Department of Education

Republic of the Philippines

Department of Education REGION III

SCHOOLS DIVISION OF TARLAC PROVINCE

1

Address: Macabulos Drive, San Roque, Tarlac City Telephone No.: (045) 982-0374 Email Address: [email protected]

QUADRATIC EQUATIONS, INEQUALITIES, AND FUNCTIONS

NO. MELC DURATION K to 12 CG Code

PAGE NO.

1 Illustrates quadratic equations Week 1

M9AL-Ia-1 4

2 Solves quadratic equations by: (a)

extracting square roots; (b) factoring; (c)

completing the square; and (d) using the

quadratic formula

M9AL-Ia-b-1

5

3 Characterizes the roots of a quadratic

equation using the discriminant

Week 2 to 3

M9AL-Ic-1 7

4 Describes the relationship between the

coefficient and the roots of a quadratic

equation

M9AL-Ic-2

9

T A B L E O F C O N T E N T S




2


5 Solves equation transformable to

quadratic equation equations (including

rational algebraic expressions).

M9AL-Ic-d-1

10

6 Solves problems involving quadratic

equations and rational algebraic

equations.

Week 4 M9AL-Ie-1

11

7 Illustrates quadratic inequalities Week 5

M9AL-If-1 14

8 Solve quadratic inequalities M9AL-If-2 16


equations and rational algebraic

equations.

M9AL-If-g-1

17

10 Models real-life situations using quadratic functions.

Week 6 M9AL-Ig-2 19

11 Represent quadratic function using: (a) table of values; (b) graph; (c) equation

M9AL-Ig-3 20

12 Transforms the quadratic function define by y = ax2 + bx + c into the form of y = a ( x – h )2 + k.

Week 7 to 8

M9AL-Ih-1

22

13 Graphs quadratic function: (a) domain; (b) range; (c) intercepts; (d) axis of symmetry; (e)vertex; (f) direction of the opening of the parabola

M9AL-Ig-h-i-1

23

14 Analyzes the effects of changing the values of a, h and k in the equation y = a ( x – h )2 + k of a quadratic function on its graph

M9AL-Ii-1

25

15 Determines the equation of a quadratic

function given: (a) a table of values; (b)

graph; (c) zeros

Week 9 M9AL-Ij-1

30


functions.

M9AL-Ii-j-2 31




3





4


L E S S O N I

ILLUSTRATING QUADRATIC EQUATIONS (Week 1)

Activity 1: Word Hunt!

Look for the following mathematical terms on the crossword puzzle. 1. quadratic 6. roots 11. formula 2. degree 7. discriminant 12. sum 3. equation 8. product 13. nature 4. factoring 9. square 14. Factors 5. completing 10. real 15. second

Activity 2: Yes or No!

Write YES if the given equation is a Quadratic Equation if not, write NO.

_________1. x2 + x + 2 = 4x4 ________ 6. y =x

x2+3

_________2. √2x2 +2

3x + 1 = 0 ________ 7. 2(x + 3)2 = 0

L E S S O N I




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_________3. 4p2 − 25 = 4p2 + p ________ 8. x2 − 2x = 5

_________4. w2 + 3x − 5 > 34 ________ 9. (x + 3) + 8 = 0

_________5. (x − 2)(x + 2) = 0 ________ 10. c = πd

Activity 3: Meet the Standard so the Values Appear.

Express the following in standard form and identify the values of a, b, and c.

Standard Form Values

1. 9x2 = x __________________ a = _____ b = _____ c = _____

2. 3x2 − 5 = 2x _________________ a = _____ b = _____ c = _____

3. 10x − 3 = 9x2 _________________ a = _____ b = _____ c = _____

4. x2 − 1 = 7x − 11 _________________ a = _____ b = _____ c = _____

5. 2x(x − 4) = 1 _________________ a = _____ b = _____ c = ____

SOLVING QUADRATIC EQUATIONS (Week 1)

Activity 1: Match Me.

Match column A to column B by solving the following quadratic equations through extracting the square roots. Write the letter of the correct answer on the space provided before each item

Column A Column B

L E S S O N 2




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Activity 2: Complete Me.

Solve the following quadratic equations by factoring then complete the table.

Factors Roots

1. 𝑥2 + 7𝑥 = 0

2. 𝑥2 + 3𝑥 + 2 = 0

3. 𝑥2 + 5𝑥 + 6 = 0

4. 𝑥2 + 3𝑥 − 10 = 0

5. 𝑥2 + 5𝑥 = 0

6. 𝑥2 − 𝑥 − 2 = 0

7. 𝑥2 + 2𝑥 − 15 = 0

8. 𝑥2 − 5𝑥 − 14 = 0

9. 6𝑥2 + 18𝑥 = 0

10. 𝑥2 − 25 = 0

Activity 3: Square Me!

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

1. x2 + 2x + 1 = ( )2

2. x2 − 4x + 4 = ( )2 3. n2 + 6n + 9 = ( )2

4. h2 − 22h + 121 = ( )2

5. r2 − 7r +49

4 = ( )2

Activity 4: Half of Me Square

Determine the b in the expression and find the number that must be added using

the formula [𝟏

𝟐𝐛] 𝟐 to the expression to make each of the following a perfect square

trinomial.

Expressions b [𝟏

𝟐𝐛] 𝟐

1. x2 + 6n + _________

2. g2 − 3g + _________




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3. f 2 − 12f + ________

4. 4t2 + 20t + _________

5. y2 +1

3t + _________

Activity 5: Find Me Completely!

Find the solutions of each quadratic equation by completing the square. Show your solution.

1. x2 − 10x − 50 = 0

2. x2 − 6x = 72

3. 2𝑥2 + 12𝑥 + 17 = 0

Activity 6: The Formula!

Complete the given table below and solve each quadratic equation using the Quadratic Formula. Item no. 1 is done for you.

Quadratic Equations A b c Roots

1. x2 − 8x + 12 = 0 1 -8 12 2, 6

2. x2 + 3x − 10 = 0

3. x2 − 2x − 2 = 0

4. 2x2 + x+= 6

5. 3𝑚2 − 13𝑚 + 14 = 0

CHARACTERIZING THE ROOTS OF A QUADRATIC EQUATION USING DISCRIMINANT (Week 2-3)

Activity 1: Count the Roots Give the value of the discriminant and determine the number of real roots.

L E S S O N 3




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Equations Discriminant Number of Real Roots 1. 𝑥2 + 3𝑥 + 1 = 0 _____________ _________________

2. 𝑦2 + 7y + 6 = 0 _____________ _________________

3. 𝑝2 − 10p + 12 = 0 _____________ _________________

4. 2𝑥2 − 7𝑥 + 16 = 0 _____________ _________________

5. 3𝑥2 + 7𝑥 − 2 = 0 _____________ _________________

Activity 2: Discriminant…

Use the discriminant to determine the nature of the roots of each quadratic equation. No. 1 is done for you.

Equation Discriminant

𝒃𝟐 − 𝟒𝒂𝒄

Nature of the Roots

1. 𝑥2 + 6𝑥 + 9 = 0 0 Roots are real, rational and equal

2. 𝑥2 + 6𝑥 = −3

3. 𝑥2 + 5𝑥 − 3 = 0

4. 2𝑥2 − 5𝑥 − 3 = 0

5. 3𝑥2 − 𝑥 = 2

Activity 3: No Real Solution?

Solve the quadratic equations using any convenient method. By using the discriminant, show that there is no real number solution. 1. 𝑥2 + 2𝑥 = 8 ____________________________________________

2. 𝑦2 − 2𝑦 = −2 ____________________________________________

3. 3𝑥2 + 𝑥 = 1 ____________________________________________

4. 𝑥(𝑥 + 5) = −3 ____________________________________________

5. 𝑥(𝑥 + 2) + 2(𝑥 − 2) = 1 ______________________________________




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DESCRIBING THE RELATIONSHIP BETWEEN THE COEFFICIENTS AND THE ROOTS OF A QUADRATIC EQUATION (Week 2-3)

Activity 1: Sum and Product….

Find the sum and product of roots of each quadratic equation.

Quadratic Equations Sum of Roots Product of Roots

1. 𝑥2 − 2𝑥 − 3 = 0

2. 𝑥2 = 10𝑥 − 25

3. 𝑥2 + 2𝑥 − 5 = 0

4. 𝑥2 − 3𝑥 + 12 = 0

5. 𝑥2 = −5𝑥 − 24

Activity 2: Add and Multiply Use the value of a, b, and c of each quadratic equations in determining the sum

and product of its roots. Verify your answers by obtaining the roots of each equations using any method. Item no. 1 is done for you.

Equations a b c Sum Product Roots

1. 1 4 3 -4 3 -1, -3

2.

3.

4.

5.

6.

7.

8.

9.

10.

Activity 3: Sum and Products on the Go

Write the quadratic equation with integral coefficient (no fractions in the coefficients) whose sum and product of the roots are given.

L E S S O N 4




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X1 + X2 X1(X2) Quadratic Equations

2 5

-3 -7

0 6 1

2 −

2

3

3

4

0

Activity 4: How Do I Look?

Write a quadratic equation whose roots are given below.

1. 5, 2 ___________________________________

2. 1, -10 ___________________________________

3. 7, 4 ___________________________________

4. -6, 1 ___________________________________

5. 2

3, 7 ________________________________

6. √2 , −√2 ___________________________________

SOLVING EQUATIONS TRANSFORMABLE TO QUADRATIC EQUATIONS (Week 2-3)

Activity 1: Transformer…

Transform each of the following equations into quadratic equation in the form ax2 + bx +c = 0. Write your answer on the table provide below.

Equation ax2 + bx +c = 0

1. 𝑥(𝑥 + 4) = 7

2. 2𝑥(𝑥 − 5) − 8 = 0

4. (𝑥 + 1)2 = 16

5. 𝑥(𝑥 + 3) = 28

6. 3𝑥(𝑥 − 2) − 12𝑥 = 0

L E S S O N 5




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7. (𝑥 − 6)2 = 15

8. (𝑥 + 1)2 + (𝑥 − 8)2 − 45 = 0

9. (𝑚 − 4)2 + (𝑚 − 7)2 = 15

10. 1

𝑥−

𝑥

6=

2

3

11. 1

𝑥+3+

𝑥

2= −2

Activity 2: Am I Right? Solve the following by any method. Write your answer on the blanks provided for. 1. 5𝑥2 = 5𝑥 + 1 __________________

2. 18𝑥 = 25 + 3𝑥2 __________________

3. 2𝑥2 − 8𝑥 = 9 __________________

4. 12𝑥2 + 5𝑥 = 2 __________________

5. 2𝑥(𝑥 − 3) = 8 __________________

Activity 3: Let Me Prove

Show the solution of each of the following equations.

1. 𝑥(𝑥 − 10) = −21

2. (𝑥 + 1)2 + (𝑥 − 3)2 = 15

3. 1

3𝑥+

4𝑥

6= 1

LESSON 6

SOLVING PROBLEMS INVOLVING QUADRATIC EQUATIONS AND RATIONAL ALGEBRAIC EQUATIONS (Week 4)

Activity 1: Representation…

Write a quadratic equation that will represent each problem.

1. Find two consecutive integers whose product is 42. __________________

L E S S O N 6




12


2. Find two consecutive even integers whose product is 624. _________________

3. If a number is added to its square, the result is 42. Find the number. _____________

4. A number less its reciprocal is 8

3 . Find the number. ___________________________

5. The length of a rectangular parking lot is 36m longer than its width. The area of the

parking lot is 5,152m2. What are its dimensions? _____________________________

Activity 2: What is the message? Solve the following word problems. Match Column A with Column B by writing the letter on the space provided. Then write this letter on the box below. Each number corresponds to a letter which will reveal a quotation if answered correctly.




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Activity 3: Solve Me Completely.

Solve the following problems completely by supplying the necessary things needed to show your solutions.

1. Find two positive real numbers that differ by 4 and have a product of 45. Given:

Asked:

Representation:

Equation:

Solution:

Checking:

2. The table used for table tennis is 4 feet longer than its width and has an area of 45 ft2. What are the dimensions of the table? Given:

Asked:

Representation:

Equation:

Solution:

Checking:

3. It takes Karla 1 hour longer to mow the lawn than it takes Sharon to mow the lawn. If they can mow the lawn in 5 hours working together, then how long would it take each girl to mow the lawn alone? (Use calculator and express your answer in decimal)

Given:

Asked:

Representation:

Equation:

Solution:

Checking:




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Activity 4: An Extra Challenge.

Find the exact solution to each problem.

1. The length of a rectangle is 2 ft longer that the width. If the area is 16 ft2, then what are the length and width?

2. A new computer can process a company’s monthly payroll in 1 hour less time than the old computer. To really save time, the manager used both computers and finished the payroll in 3 hours. How long would it take the new computer to do the payroll by itself?

3. One number is 5 greater than another number. If the sum of their squares is 5 times the square of the smaller number, what are the numbers?

LESSON 7

ILLUSTRATING QUADRATIC INEQUALITIES (Week 5)

Activity 1: Quadratic Inequalities or Not?

Shade the mathematical sentences which are not quadratic inequalities. Form the

hidden message by getting the corresponding letters from the unshaded mathematical

sentences.

What is the hidden message? _______________________________________________

L E S S O N 7




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Activity 2: Describe My Solutions!

Find the solution set of each of the following quadratic inequalities then graph. 1. 𝑥2 − 5𝑥 − 14 > 0 2. 3𝑥2 − 8𝑥 + 5 ≥ 0

Activity 3: Am I a Solution or Not?

Determine whether or not each of the following points is a solution of the inequality 𝑦 < 2𝑥2 + 3𝑥 − 5. Justify your answer.




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1. A ( 3, 6) 2. B ( - 3, 4) 3. C (-6, -2)

Activity 4: Make It Real!

Read the situation below then answer the questions that follow.

The floor of a conference hall can be covered completely with tiles. Its length is 36 ft. longer than its width. The area of the floor is less than 2,040 square feet.

1. How would you represent the width of the floor? How about its length? 2. What mathematical sentence would represent the given situation? 3. What are the possible dimensions of the floor? How about the possible areas of the floor?

LESSON 8

SOLVING QUADRATIC INEQUALITIES (Week 5)

Activity 1: Mathematical Word Power.

For each mathematical term, choose the correct meaning.

1. quadratic equation

a. ax + b = c with a ≠ 0 c. ax + b = 0 with a ≠ 0

b. ax2 + bx + c = 0 with a ≠ 0 d. a/x2 + b/x = c with a ≠ 0

2. quadratic inequality a. ax2 + bx + c > 0 with a ≠ 0or with ≥, < or ≤ b. ax + b = 0 with a ≠ 0 or with ≥, < or ≤ c. completing the square d. the Pythagorean theorem 3. quadratic in form

a. ax2 + bx + c = 0 b. a parabola c. an equation that is quadratic after a substitution d. having four equal sides 4. rational inequality a. an inequality involving a rational expression(s) b. a quadratic inequality

L E S S O N 8




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c. an inequality with rational exponents d. an inequality that compares two fractions 5. test point a. the end of a chapter b. to check if a point is in the right location c. a number that is used to check if an inequality is satisfied d. a positive integer

Activity 2: Truth or Consequence?

True or False. If your answer is False, explain why.

1. The solution set to 𝑥2 > 4 is (2,∞).

2. The inequality 𝑥

𝑥−3> 2 is equivalent to 𝑥 > 2𝑥 − 6.

3. The inequality (x – 1)(x + 2) < 0 is equivalent to x – 1 < 0 or x + 2 < 0. 4. We cannot solve quadratic inequalities which are not factored. 5. One technique for solving quadratic inequalities is identifying the 3 intervals

on the number line? 6. In solving quadratic or rational inequalities, we always get 0 on one side.

7. The inequality 𝑥

2> 3 is equivalent to x > 6.

Activity 3: Find the Truth in Me

Find the solution set of each of the following inequality to make it TRUE.

1. 𝑥2 + 𝑥 − 6 < 0 ________________________ 2. 𝑥2 − 3𝑥 − 4 ≥ 0 ________________________ 3. 𝑥2 + 25 < 10𝑥 ________________________ 4. 𝑥2 − 2𝑥 − 5 ≤ 0 ________________________ 5. 2𝑥2 − 8𝑥 + 3 < 0 ________________________

LESSON 9

SOLVING PROBLEMS INVOLVING QUADRATIC INEQUALITIES (Week 5)

Activity 1: What’s the Problem? Solve these problems.

L E S S O N 9




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1. A projectile is fired straight up from the ground with an initial velocity of 80 feet per seconds. Its height s(t) in feet at any time t is given by the function s(t) = -16t2 + 80t. Find the interval of time which the height of the projectile is greater than 96 feet.

2. If the initial velocity in No. 2 is 128 feet per second, when is the object is 112 feet or higher?

3. If the initial velocity in No. 2 is 240 feet per second, when is the object 112 feet or higher?

Activity 2: More Problems? “Practice makes perfect” as they say. So, here are some more problems to solve. 1. The total profit function P(x) for a company producing x thousands of pens is given by P(x) = -2x2 + 26x + 44. Find the values of x for which the company makes a profit. [Hint: The company makes a profit when P(x) > 0]. For Nos. 2 -3. An object tossed downward with initial velocity of V0 will travel a distance of s meters, where s = 4.9t2 + V0t and t is measured in seconds. 2. A screw falls off an airplane at a height of 500 meters. Approximately how long does it take the screw to reach the ground? 3. A coin is dropped from a helicopter at an altitude of 75 meters. Approximately how long does it take the coin to reach the ground?

Activity 3: Am I Improving? “Patience is a virtue” as they say. Solve the following problems to show your patience. 1. The height in feet for a ball thrown upward at 48 feet per second is given by s(t) = -16t + 48t, where t is the time in seconds after the ball is tossed. What is the maximum height that the ball will reach? 2. The monthly profit P (in Pesos) that Big Brother makes on the sale of x mobile

homes, is determined by the formula 𝑷 = 𝒙𝟐 + 𝟓𝒙 − 𝟓𝟎. For what values of x is his profit positive?




19


3. Susan’s revenue R (in Pesos) on the sale of x cupcakes is determined by the formula R = 50x – x2. Her cost C (in Pesos) for producing x cupcakes is given by the formula C = 2x + 40. For what values of x is Susan’s profit positive? (Profit = revenue – cost)

LESSON 10

MODELING REAL-LIFE SITUATIONS USING QUADRATIC FUNCTIONS (Week 6)

Activity 1: How could these be Real?

Identify a real-life experience that you can associate or relate with the given pictures. After which, answer the process questions that follow.

Process Questions: Write your answers in the box. 1. Name real-life situations that are represented by each picture.

L E S S O N I0




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2. What is common among the pictures? 3. How does each picture show the use of quadratic equations? 4. Can you name other real-life experiences that use the concepts of quadratic equations like the ones shown in the picture? 5. Name real-life situations or experiences or natural objects near you or around you that model quadratic equations.

Activity 2: Photo Collections.

On a bond paper, make a picture collage of real-life situations that models quadratic equations that you have seen or experienced. You may take screen shots, cut pictures or draw them.

LESSON 11

REPRESENTING QUADRATIC FUNCTIONS (Week 6)

Activity 1: Showing Justification. Determine which of the following are quadratic functions. Write QUADRATIC FUNCTION if it is. Otherwise write NOT A QUADRATIC FUNCTION. Show your solution/Justify your answer.

1.

𝑥 −2 −1 0 1 2

𝑦 −10 −5 0 5 10

2.

L E S S O N I1




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𝑥 0 1 2 3 4

𝑦 11 5 3 5 11

3.

4.

Activity 2: Give Justice to your Decisions. State whether each of the following functions represents a quadratic function or not. Then, justify your answer.

Activity 3: Which is Which.

Determine which of the following graphs represent a function. Tell which represents a Quadratic Function, if not just write FUNCTION, if it is not a function. just write MERELY RELATION.

𝑥 −4 −3 −2 -1 0

𝑦 2 3 6 11 18

𝑥 −4 −3 −2 -1 0

𝑦 2 3 6 11 18

Functions Yes or

No Justification/Reason

1. 𝑓(𝑥) = 𝑥2 + 2

2. 𝑔(𝑥) = 2𝑥 − 10

3. ℎ(𝑥) = 9 − 2𝑥2

4. 𝐽(𝑥) = 2𝑥 + 7

5. 𝑘(𝑥) = 𝑥3 + 4𝑥2

6. 𝐹(𝑥) = 2𝑥2

7. 𝑦2 = 𝑥2 − 9

8. 𝑚(𝑥) =4

𝑥2 − 1

9. 𝑓(𝑥) = (𝑥 − 2)(𝑥 + 3)

10. 𝑔(𝑥) = (𝑥 − 5)(𝑥 + 1) − 𝑦




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1. 2. 3.

4. 5. 6.

TRANSFORMING QUADRATIC FUNCTIONS DEFINED BY y = ax2 + bx + c INTO THE FORM y = a(x – h)2+ K (Week 7-8)

Activity 1: Standard to Vertex form.

Write each function in vertex form.

1. 𝑦 = 𝑥2 + 5𝑥 ________________ 6. 𝑦 = 2𝑥2 + 6𝑥 + 10 _____________

2. 𝑦 = 2𝑥2 − 6 ________________ 7. 𝑦 = 𝑥2 + 7𝑥 + 1 ________________

3. 𝑦 = −2𝑥2 − 8𝑥 _______________ 8. 𝑦 = 𝑥2 + 4𝑥 − 3 ________________

4. 𝑦 = 𝑥2 + 4𝑥 _________________ 9. 𝑦 = −3𝑥2 − 𝑥 − 8 _______________

5. 𝑦 = 𝑥2 − 4𝑥 + 4 ______________ 10. 𝑦 = 2𝑥2 + 8𝑥 + 3 _______________

L E S S O N I2




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Activity 2: Vertex form to Standard.

Write each function in standard form.

1. 𝑦 = (𝑥 − 1)2 + 2 ______________ 5. 𝑦 = −3(𝑥 − 2)2 + 4 _____________

2. 𝑦 = 3(𝑥 − 2)2 − 4 ______________ 6. 𝑦 = (𝑥 − 1)2 + 2 _ ______________

3. 𝑦 = 2(𝑥 − 1)2 − 1 ______________ 7. 𝑦 = −2(𝑥 + 5)2 − 3 ____________

4. 𝑦 = 4(𝑥 − 5)2 + 1 ______________ 8. 𝑦 = −5(𝑥 + 2)2 + 5 ____________

LESSON 13

GRAPHS OF QUADRATIC FUNCTIONS (Week 7-8)

Activity 1: What is In Me?

Graph the following given the table of values. Then answer what is asked. 1.

𝑥 −3 −2 −1 0 1

𝑦 4 1 0 1 4

Vertex: ____________

Axis of Symmetry: ______________________

x – intercepts: __________

y – intercept: ___________

Domain: ______________

Range: ________________

L E S S O N I3




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2.

Activity 2: The Beauty Inside Me.

Given the following graphs, analyze the components of the quadratic function.

1.

𝑥 −2 −1 0 1 2

𝑦 6 3 2 3 6

Direction of Opening: ___________________

Minimum/Maximum: ___________________

Vertex: ____________

Axis of Symmetry: ____

Minimum/Maximum Value: ______________

x – intercepts:________

y – intercept: ________

Domain: ____________

Range: _____________

Reflection of the y-intercept: __________

Vertex: ____________

Axis of Symmetry: ______

x – intercepts: _________

y – intercept: __________

Domain: ______________

Range: _______________




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2.

Activity 3: I am Proud of Me. Graph the quadratic function f(x) = x2 + 4x – 5 to approximate the following.

Direction of Opening:

______________________

Minimum/Maximum:

______________________

Vertex: ____________

Axis of Symmetry: _______

Minimum/Maximum

Value: ________________

x – intercepts:__________

y – intercept: ___________

Domain: _______________

Range: ________________

Reflection of the y-

intercept: _____________

• For what x-values is the curve increasing?

decreasing? (Write your answer as

inequality)________________________

_________________________________

• Vertex: _________________________

• Is the vertex the minimum or maximum

value? __________________________

• x – intercepts: ____________________

• y – intercept: _____________________

• Axis of Symmetry: _________________

• Reflection of the y-intercept: ________

• f(-5): ____________________________

• Domain: _________________________

• Range: ___________________________




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LESSON 14

ANALYZING THE EFFECTS OF CHANGING THE VALUES OF a, h, and k IN THE EQUATION y = a (x – h)2 + k OF A QUADRATIC FUNCTION ON ITS GRAPH (Week 7-8)

Activity 1. Graph Me and See What I can Be.

A. Graph the following functions in ONE CARTESIAN PLANE. Show your solutions in solving for the value of 𝑦.

1. 𝑦 = 𝑥2

2. 𝑦 = 𝑥2 + 3 Solution: 3. 𝑦 = 𝑥2 − 3 Solution:

𝑥 −2 −1 0 1 2

𝑦 4 1 0 1 4

𝑥

𝑦

𝑥

𝑦

L E S S O N I4




27


B. Complete the table.

Function Vertex Axis of Symmetry Minimum/Maximum Value

1. 𝑦 = 𝑥2

2. 𝑦 = 𝑥2 + 3

3. 𝑦 = 𝑥2 − 3

C. Short-response:

What happens to the graph when the value of 𝑘 is positive? negative? _______________________________________________________________ _______________________________________________________________ _______________________________________________________________




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Activity 2. Mixed Up? Fill up the following table of values and graph the functions in ONE CARTESIAN PLANE.

1. 𝑦 = 𝑥2

𝑥 −2 −1 0 1 2

𝑦 4 1 0 1 4

2. 𝑦 = 4𝑥2

𝑥 −2 −1 0 1 2

𝑦

3. 𝑦 = −𝑥2

𝑥 −2 −1 0 1 2

𝑦 −4 −1 0 −1 −4

4. 𝑦 = −1

2𝑥2

𝑥 −2 −1 0 1 2

𝑦 −2 −1

2 0 −

1

2 −2

5. 𝑦 = −2𝑥2

𝑥 −2 −1 0 1 2

𝑦




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B. Complete the table.

Function Vertex Direction of Opening Minimum/Maximum

1. 𝑦 = 𝑥2

2. 𝑦 = 4𝑥2

3. 𝑦 = −𝑥2

4. 𝑦 = −1

2𝑥2

5. 𝑦 = −2𝑥2




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C. Short Response: Write all your observations based on the graphs and other information you got above. ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

ACTIVITY 3. Don’t Be Misled! A. Determine which of the given 2 functions in each number is narrower. Write NARROWER beside the function.

1. 𝑦 = 3𝑥2 2. 𝑦 = −2𝑥2 3. 𝑦 =1

3𝑥2

𝑦 = 7𝑥2 𝑦 = 𝑥2 𝑦 = 4𝑥2 B. Determine which of the given 2 functions in each number is wider. Write WIDER beside the function.

1. 𝑦 = −8𝑥2 2. 𝑦 = 9𝑥2 3. 𝑦 =1

5𝑥2

𝑦 = −6𝑥2 𝑦 = 11𝑥2 𝑦 = 5𝑥2

LESSON 15

DETERMINING THE EQUATION OF A QUADRATIC FUNCTION (Week 9)

Activity 1: Who Am I?

To know the name of this Mathematician, Use the table of values below to determine the equation of the function in each item. Write the letter of the corresponding equation in the box above the corresponding answer.

𝑓(𝑥) = 𝑥2 − 5𝑥 + 6 𝑓(𝑥) = 𝑥2 − 2𝑥 − 8 𝑓(𝑥) = 𝑥2 − 3𝑥 + 2 𝑓(𝑥) = 𝑥2 − 2𝑥 − 8

L E S S O N I5




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𝑓(𝑥)

= 𝑥2 + 3𝑥+ 2

𝑓(𝑥)

= 𝑥2 − 2𝑥− 8

𝑓(𝑥)

= 2𝑥2 + 15− 8

𝑓(𝑥)

= 𝑥2 − 2𝑥− 3

𝑓(𝑥)

= 𝑥2 + 𝑥− 2

𝑓(𝑥)

= 𝑥2 − 5𝑥+ 6

𝑓(𝑥)

= 3𝑥2 + 9𝑥− 12

𝑓(𝑥)

= 𝑥2 − 2𝑥− 8 + 6

𝑓(𝑥)

= 2𝑥2

+ 15𝑥 − 8

Activity 2: A-maze Me!

Determine the equation of the quadratic function given the zeros. Follow the path of correct answers to complete the maze. START HERE




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SOLVING PROBLEMS INVOLVING QUADRATIC FUNCTIONS (Week 9)

Activity 1: Applications!

Solve the following word problems.

1. After t seconds, a ball tossed in the air from the ground level reaches a height of h feet given by the function h(t) = 144t – 16t2. a. What is the height of the ball after 3 seconds? b. What is the maximum height the ball will reach? c. After how many seconds will the ball hit the ground before rebound?

2. A rocket carrying fireworks is launched from a hill 80 feet above a lake. The rocket will fall into the lake after exploding at its maximum height. The rocket’s height above the surface of the lake is given by the function h(t) = -16t2 + 64t + 80. a. What is the height of the rocket after 1.5 seconds? b. What is the maximum height reached by the rocket? c. After how many seconds after it is launched will the rocket hit the lake?

3. Using the graph at the right, It shows the height h in feet of a small rocket t seconds after it is launched. The path of the rocket is given by the equation: h = -16t2 a. How long is the rocket in the air? b. What is the greatest height the rocket reaches? c. About how high is the rocket after 1 second? d. After 2 seconds, about how high is the rocket? is the rocket going up or going down? e. After 6 seconds, about how high is the rocket? is the rocket going up or going down? f. Do you think the rocket is traveling faster from 0 to 1 second or from 3 to 4 seconds? Explain your answer. g. Using the equation, find the exact value of the height of the rocket at 2 seconds. h. What is the domain of the graph? i. What is the range of the graph? j. Express the interval over which the graph is increasing.

L E S S O N I6




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k. Express the interval over which the graph is decreasing

Activity 2: Be the Problem Solver Master Solve the following problems completely.

1. A rock is thrown from the top of a tall building. The distance, in feet, between the rock and the ground t seconds after it is thrown is given by d(t) = -16t2 – 4t + 382. How long after the rock is thrown is it 370 feet from the ground?

2. Find the number of computer units x that needs to be sold to produce a maximum profit P (in pesos) if P(x) = 900x – 0.1x2.

3. The Perimeter of a rectangular ranch is 400 meters. Find the dimensions of the ranch that will contain the greatest area?

4. A cellular phone store can sell 1000 phone kits per month for Php 5,000 each. Research indicates that the store can sell 50 more kits per month for each Php 100 decrease in price. What price per kit will give the greatest monthly sales? What is the sales amount?

5. A skyrocket is shot into the air. It’s altitude h in feet after t seconds is given by the function h = - 16t2 + 128t. In how many seconds does the skyrocket reach its maximum height? What is its maximum height?

Activity 3. Blueprint in Solving Problems Learning must not STOP in this time of pandemic. No matter what kind of learning modalities we have, we need to embrace the changes brought about of what we call the “New Normal”. From what you have experienced in learning Math for the first quarter, you must have thought of some realizations brought about by the things you learned about your lessons this quarter. Here are some questions for you to answer and please answer them as honestly as possible.

1. How were the different math concepts you learned from this quarter enhanced your understanding about life? Did they make sense?

2. What is the essence of quadratic functions in relation to real-life experiences?




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3. Why do we use different methods to solve math problems?

4. Why do we need to use appropriate method in solving math problems?

5. What are the type pf problems that require minimum values?

6. What are the type of problems that require maximum values?

7. How were the activities enhances your self-confidence in solving problems?

A P P R O V A L S H E E T

CONTENT WRITERS:

VALENTINE T. AGUILAR ROMEO P. AGULLANA

AMELIA A. BONDOC LYDIA L. BUSTOS

RENELYN A. LLAPITAN CECILLE H. VILLAMAYOR

Reviewed by:

AUGUSTO L. BALLESTEROS DR. BOBBY P. CAOAGDAN EPSvr - Math Secondary EPSvr – LRMDS

Recommended by: DR. PAULINO D. DE PANO DR. MELISSA S. SANCHEZ Chief-Education Supervisor – CID Asst. Schools Division Superintendent

APPROVED:

RONALDO A. POZON, Ph. D., CESO V Schools Division Superintendent




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