Mathematics 9 Quarter 1 Module 22: Equation of Quadratic Function Given Graphs and Zeros

Mathematics Quarter 1 – Module 22

Equation of Quadratic Function Given Graphs and Zeros Week 9 Learning Code – M9AL-Ij-15.2

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GRADE 9 Learning Module for Junior High School Mathematics

Quarter 1 – Module 22 – New Normal Math for G9 First Edition 2020 Copyright © 2020

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

Development Team of the Module

Writers: Analynn M. Argel- MTII Marvin G. Sollera -MTI

Editors: Sally C. Caleja- Head Teacher VI David Bowie U. Montales – Head Teacher III Cristina R. Solis – 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

GRADE 9 Learning Module for Junior High School Mathematics

WHAT I NEED TO KNOW

LEARNING COMPETENCY/IES The learners will be able to:

• Determine the equation of a quadratic function given a graph or zeros of the function. M9AL-Ij-15.2

WHAT I KNOW

Find out how much you already know about the equation of quadratic function given graphs and zeros. 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. What do you call the graph of the quadratic function?
   a. A) Circle B) Hyperbola C) Line D) Parabola
2. What are the zeros of the quadratic function = x² − 2x − 63 ?
   A) – 9 & – 7 B) – 9 & 7 C) – 7 & 9 D) 7 & 9
   [transcriber’s note: the printed function begins with “=” and no “y” is shown before it]
3. Which quadratic function has no zeros?
   A) y = x² − 7 C) y = −x² + 7
   B) y = −x² − 7 D) y = x² + 7x

For items 4 – 7, refer to the graph at the right.

4. What is the vertex of the graph?
   A) (0, –1)
   B) (0, 1)
   C) (– 1. 0)
   D) (1, 0)

[chart: graph of a downward-opening parabola with its vertex on the x-axis near x = −1, drawn on a grid with x-axis ticks from −4 to 3 and y-axis ticks from 1 to −7; no values are printed on the chart beyond the axis ticks]

[transcriber’s note: the decorative banner printed beside this heading reads “PPREPREVIER!”]

GRADE 9 Learning Module for Junior High School Mathematics

WHAT I KNOW (continued)

5. What is the zero(s) of the graph?
   A) – 1 B) 0 C) 0 and – 1 D) None
6. What is the y-intercept of the graph?
   A) – 1 B) 0 C) 1 D) None
7. Which equation define the given graph?
   A) y = −x² − 2x − 1 C) y = −x² + 2x − 1
   B) y = −x² − 2x + 1 D) y = −x² + 2x + 1
8. What quadratic function that passes the point (1, 1) and to its only zero of 2?
   A) y = x² − 4x − 4 C) y = x² − 4x + 4
   B) y = x² + 4x − 4 D) y = x² + 4x + 4
9. Find the equation of the quadratic function that doesn’t cross the x-axis and passes through the point (1, 7).
   A) y = x² + 5 C) y = x² − 5
   B) y = 2x² + 5 D) y = 2x² − 5
10. What quadratic function contain the points (–1, 0), (1, 0) and (2, 3)?
    A) y = −x² − 1 C) y = x² − 1
    B) y = −x² + 1 D) y = x² + 1

WHAT’S IN

It is easier to understand this module if you know how to find the coordinates of points, particularly the zeroes, on a graph of a quadratic function. Recall that zeros of the quadratic function are actually the x-intercepts of the parabola that corresponds to it.

Determine the zeroes of the quadratic function whose graphs are given below:

1. [chart: graph of an upward-opening parabola crossing the x-axis near x = −2 and x = 2, with its lowest point just below the origin; grid with x-axis ticks from −4 to 4 and y-axis ticks from 4 to −4; no values are printed beyond the axis ticks]

   
   

2. [chart: graph of a downward-opening parabola crossing the x-axis near x = −1 and x = 4, with its highest point near y = 4; grid with x-axis ticks from −2 to 6 and y-axis ticks from 4 to −4; no values are printed beyond the axis ticks]

   
   

GRADE 9 Learning Module for Junior High School Mathematics

3. [chart: graph of an upward-opening parabola with its lowest point near the origin’s left side; grid with x-axis ticks from −5 to 3 and y-axis ticks from −8 to 1; no values are printed beyond the axis ticks]

   
   

4. [chart: graph of a downward-opening parabola with its vertex on the x-axis near x = −2; grid with x-axis ticks from −4 to 4 and y-axis ticks from 1 to −7; no values are printed beyond the axis ticks]

   
   

5. [chart: graph of an upward-opening parabola with its lowest point near x = −2, y = 2; grid with x-axis ticks from −5 to 3 and y-axis ticks from 1 to 8; no values are printed beyond the axis ticks]

   
   

6. [chart: graph of an upward-opening parabola with its lowest point near x = 1, y = 1; grid with x-axis ticks from −4 to 4 and y-axis ticks from 1 to 8; no values are printed beyond the axis ticks]

   
   

How did you find the activity? Were you able to spot the zeroes of the functions? Can you give other points on the parabola aside from the zeroes?

WHAT’S NEW

The graph of a quadratic function is a smooth curve called a parabola. A parabola may intersect the x–axis at two points, one point or no point at all.

[chart: three diagrams of a parabola with a vertical and horizontal axis line; the first intersects the horizontal line at two points, the second touches it at one point, the third does not meet it; labeled “two points”, “one point”, “no point”]

The x-coordinates of the points of intersection of the parabola and the x-axis are the zeroes of the quadratic function if it exists. Graphically, these are the x-

GRADE 9 Learning Module for Junior High School Mathematics

intercepts of the graph of a quadratic function. An x-intercept is a point on the graph that intersects the x-axis.

How will the points on the parabola, particularly the x-intercepts or the zeroes of the function, help in determine the equation of the quadratic function represented by the parabola?

WHAT IS IT

Deriving Quadratic Function Given Its Graph

If the graphs representing quadratic function y = ax² + bx + c is given, the equation representing the graph can be determined given any of the following conditions: a. Three points on the parabola none of which is an x-intercepts nor a vertex. b. The x-intercepts and a point on the parabola for graphs where the coordinates of the vertex are not integers. c. A vertex and a point on the parabola

Take note that it is more convenient if the points to consider are integral values. Study the examples below.

Example 1: Determine the quadratic function y = ax² + bx + c represented by the given graphs.

a. [chart: graph of an upward-opening parabola crossing the x-axis near x = −2 and x = 1, with its lowest point just below the origin; grid with x-axis ticks from −4 to 4; no values are printed beyond the axis ticks]

b. [chart: graph of a downward-opening parabola with its vertex at the origin’s left, near x = −1, y = 0; grid with x-axis ticks from −4 to 4 and y-axis ticks from 1 to −7; no values are printed beyond the axis ticks]

Solution: a. In the graph, the most convenient points to considers are the x-intercepts, (-2, 0) and (1,0) and the y-intercept (0, -2)

Given three points on the graph, you can find the equation of quadratic function creating systems of equation in three unknowns. However, if the x-intercepts are given, you can have an alternate solution:

GRADE 9 Learning Module for Junior High School Mathematics

If x = -2 and x = 1 are the x-intercepts, then (x + 2) and (x – 1) are factors of the equation of quadratic function. Thus,

y = a (x + 2) (x -1)

Take note that the value of a is not always 1 and the third point on the graph can help solve for the value of a. Substitute (0, -2) to the equation,

-2 = a (0 + 2) (0 – 1) -2 = -2a a = 1

Substitute the obtained value to the equation and simplify,

y = 1(x + 2) (x – 1) y = x² + x – 2

b. In this graph, the most convenient points to consider are the vertex (-1, 0), the y-intercept (0, -1) and any of the points (-3, -4) or (1, -4). However, if the vertex is given, you only need one of the point mentioned above.

Substitute the coordinates of the vertex (-1, 0) in the vertex form of the equation of quadratic function,

y = a (x+1)² + 0

Find the value of a by choosing any of the points (0, -1), (-3, -4) or (1, -4), then substitute:

-1 = a (0 + 1)² -1 = a

Substitute a = -1 in the equation, then transform it into y = ax² + bx + c;

y = -1 (x + 1)² → y = -x² – 2x – 1

Deriving Quadratic Function Given Its Zeroes

If the zeroes are given, then the equation of the quadratic function can be determined. Recall that zeroes, x₁ and x₂, of the function are the roots, of the equivalent quadratic equation,

y = (x – x₁) (x – x₂) → 0 = (x – x₁) (x – x₂)

Example 2: Determine the quadratic function given the following zeroes: a. 5, -3 b. 4, 2/3 c. 1 + √3

GRADE 9 Learning Module for Junior High School Mathematics

Solution: a. If 5 and -3 are zeroes of quadratic function y = ax² + bx + c, then (x – 5) and [x –(-3)] are factors of y, y = (x – 5) (x + 3) → y = x² – 2x – 15

b. If 4 and 2/3 are zeroes of quadratic function y = ax² + bx + c, then (x – 4) and (x – 2/3) are factors of y, y = (x – 4) (x – 2/3) y = (x – 4) (3x – 2) y = 3x² – 14x + 8

c. If 1 + √3 is a zero of quadratic function y = ax² + bx + c, then its conjugate 1 – √3 is also a zero of the function, thus [x + (1 + √3 )] and [x – (1 – √3 )] are factors of y,

y = [x – (1 + √3 )] [x – (1 – √3 )] y = [(x – 1) – √3 ] [ (x – 1) + √3 ] y = (x – 1)² – (√3)² y = x² – 2x + 1 – 3 y = x² – 2x – 2

How did you find the given examples? Were you able to understand how to derive equations given the graph or zeros of a quadratic function? If not, go back to those parts that you find challenging and study further.

WHAT’S MORE

A. Determine the quadratic function y = ax² + bx + c represented by the given graphs.

1. [chart: graph of an upward-opening parabola with its lowest point near x = −2, y = 2; grid with x-axis ticks from −5 to 3 and y-axis ticks from 1 to 8; no values are printed beyond the axis ticks]

   
   

2. [chart: graph of an upward-opening parabola with its lowest point near x = 0, y = −3; grid with x-axis ticks from −4 to 4 and y-axis ticks from 6 to −4; no values are printed beyond the axis ticks]

   
   

B. Determine the quadratic function given the following zeroes:

1. 2, -3 11. ±√3
2. 5, 7 12. 2 – √3
3. -1, -4 13. 1 + √5
4. 6, -5 14. -3 + √2
5. 8, 1 15. -4 – √7
6. 1/2, 2
7. 5, −1/3
8. 1/4, 3/4
9. 3/5, −1/5
10. −3/4, 4/3

How did you find the activity? Did you get all the equations of quadratic functions from the graphs and zeros? If not, in which part did you find challenging? How did you cope up with it?

WHAT I HAVE LEARNED

1. If the graphs representing quadratic function y = ax² + bx + c is given, the equation representing the graph can be determined given any of the following conditions:
   a. Three points on the parabola, not x-intercepts nor vertex
   b. The x-intercepts and a point on the parabola, if coordinates a vertex are not integral
   c. A vertex and a point of the parabola
2. The zeroes, x₁ and x₂, of the function are the roots, of the equivalent quadratic equation. That is,
   y = (x – x₁) (x – x₂) → 0 = (x – x₁) (x – x₂)

Now that you are equipped with knowledge on determining the equation of quadratic function given graphs and zeros, it’s about time to find out what you can do.

WHAT I CAN DO

Critical Thinking

A. Determine the quadratic function y = ax² + bx + c represented by the given graphs

1. [chart: parabola graph opening upward; item 1 of WHAT I CAN DO]
2. [chart: parabola graph opening downward; item 2 of WHAT I CAN DO]
3. [chart: parabola graph opening upward; item 3 of WHAT I CAN DO]
4. [chart: parabola graph opening downward; item 4 of WHAT I CAN DO]

[transcriber’s note: the graphs have axes and grid lines but no printed values, so no coordinates are given for them]

5. [chart: parabola graph opening upward; item 5 of WHAT I CAN DO]
6. [chart: parabola graph opening upward; item 6 of WHAT I CAN DO]

B. Determine the quadratic function given the following zeroes:

1. 5, 8 6. 2/3, −1/6
2. -2, 7 7. ±√5
3. 11, -4 8. 1 ± √2
4. 3/2, 1 9. -2 ±3√3
5. 1/2, 3/2 10. (3 ± √17)/4

ASSESSMENT

Read each item carefully. Identify the choice that best completes the statement or answers the question

1. What do you call the graph of the quadratic function?
   A) Circle B) Hyperbola C) Line D) Parabola

   
   

2. What are the zeros of the quadratic function = 2𝑥² − 7𝑥 + 3 ?
   A) – 3 & –1/2 B) – 3 & 1/2 C) –1/2 & 3 D) 1/2 & 3

   
   

3. Which quadratic function has no zeros?
   A) 𝑦 = 𝑥² − 4 C) 𝑦 = −𝑥² + 4
   B) 𝑦 = −𝑥² − 4 D) 𝑦 = 𝑥² + 4𝑥

   
   

For items 4 – 7, refer to the graph at the right.

4. What is the vertex of the graph?
   A) (0, –1)
   B) (0, 1)
   C) (– 1. 0)
   D) (1, 0)

[chart: parabola graph opening downward with vertex near the origin; referenced by items 4–7]

5. How many zero(s) does the graph have?
   A) 0 B) 1 C) 2 D) None

   
   

6. What is the range of the graph?
   A) (–∞, 0) B) (–∞, 0] C) (0, ∞) D) [0, ∞)

   
   

7. Which equation define the given graph?
   A) 𝑦 = −𝑥² − 2𝑥 − 1 C) 𝑦 = −𝑥² + 2𝑥 − 1
   B) 𝑦 = −𝑥² − 2𝑥 + 1 D) 𝑦 = −𝑥² + 2𝑥 + 1

   
   

8. What quadratic function that passes the point (2, 1) and to its only zero of 3?
   A) 𝑦 = 𝑥² − 6𝑥 − 9 C) 𝑦 = 𝑥² − 6𝑥 + 9
   B) 𝑦 = 𝑥² + 6𝑥 − 9 D) 𝑦 = 𝑥² + 6𝑥 + 9

   
   

9. Which of the following equations of the quadratic function that doesn’t cross the x-axis and passes through the point (1, 4).
   A) 𝑦 = 𝑥² − 2𝑥 + 5 C) 𝑦 = 3𝑥² − 5𝑥
   B) 𝑦 = 2𝑥² − 3𝑥 + 1 D) 𝑦 = 𝑥² + 2𝑥 − 5

   
   

10. What quadratic function contain the points (–5, 0), (0, 0) and (2, 14)?
    A) 𝑦 = −𝑥² − 5𝑥 C) 𝑦 = 𝑥² − 5𝑥
    B) 𝑦 = −𝑥² + 5𝑥 D) 𝑦 = 𝑥² + 5𝑥

    
    

ADDITIONAL ACTIVITIES

Communication, Critical Thinking, Creativity and Character Building

Activity 1: Parabola Around Us

There are a lot of objects around us that looks like a parabola like the architectural design of Mactan Cebu international airport.

For this activity, you will explore the rich arts and architectures of the Philippines that have parabolic design.

[photo: interior of an airport terminal with a curved, arched roof of glass and steel]

1. Find an example of a real parabolic curve found in the Philippines
2. Research about its origin and its actual size.
3. Using any graphing applications (android apps, internet app, computer software, and a like), insert the picture into the rectangular plane and adjust the scale to its actual size.
4. Insert a point for the vertex of your picture and determine its coordinates.
5. Locate for additional points that your graph pass through. Determine the coordinates of this points.
6. Using these points, find the equation in general form of the parabola formed by your picture.

PROBLEM – BASED WORKSHEET

A soccer player passes the ball to a teammate, and the teammate kicks the ball. The graph displays the relation of the height in feet of the ball, in feet, t seconds after it is kicked.

[chart: parabola graph of height of the ball; horizontal axis -1, -0.5, O, 0.5, 1, 1.5, 2; vertical axis labeled y with values 5, 6, 7, 8 marked; a soccer player and ball are drawn beside the curve]

Let’s Analyze!

1. What is the maximum height that the ball could reach?
2. After how many seconds did the ball reached the maximum height?
3. Determine the function h(t) from the given graph.

E-Search

You may also check the following link for your reference and further learnings on determining quadratic function given table of values.

• https://www.youtube.com/watch?v=jLzkaJk0iZ0
• https://www.youtube.com/watch?v=vAPPYoBV2Ow
• https://www.youtube.com/watch?v=OXViZtD2BTE
• https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadraticfunctions-equations/x2f8bb11595b61c86:quadratic-formsfeatures/e/rewriting-expressions-to-reveal-information

REFERENCES

MacKeague, Charles, Intermediate Algebra, Concepts and Graphs. Saunders College Publishing, USA

Mathematics 9 Learner’s Material, Department of Education

Ogena, Ester, et. al. Our Math Grade 9. Mc Graw Hill, Vibal Group. Inc

https://www.radford.edu/rumath-smpdc/Performance/src/Leslie%20Cumbow%20-%20Discovering%20Quadratics.pdf

Illustrations:

https://www.freepik.com/free-vector/woman-with-long-hair-teachingonline_7707557.htm https://www.freepik.com/free-vector/kids-having-online-lessons_7560046.htm https://www.freepik.com/free-vector/illustration-with-kids-taking-lessons-onlinedesign_7574030.htm http://www.usccg.com/tag/quality-control/ http://www.erpbusinessschool.lk/programmes/professional-diploma-in-enterpriseresource-planning/careers_chemd_guygirllab_0/ https://www.dezeen.com/2018/12/03/mactan-cebu-international-airportphilippines-integrated-design-associates/

[transcriber’s note: this page is printed upside down; it was read by rotating it. Readings marked [?] are uncertain]

ANSWER KEY

WHAT I KNOW

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

WHAT’S IN

1. -2, 1 2. -1, 3 3. -3, 1 4. -1 5. none 6. none

WHAT’S MORE

A.

1. y = x² + 4x + 6
2. y = x² − x − 2
3. y = x² − 6x + 7 [?]
4. y = x² + 2x + 1
5. y = 2x² − 4x + 4 [?]
6. y = x² − 2x + 2 [?]

B.

1. y = x² + x − 6
2. y = x² − 12x + 35
3. y = x² + 5x + 4
4. y = x² − x − 30
5. y = x² − 9x + 8
6. y = 2x² − 5x + 2
7. y = 3x² − 14x − 5
8. y = 9x² − 16x + 3 [?]
9. y = 25x² − 10x − 3
10. y = 12x² − 7x − 12
11. y = x² − 3
12. y = x² − 4x + 1
13. y = x² − 2x − 4
14. y = x² + 6x + 7
15. y = x² + 8x + 9

WHAT I CAN DO

A.

1. y = −x² − 6x + 11 [?]
2. [illegible]
3. [illegible]
4. [illegible]
5. [illegible]
6. [illegible]

B.

1. y = x² − 13x + 40 [?]
2. y = x² − 5x − 14 [?]
3. y = x² − 7x − 44 [?]
4. [illegible]
5. [illegible]
6. [illegible]
7. [illegible]
8. [illegible]

ASSESSMENT

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

ADDITIONAL ACTIVITIES

Use this rubric for checking your output.

[table: RUBRIC FOR ACTIVITY]

Element	0	1	2
Picture is of a parabola	No picture is turned in	Picture is of a parabola	Picture is of a parabola
Dimensions of parabola are noted	No dimensions given	Dimensions given incorrectly	Dimensions of parabola are noted
Picture is inserted into graph	No picture inserted	N/A	Picture inserted correctly
Vertex and second point are identified	Points are not correctly identified	One point is correctly identified	Both points are identified correctly
Equation in vertex form is computed	Equation is not done	Equation is computed with error	Equation is computed correctly

PROBLEM – BASED WORKSHEET

1. 8 ft
2. After 0.5 second of ½ second
3. h(t) = −16t² + 16t + 4