1. Taylor H. Asia J. Period 1 :)
Quadratic Functions
Taylor H.
Asia J.
Period 1 :)

2. Definition ;
Quadratic function; a function that can be described by an equation of a form f(x)=ax^2+bx+c

3. Examples; f(x) = -2x2 + x - 1 f(x) = x2 + 3x + 2 f(x) = x2 + 2x - 3
g(x) = 4x2 - x + 1
h(x) = -x2 + 4x + 4

4. Real World Examples
Quadratic equations not only described the orbits along which the planets moved round the Sun, but also gave a way to observe them more closely.

5. Real World Examples 2
Building a 72-story skyscraper like Trump Tower in New York City is no easy feat. That’s where structural engineers like Ysrael Seinuk come in. This movie explains how shapes are at the foundation of structural engineering

6. Real World Examples 3
The science of optics teaches us how to make "bigger eyes", and if you want to look at something very far away, or very faint, you're going to need a really huge eye--such as the one used at Kitt Peak by astronomers studying distant galaxies

7. Nature Examples

8. Nature Examples Quadratic Equations
An example of a Quadratic Equation:
                                                                                                 
The name Quadratic comes from "quad" meaning square, because the variable gets squared (like x2).
It is also called an "Equation of Degree 2" (because of the "2" on the x)
The Standard Form of a Quadratic Equation looks like this:
                                                                                           
The letters a, b and c are coefficients (you know those values). They can have any value, except that a can't be 0.
The letter "x" is the variable or unknown (you don't know it yet)
Here are some more examples:
In this one a=2, b=5 and c=3
This one is a little more tricky:
Where is a? In fact a=1, as we don't usually write "1x2"
b = -3
And where is c? Well, c=0, so is not shown.
Oops! This one is not a quadratic equation, because it is missing x2 (in other words a=0, and that means it can't be quadratic)
Nature Examples
Hidden Quadratic Equations!
So the "Standard Form" of a Quadratic Equation is
ax2 + bx + c = 0
But sometimes a quadratic equation doesn't look like that! For example:
In disguise →
In Standard Form
a, b and c
x2 = 3x -1
Move all terms to left hand side
x2 - 3x + 1 = 0
a=1, b=-3, c=1
2(w2 - 2w) = 5
Expand (undo the brackets), and move 5 to left
2w2 - 4w - 5 = 0
a=2, b=-4, c=-5
z(z-1) = 3
Expand, and move 3 to left
z2 - z - 3 = 0
a=1, b=-1, c=-3
5 + 1/x - 1/x2 = 0
Multiply by x2
5x2 + x - 1 = 0
a=5, b=1, c=-1
                          
Have a Play With It
I have a "Quadratic Equation Explorer" so you can see:
the graph it makes, and
the solutions (called "roots").
How To Solve It?
The "solutions" to the Quadratic Equation are where it is equal to zero. There are usually 2 solutions (as shown in the graph above).
They are also called "roots", or sometimes "zeros"
There are 3 ways to find the solutions:
1. You can Factor the Quadratic (find what to multiply to make the Quadratic Equation)
2. You can Complete the Square, or
3. You can use the special Quadratic Formula:
                                                                              
Just plug in the values of a, b and c, and do the calculations.
We will look at this method in more detail now.

9. Nature Examples

10. Graphs Of Functions

11. Graphs Of Functions

12. Graph Of Quadratic Function

13. Sources

14. Sources

15. Sources