MODULE 12

• DISCIPLINE • GOOD TASTE • EXCELLENCE
• Writer


  Florefe T. Narne
• Cover Illustrator

  Joel J. Estudillo
• MATHEMATICS Quarter 1: Module 12 Finding the Equations of Quadratic Functions
• Department of Education National Capital Region SCHOOLS DIVISION OFFICE MARIKINA CITY
• Quadratic function


  A function that can be written in the form f(x) = ax^2 + bx + c, where a, b, and c are real numbers and a ≠ 0
• In this module, you will learn how to:

  1. Formulate the equation of a quadratic function given its

  1. table of values,
  2. graph and
  3. zeros and

  2. Find the zeros of quadratic functions.
• The table of values below describes a quadratic function

  x -1 0 1 2 3

  y 2 -1 -2 -1 2
• To formulate the equation of a quadratic function given its table of values:

  Choose three (3) ordered pairs (x, y) from the table.

  2. Substitute each chosen ordered pair in y = ax^2 + bx + c.

  3. Solve for the values of a, b and c.

  4. Substitute a, b and c in y = ax^2 + bx + c.
• The COVID 19 cases are rapidly increasing each day as shown on the table below.

  No. of days 1 2 3 4 5 6 7 8 9

  No. of cases 5 11 21 35 53 75 101 131 165
• One way of finding the equation of a quadratic function is a table of values. A table of values describes a quadratic function if equal differences in the values of the independent variable x produces equal second differences in the values of the dependent variable y.
• To determine the equation of a quadratic function given its table of values:

  Choose three (3) ordered pairs (x, y) from the table.

  2. In one ordered pair, substitute x and y in y = ax^2 + bx + c. Repeat step 2 in the remaining two ordered pairs.

  3. Solve a, b, and c using the derived system of linear equations.

  4. Substitute a, b and c in y = ax^2 + bx + c.
• Steps to formulate the equation:

  Step 1: Choose 3 ordered pairs (1,20), (3,46), (4,68)

  Step 2: Substitute the 3 ordered pairs in y = ax^2 + bx + c

  Step 3: Solve the system of linear equations to find a, b, c

  Step 4: Substitute a, b, c in y = ax^2 + bx + c
• The table shows the first few terms of a sequence.

  Term number, (n) 1 2 3 4 5

  Term of sequence, f(n) 1 2 6 13 23
• Steps to formulate the equation of a quadratic function:

  Step 1:

  Step 2:

  Step 3:

  Step 4:

  Step 5:
• Number of sides (x) and number of diagonals (y) of a polygon


  • x y

  3 0

  4 2

  5 5

  6 9

  7 14

  8 20
• Formulating the equation of a quadratic function given its zeros

  1. Step 1: Set each zero equal to x

  2. Step 2: Equate each to zero

  3. Step 3: Set the product of the factors equal to zero

  4. Step 4: Multiply the factors

  5. Step 5: Set f(x) = 0
• The x-intercepts of quadratic functions are the zeros of the function
• To find the equation of the quadratic equation given its zeros, set each zero equal to x, equate each to zero, set the product of the factors equal to zero, multiply the factors, and set f(x) = 0
• There is a mathematical concept applied in the picture of Ted's smile. Explain.
• 1. intercepts


  The zeros of the quadratic function
• To find the equation of the quadratic equation given its zeros

  1. Set each zero equal to x

  2. Equate each to zero

  3. Set the product of the factors equal to zero

  4. Multiply the factors

  5. Set f(x) = 0
• Zeros of quadratic functions

  • 2 and -3
  • √5 and -√5
  • 1/3 and 2/3
• To find the equation of the quadratic function with given zeros

  1. Set each zero equal to x

  2. Equate each to zero using APE

  3. Express the equations in factored form and equate to zero

  4. Multiply

  5. Simplify

  6. Set f(x) = 0
• The equation of the quadratic function with zeros 2 and -2 that passes through 1 and -6 is y = 2(x^2 - 4)
• Given the zeros of the quadratic functions f(x) = a(x - z1)(x - z2) where a = 1

  • -1 and 3
  • 2 and -5
• The process for finding the zeros of a quadratic function includes extracting the square root, factoring, completing the square, and using the quadratic formula
• Quadratic functions


  • f(x) = x^2 - 6x + 5
  • f(x) = x^2 + 6x + 5
  • f(x) = x^2 - 4x - 5
  • f(x) = (x+2)(x-3)
  • f(x) = (x-2)(x-3)
• The zeros of a quadratic function can be determined by setting f(x) = 0 or y = 0
• Methods to find the zeros of a quadratic function

  1. Extracting the square root

  2. Factoring

  3. Completing the square

  4. Using the quadratic formula
• Quadratic functions

  • f(x) = x^2 - 5
  • y = x^2 + 6x - 7
• The zeros of the quadratic function f(x) = x^2 - 5 are ±√5
• The zeros of the quadratic function y = x^2 + 6x - 7 are 1 and -7
• Quadratic functions

  • f(x) = x^2 + 19x - 20
  • g(x) = x^2 - 4x - 28
  • h(x) = x^2 + 14x + 49
  • s(x) = x^2 + 6x - 27
  • t(x) = 2x^2 - 24x + 72
• Zeros of quadratic functions

  • 6, 6
  • 1, -20
  • -7, -7
  • -9, 3
  • 7, -4
• Quadratic functions


  • y = x^2 - 8
  • y = 9x^2 + 6x + 1
  • f(x) = 2x^2 + 6x + 2
  • y = x^2 + 2x - 15
• The methods to find the zeros of a quadratic function are: extracting the square root, factoring, completing the square, and using the quadratic formula
• The zeros of the quadratic function f(x) = x^2 + 3x - 2 are
• Zeros

  • f(x) = x^2 + 19x - 20 (6, 6)
  • g(x) = x^2 - 4x - 28 (1, -20)
  • h(x) = x^2 + 14x + 49 (7, -4)
  • s(x) = x^2 + 6x - 27 (-9, 3)
  • t(x) = 2x^2 - 24x + 72 (-7, -7)
• Find the zeros using the specified method

  1. y = x^2 - 8, extracting the square root

  2. y = 9x^2 + 6x + 1, factoring

  3. f(x) = 2x^2 + 6x +2, completing the square

  4. y = x^2 + 2x - 15, quadratic formula