“Parabola and Your Life”

Quadratic Functions

The Zeros of Quadratic Functions

The zeros of Quadratic Function f(x) = ax2 + bx + c can be found by letting f(x) = 0. Thus,

0 = ax2 + bx + c

Connecting to Real Life

A bookstore uses the quadratic function y = f(x) = - 0.00025x2 + 0.105x – 1.025 as the formula that approximates the profit y for the number of books sold x. At what point does the bookstore start to lose money?

solutionSet f(x) = 0F(x)= -0.00025x2 + 0.105x – 1.025 = 0

= -25x2 + 10500x – 102 500 = 0 = -x2 + 420x – 4 100 = 0 = x2 – 420x + 4100 = 0 = (x-10)(x-410) = 0

X - 10 = 0 or x – 410 = 0X = 10 x = 410

The company loses money if they sell fewer tha 10 books or more than 410 books.

http://www.youtube.com/watch?v=1Pva-Iv43Nc

Methods Used to Solve Quadratic Equations

1. Factoring

2. Completing the Square

3. Quadratic Formula

Why so many methods?- Some methods will not work for all equations.

-Variety is the- spice of life.

- Some equations are much easier to solve using a particular method.

FactoringFactoring is typically one of the easiest and quickest ways to solve quadratic equations;

however,

not all quadratic polynomials can be factored.

This means that factoring will not work to solve many quadratic equations.

Factoring (Examples)

Example 1

x2 – 2x – 24 = 0

(x + 4)(x – 6) = 0

x + 4 = 0 x – 6 = 0

x = –4 x = 6

Example 2

x2 – 8x + 11 = 0

x2 – 8x + 11 is prime; therefore, another method must be used to solve this equation.

Completing the SquareThis method will work to solve ALL quadratic equations;

however,

it is “messy” to solve quadratic equations by completing the square if a ≠ 1 and/or b is an odd number.

Completing the square is a great choice for solving quadratic equations if a = 1 and b is an even number.

Completing the Square (ExamplesExample 1

a = 1, b is even

x2 – 6x + 13 = 0

x2 – 6x + 9 = –13 + 9

(x – 3)2 = –4 x – 3 = ± 2i

x = 3 ± 2i

Example 2

a ≠ 1, b is not even

3x2 – 5x + 2 = 0

2 5 2 03 3

x x

2 5 25 2 253 36 3 36

x x

25 16 36

x

5 16 6

x

5 16 6

x

5 16 6

x

OR

x = 1 OR x = ⅔

Quadratic FormulaThis method will work to solve ALL quadratic equations;

however,

for many equations it takes longer than some of the methods discussed earlier.

The quadratic formula is a good choice if the quadratic polynomial cannot be factored, the equation cannot be written as (x+c)2 = n, or a is not 1 and/or b is an odd number.

Quadratic Formula (Example)

x2 – 8x – 17 = 0

a = 1b = –8c = –17

28 ( 8) 4(1)( 17)

2(1)x

8 64 68

2x

8 132

2x

8 2 33

2x

4 33

How importance is parabola to you? One of the benefits of Quadratic Functions to us is when we play basketball.

http://www.youtube.com/watch?v=dSRWY5vUHCU

What are your “UPS” and “DOWNS” in life and how these affect you?

UPS

• complete family

•3 times a day to eat

•Studying

• When I passed the subject

• when I represented the school from quiz bee

DOWNS

• Death of my mom, niece, and puppy.

•These help me not to give up but help me to become a better person

Resources:

• http://www.youtube.com/watch?v=1Pva-Iv43Nc

• Advanced algebra with trigonometry and statistics by Buzon, Olivia N.

• http://plus.maths.org/content/101-uses-quadratic-equation-part-i

ROLES

Jomar Lapayag – Secretary, Researcher

Jomar Vitor Lapayag – Researcher, Presentor

Jomarvelous Lapayag – Presentor, Analyst

Jomarvelous Vitor Lapayag – Analyst, Powerpoint Presentation maker