“parabola and your life” quadratic functions. the zeros of quadratic functions the zeros of...
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“Parabola and Your Life”
Quadratic Functions
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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
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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?
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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.
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http://www.youtube.com/watch?v=1Pva-Iv43Nc
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Methods Used to Solve Quadratic Equations
1. Factoring
2. Completing the Square
3. Quadratic Formula
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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.
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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.
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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.
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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.
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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
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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.
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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
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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
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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
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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
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ROLES
Jomar Lapayag – Secretary, Researcher
Jomar Vitor Lapayag – Researcher, Presentor
Jomarvelous Lapayag – Presentor, Analyst
Jomarvelous Vitor Lapayag – Analyst, Powerpoint Presentation maker