Real World Application #1 ( Parabola)

1) Parabola: The set of all points that are equidistant from a point ( Focus/Foci) & a line ( Directrix ).

Vertical

https://people.richland.edu/james/lecture/m116/conics/translate.html

Horizontal

https://people.richland.edu/james/lecture/m116/conics/translate.html

http://en.wikipedia.org/wiki/Eccentricity_(mathematics)

2) The Parabola is a very common conic section found almost anywhere in the world. The naked eye is oblivious to it, but once you know what too look for, it'll be quite common.

http://en.wikipedia.org/wiki/Parabola

The Parabola has four basic parts that help define what type of parabola and its size. Those parts are the Axis of Symmetry, Focus, Vertex, and the Directrix. 

Axis of Symmetry:  The x-coordinate of the vertex is the equation of the axis of symmetry on the parabola.
Focus: The focus of a parabola is a fixed point on the interior of a parabola used in the formal definition of the curve.
Vertex: A vertex typically means a corner or a point where lines meet. The plural form of vertex is vertices.
Directrix: A line perpendicular to the axis of symmetry used in the definition of a parabola

The Equation for Parabolas is also different. These two forms are known as standard.

Parabolas: Standard Form (Vertex Form)

Vertical:

y = a (x - h)2 + k

If "a" is positive the parabola opens up and has a minimum value.

If "a" is negative the parabola opens down and has a maximum value.

|a| = 1; normal width

|a| > 1; narrow width (vertical stretch)

|a| < 1; wider width (vertical shrink)

vertex (h, k)

axis of symmetry: x = h

The minimum or maximum is the same as the "y" value of the vertex.

The vertex is the midpoint between the focus and the directrix.

The eccentricity of a parabola is the distance from the focus to any point on the graph divided by the distance from that same point on the graph to the directrix.  Since this the same for a parabola, the eccentricity, e = 1.

 Horizontal:

x = a (y - k)2 + h

If a is positive the parabola opens right

If a is negative the parabola opens left

|a| = 1 normal width

|a| > 1 narrow width (horizontal stretch)

|a| < 1 wider width (horizontal shrink)

vertex (h, k)

axis of symmetry: y = k

The vertex is the midpoint between the focus and the directrix.

The eccentricity of a parabola is the distance from the focus to any point on the graph divided by the distance from that same point on the graph to the directrix.  Since this the same for a parabola, the eccentricity, e = 1.

How Does the Focus relate to Eccentricity?

In mathematics, the eccentricity, denoted e  is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular.
In particular,
The eccentricity of a circle is zero.
Ellipses, hyperbolas with all possible eccentricities from zero to infinity and a parabola on one cubic surface.
The eccentricity of an ellipse which is not a circle is greater than zero but less than 1.
The eccentricity of a parabola is 1.
The eccentricity of a hyperbola is greater than 1.


3)  Real Life example

http://algebraproject07.wikispaces.com/What+is+a+catenary+and+what+are+some+real+examples%3F

I chose this as a real life examplse since it has multiple parabolas aother conic sections working together to interacrt as one giant maze of lines, with their foci being directed by the light/shade.

Youtube:

http://youtu.be/vj7u6RsoX6g

Links:

http://algebraproject07.wikispaces.com/What+is+a+catenary+and+what+are+some+real+examples%3F

http://en.wikipedia.org/wiki/Parabola

http://en.wikipedia.org/wiki/Eccentricity_(mathematics)

https://people.richland.edu/james/lecture/m116/conics/translate.html