28	12	0	-8	-12	-12	-8	0

A. f(x) = x² − 3x − 4 C. f(x) = 2x² + 6x − 8 B. f(x) = 4x² − 12x − 16 D. f(x) = 2x² − 6x − 8

5. Which of the following represents the graph for your answer in number 4?

A. [illustration: coordinate graph of an upward-opening parabola with its lowest point near (1, −4)]

C. [illustration: coordinate graph of a narrow upward-opening parabola with its lowest point near (1, −12)]

GRADE 9

Learning Module for Junior High School Mathematics

B. [illustration: coordinate graph of an upward-opening parabola with its lowest point near (1, −12)]

D. [illustration: coordinate graph of an upward-opening parabola with its lowest point near (1, −4)]

6. Which of the following is true about this given table of values?

x	1	2	3	4	5
y	5	12	19	26	33

A. The y values are all positive. Hence, the table represents a quadratic function. B. The y values are all positive. Hence, the table does not represent a quadratic function. C. The first and second differences of the y values are equal. Hence, the table represents a quadratic function. D. The first differences of the y values are already equal. Hence, the table does not represent a quadratic function

7. If an object is dropped from a height of 39 feet, the function h(t) = −16t² + 39 gives the height of the object after t seconds. Graph the function.

A. [chart: line graph of h against t; h axis labeled 10 to 50, t axis labeled 1 to 4; curve decreasing from about 39 at t = 0 to 0 near t = 3]

C. [chart: line graph of h against t; h axis labeled 10 to 50, t axis labeled 1 to 4; curve rising steeply from 0 near t = 0]

B. [chart: line graph of h against t; h axis labeled 10 to 50, t axis labeled 1 to 4; curve decreasing from about 39 and reaching 0 near t = 2]

D. [chart: line graph of h against t; h axis labeled 10 to 50, t axis labeled 1 to 4; curve decreasing from about 40 toward 0 and flattening out]

GRADE 9

Learning Module for Junior High School Mathematics

8. Identify the graph of that best represent the equation y = x² − 4

A. [illustration: coordinate graph of a downward-opening parabola crossing the x-axis at −2 and 2, with y-intercept near 4]

C. [illustration: coordinate graph of an upward-opening parabola crossing the x-axis at −2 and 2, with y-intercept −4]

B. [illustration: coordinate graph of an upward-opening parabola crossing the x-axis at −2 and 2, with y-intercept −4 and a steeper curve]

D. [illustration: coordinate graph of an upward-opening parabola with its vertex near (0, 4) and no x-intercepts shown]

For Items 9 – 10, use the situation below

The equation y = x² – 12x + 45 models the number of books y sold in a bookstore x days after an award-winning author appeared at an autograph-signing reception.

9. Determine the table and graph that best describes the given situation.

A.

x	1	2	3	4	5	6	7
y	34	25	18	13	10	9	10

[illustration: coordinate graph of an upward-opening parabola with its lowest point near x = 6, y = 9]

C.

x	1	2	3	4	5	6	7
y	34	25	18	13	10	9	10

[illustration: coordinate graph of an upward-opening parabola with its lowest point near x = −6, y = 9]

GRADE 9

Learning Module for Junior High School Mathematics

B.

x	1	2	3	4	5	6	7
y	5	2	1	2	5	10	17

[illustration: coordinate graph of an upward-opening parabola with its lowest point near x = 3, y = 1]

D.

x	1	2	3	4	5	6	7
y	5	2	1	2	5	10	17

[illustration: coordinate graph of an upward-opening parabola with its lowest point near x = 3, y = 1]

10. What was the day that exactly 90 copies of the book were sold?
    A. A .12th day
    B. B .13th day
    C. C .14th day
    D. 15th day

WHAT’S IN

[handwritten: RECALL: If you are making table of value, just substitute the x value to get y]

COMPLETING TABLE OF VALUES

Complete the following table using the indicated function.

1. y = 2x − 1

[illustration: cartoon character wearing a cap]

x	y
-2
-1
0
1
2

Solution: Since y = f(x), to get the values of y and complete the table we need to do this

if x = -2, y = 2x − 1 f(x) = 2x − 1 f(−2) = 2(−2) − 1 f(−2) = −4 − 1 f(−2) = −5 then y = -5

if x = -1, y = 2x − 1 f(x) = 2x − 1 f(−1) = 2(−1) − 1 f(−1) = −2 − 1 f(−2) = −3 [transcriber’s note: the last line of this solution prints f(−2) where f(−1) is expected; transcribed as printed] then y = -3

GRADE 9

Learning Module for Junior High School Mathematics

if x = 0 y = 2x − 1 f(x) = 2x − 1 f(0) = 2(0) − 1 f(0) = 0 − 1 f(0) = −1 then y = -1

if x = 1, y = 2x − 1 f(x) = 2x − 1 f(1) = 2(1) − 1 f(1) = 2 − 1 f(1) = 1 then y = 1

if x = 2 y = 2x − 1 f(x) = 2x − 1 f(2) = 2(2) − 1 f(2) = 4 − 1 f(2) = 3 then y = 3

2. y = 2x² − 8x + 9

x	y
0
1
2
3
4

Solution: Since y = f(x), to get the values of y and complete the table we need to do this

if x = 0, y = 2x² − 8x + 9 f(x) = 2x² − 8x + 9 f(0) = 2(0)² − 8(0) + 9 f(0) = 2(0) − 0 + 9 f(0) = 0 − 0 + 9 f(0) = 9 then y = 9

if x = 1, y = 2x² − 8x + 9 f(x) = 2x² − 8x + 9 f(1) = 2(1)² − 8(1) + 9 f(1) = 2(1) − 8 + 9 f(1) = 2 − 8 + 9 f(1) = 3 then y = 3

if x = 2, y = 2x² − 8x + 9 f(x) = 2x² − 8x + 9 f(2) = 2(2)² − 8(2) + 9 f(2) = 2(4) − 16 + 9 f(2) = 8 − 16 + 9 f(2) = 1 then y = 1

if x = 3, y = 2x² − 8x + 9 f(x) = 2x² − 8x + 9 f(3) = 2(3)² − 8(3) + 9 f(3) = 2(9) − 24 + 9 f(3) = 18 − 24 + 9 f(3) = 3 then y = 3

if x = 4, y = 2x² − 8x + 9 f(x) = 2x² − 8x + 9 f(4) = 2(4)² − 8(4) + 9 f(4) = 2(16) − 32 + 9 f(4) = 32 − 32 + 9 f(4) = 9 then y = 9

You try!

Complete the following table using the indicated function

1. y = 2x + 1

x	y
-2
-1
0
1
2

2. y = x² + 2x − 1

x	y
-2
-1
0
1
2

3. y = 2x² + 4x − 6

x	y
-2
-1
0
1
2

WHAT’S NEW

Communication, Creativity, Critical Thinking and Collaboration

QUADRATIC OR NOT?

Activity #1: “Let’s Play some Blocks”

Examine the number of blocks per column. Predict the number of blocks on the last position.. You can share each other’s idea.

[transcriber’s note: the header “Position (x)” spans columns 1 to 5. The vertical label “N U M B E R Of B L O C K S (y)” spans all three rows and is placed in the first row only.]

Position (x)	1	2	3	4	5
N U M B E R Of B L O C K S (y)	[illustration: block pattern 1, position 1]	[illustration: block pattern 1, position 2]	[illustration: block pattern 1, position 3]	[illustration: block pattern 1, position 4]	_____
	[illustration: block pattern 2, position 1]	[illustration: block pattern 2, position 2]	[illustration: block pattern 2, position 3]	[illustration: block pattern 2, position 4]	_____
	[illustration: block pattern 3, position 1]	[illustration: block pattern 3, position 2]	[illustration: block pattern 3, position 3]	[illustration: block pattern 3, position 4]	_____

Which of this block patterns represents a Quadratic Function?

Activity #2: “Have you ever Wonder?”

[photo: basketball free-throw shot in an arena, with printed labels v = 8.552 m/s, θ = 51.89°, R = 7.239 m, h = 3.048 m]

[illustration: skateboarder in the air above a curved skate ramp, with the height labeled h]

Have you ever asked yourself why PBA star players are good in free throws?

Have you ever asked yourself why the ramp of skate boarding is designed like this?

What is common between these two pictures?

Which of this block patterns represents a Quadratic Function?

WHAT IS IT

QUADRATIC FUNCTION: TABLE OF VALUES

Equal Differences Method

A function represents a table of values having the differences of the y values. If the second differences of the y values are equal, then the given function is a quadratic function.

Example 1: Here are the answers for activity #1

Position (x)	1	2	3	4
N U M B E R	3 [illustration: block figure, position 1]	8 [illustration: block figure, position 2]	15 [illustration: block figure, position 3]	24 [illustration: block figure, position 4]	35
Of	1 [illustration: block figure, position 1]	3 [illustration: block figure, position 2]	5 [illustration: block figure, position 3]	7 [illustration: block figure, position 4]	9

[transcriber’s note: table continued from the previous page]

Position (x)	1	2	3	4
B L O C K S (y)	1 [illustration: block pattern 3, position 1]	3 [illustration: block pattern 3, position 2]	6 [illustration: block pattern 3, position 3]	10 [illustration: block pattern 3, position 4]	15

Let’s Investigate!

Block pattern #1

Position (x)	1	2	3	4	5
Number of blocks (y)	3	8	15	24	35

First differences: 5, 7, 9, 11 (The first differences are different)

Second differences: 2, 2, 2 (The second differences are equal)

The second differences are equal. Hence, this table represents a quadratic function

Block pattern #2

Position (x)	1	2	3	4	5
Number of blocks (y)	1	3	5	7	9

First differences: 2, 2, 2, 2 (The first differences are equal)

The first differences are already equal. Hence, this table does not represent a quadratic function

Block pattern #3

Position (x)	1	2	3	4	5
Number of blocks (y)	1	3	6	10	15

First differences: 2, 3, 4, 5 (The first differences are different)

Second differences: 1, 1, 1 (The second differences are equal)

The second differences are equal. Hence, this table represents a quadratic function

QUADRATIC FUNCTION: GRAPH

The graph of a quadratic function is called a parabola. It is basically a curved shape opening up or down.

When you have a quadratic function in form y = ax² + bx + c

if a > 0, then the parabola opens upward ∪,

if a < 0, then the parabola opens down ∩.

Both pictures on activity #2 shows curved shape like a parabola

[illustration: curved arc drawn over the basketball free-throw photo]

The other one is opening downward

One is opening upward

[illustration: skate ramp with a skateboarder, from activity #2]

How to represent a graph of a quadratic function?

Example 2:

Sketch the graph of the given quadratic function in the WHAT’S IN part.

y = 2x² − 8x + 9

Use the filled up table below:

x	y
0	9
1	3
2	1
3	3
4	9

Step 1: Complete the table

We already have solution in this part. Please see the WHAT’S IN part

Step 2: Plot the points on a Cartesian plane

[chart: Cartesian plane with plotted points; labels A = (0, 9), B = (1, 3), C = (2, 1), D = (3, 3), E = (4, 9)]

The coordinates are: (x,y)

A. (0,9)

B. (1,3)

C. (2,1)

D. (3,3)

E. (4,9)

Step 3: Connect the plotted points

[chart: parabola graph of y = 2x² − 8x + 9 on a Cartesian plane; labels A = (0, 9), B = (1, 3), C = (2, 1), D = (3, 3), E = (4, 9)]

y = 2x² − 8x + 9

Don’t forget to curve (smooth curve) it perfectly like a parabola

WHAT’S MORE

Complete the table of values and sketch the graph of the following quadratic functions: (Use a separate graphing paper for your solutions and graphs.)

1. y = −x²

x	y
0
1
2
3
-1
-2
-3

2. y = 2x² − 4x − 3

x	y
-3
-2
0
1
2
3
4
5

3. y = −2x² − 8x + 5

x	y
-2
-3
-1
-4
0
-5
1

WHAT I HAVE LEARNED

• A function is represented by a table of values with common differences of the y values. If the second differences of the y values are equal, then the given function is a quadratic function.
• The graph of a quadratic function is called a parabola. It is basically a curved shape opening up or down.
• When you have a quadratic function in the form y = ax² + bx + c
  • ✓ if a > 0, then the parabola opens upward ∪,
  • ✓ if a < 0, then the parabola opens down ∩.

WHAT I CAN DO

A. Determine which of the following table of values represents a quadratic function. Write Q if Quadratic Function and NQF if not.

x	-2	-1	0	1	2
y	5	2	1	2	5

x	-2	-1	0	1	2
y	-4	-1	2	5	8

x	-2	-1	0	1	2
y	1	0	1	4	9

x	1	2	3	4	5
y	5	10	15	20	25

x	-2	-1	0	1	2
y	0	-4	-6	-6	-4

B. Sketch the graph of each function. Use the allocated grids below:

1. y = 3x²

Use this table

x	y
-2
-1
0
1
2

2. y = 2x² − 16x + 33

Use this table

x	y
2
3
4
5
6

3. y = −x² + 2x + 1

Use this table

x	y
-1
0
1
2
3

[illustration: blank coordinate grid for item 1, with labeled x and y axes]

[illustration: blank coordinate grid for item 2, with labeled x and y axes]

[illustration: blank coordinate grid for item 3, with labeled x and y axes]

ASSESSMENT

Write the letter of the correct answer on your answer sheet. If your answer is not among the choices, write E together with your final answer.

1. Identify the table that represents the graph of quadratic function.

A.

x	-3	-2	-1	0	1
y	0	4	6	4	0

B.

x	-3	-2	-1	0	1
y	0	6	8	6	0

C.

x	-4	-2	0	2	4
y	0	4	6	4	0

D.

x	-4	-2	0	2	4
y	0	6	8	6	0

[chart: parabola opening downward on a Cartesian plane with x and y axes; tick labels printed on the x-axis include -6, -4, -2, 2, 4 and on the y-axis include 10, 8, 6, 4, 2, -2, -4. Values are not labeled on the chart.]

2. Which of the following table of values does not represents a quadratic function?

A.

x	-2	-1	0	1	2
y	-6	-6	-4	0	6

B.

x	0	1	2	3	4
y	0	1	4	9	16

C)

x	0	1	2	3	4
y	1	0	3	10	21

D)

x	-2	-1	0	1	2
y	-4	-1	2	5	8

3. Given the equation y = −2x² + 4x + 7, what value of m and n will make the table complete,

x	-3	-2	-1	0	1	2	3	4
y	m	-9	1	7	9	n	1	-9

A. -23 and 7

B. -13 and 7

C. -23 and 1

D. 1 and 7

4. Which of the following statements is FALSE about this given table of values?

x	0	1	2	3	4
y	-2	-1	2	7	14

A. The equation represented by the given table of values is y = x² − 2

B. The second differences of the y values of the given table are all equal to 2.

C. The first differences of the y values of the given table are different. Thus, the given table represents a quadratic function

D. The given table represents a quadratic function.

5. Which quadratic function opens downward?

A. y = 2x² + 3 B. y = x² + 4x − 6 C. y = −3x² + 3x + 7 D. 3y = 2x² − 4x + 9

6. Which graph best describe this table of values?

x	-3	-2	-1	0	1
y	18	8	0	-6	-10

A. [chart: graph; no number or title printed] B. [chart: graph; no number or title printed] C. [chart: graph; no number or title printed] D. [chart: graph; no number or title printed]

7. What function models the graph on the right?

[chart: graph; item 7, no number or title printed]

A. y = x² + 8x + 18 B. y = x² + 4x + 8 C. y = x² − 8x + 18 D. y = x² − 8x + 14

8. Identify the graph that best represent the equation y = x² + 4x − 7.

A. [chart: graph; choice A, no number or title printed] B. [chart: graph; choice B, no number or title printed] C. [chart: graph; choice C, no number or title printed] D. [chart: graph; choice D, no number or title printed]

For Items 9 – 10, use the situation below

Suppose you have 56 feet of fencing to enclose a rectangular dog pen. The function A = 28x − x², where x is the width, gives you the area of the dog pen in square feet.

9. Determine the table that best describes the given situation.

A.

Width (x)	10	11	12	13	14	15
Area (A)	180	187	192	195	196	195

B.

Width (x)	10	11	12	13	14	15
Area (A)	280	308	336	364	392	420

C.

Width (x)	10	11	12	13	14	15
Area (A)	100	121	144	169	196	225

D.

Width (x)	10	11	12	13	14	15
Area (A)	180	196	195	195	196	180

10. What width gives you the maximum area? What is the maximum area?

A) width = 28 ft; area = 196 ft² B) width = 28 ft; area = 420 ft² C) width = 14 ft; area = 588 ft² D) width = 14 ft; area = 196 ft

ADDITIONAL ACTIVITIES

Communication, Creativity, Critical Thinking and Character Building

THINK IT UP!

Think of an application of quadratic function in your life and perform the following task.

1. Create a quadratic function relating your real-life application
2. State a scenario to create a table if values using your given quadratic function.
3. Graph your quadratic function.

REFLECT!

What are the characteristics of a parabola that adds up to the skills of a basketball player in shooting accurately?

PROBLEM – BASED WORKSHEET

[illustration: seven photographs of wonders of the world, with labels]

Wonders of the world

• Chichen Icha
• Christ the Redeemer
• Great Wall of China
• Machu Picchu
• Petra
• Taj Mahal
• The Roman Colosseum

Let’s Analyze!

1. How many wonders of the world we have? _____________________________.
2. What do we call the graph of a quadratic function? _________________________.
3. In which of these wonders of the world you can see a parabola? _____________.

You may also check the following link for your reference and further learnings on representing quadratic function through table of values and graphs:

• https://www.youtube.com/watch?v=LCjA3jPDvAs
• https://www.youtube.com/watch?v=p0CEGUg8PXU

E-Search

REFERENCES

Morgan, F. M., & Paige, B. L. (n.d.). Algebra 2. pp 98-102. America: Henry Holt and Company

https://cdn.instructables.com/FJI/KOSW/IWQV486H/FJIKOSWIWQV486H.LARGE.jpg?auto=webp&frame=1&width=1024&height=1024&fit=bounds

https://i.ytimg.com/vi/fXyuzo0NRvY/maxresdefault.jpg

https://www.geogebra.org/graphing?lang=en

https://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut34_quadfun.htm

https://www.coursehero.com/file/37800346/math-9-lm-draft-3242014pdf/

https://cdn.kutasoftware.com/Worksheets/Alg1/Graphing%20Quadratic%20Functions.pdf

[transcriber’s note: this page is printed upside down; the content below is transcribed in reading order after rotation]

WHAT I CAN DO

A.

1. Q
2. NQF
3. NQF
4. Q
5. Q

B.

1. y = 3x²

x	y
-2	12
-1	3
0	0
1	3
2	12

2. y = 2x² − 16x + 33

x	y
2	9
3	3
4	1
5	3
6	9

3. y = −x² + 2x + 1

x	y
-1	-2
0	1
1	2
2	1
3	-2

[chart: graph; items 1 to 3 of part B, no number or title printed]

ASSESSMENT

1. B 2. A 3. A 4. C 5. C 6. D 7. C 8. B 9. C 10. D

PROBLEM – BASED WORKSHEET

1. 7
2. Parabola
3. The Roman Colosseum, Taj Mahal,

ANSWER KEY

WHAT I KNOW

1. C 2. B 3. A 4. D 5. C 6. D 7. B 8. C 9. A 10. D

WHAT’S IN

1. y = 2x + 1

x	y
-2	-3
-1	-1
0	1
1	3
2	5

2. y = x² + 2x − 1

x	y
-2	-1
-1	-2
0	-1
1	2
2	7

3. y = 2x² + 4x − 6

x	y
-2	-6
-1	-8
0	-6
1	0
2	10

WHAT’S MORE

1. y = −x²

x	y
-3	-9
-2	-4
-1	-1
0	0
1	-1
2	-4
3	-9

2. y = 2x² − 4x − 3

x	y
0	-3
1	-5
2	-3
3	3
4	13
5	27

3. y = −2x² − 8x + 5

x	y
-5	-5
-4	5
-3	11
-2	13
-1	11
0	5
1	-5

[transcriber’s note: this table is printed upside down; the row order was read from the rotated image and the values were checked against the equation, so verify the row order against the original PDF]

[chart: graph; Graphs 1 to 3 (shown under the heading “Graphs”), no values or titles printed]