RWA #1: Unit M Concepts 4-6 - Conic Sections in real life (parabola)

1. Mathematical Definition of a Parabola- "The set of all points equidistant from a given point known as the focus and a given line known as the directrix." (http://www.lessonpaths.com/learn/i/unit-m-conic-section-applets/parabola-drawn-from-definition-geogebra-dynamic-worksheet)

2. Algebraically: The equation for a vertical hyperbola is (x-h)^2=4p(y-k).
                           The equation for a horizontal hyperbola is (y-k)^2=4p(x-h).
When the vertex of a parabola is at the origin,  you must see if the graph is y^2 or x^2. Then, you must put in the right value for p based on if you are given the focus or directrix. The standard form would then be (x-h)^2= or (y-k)^2=, which is the equation. In the equation, h and k represent the vertex, or center of the graph. P tells us what way the graph goes. The term that is squared tells us the direction of the parabola.

This link explains the parts that parabolas have and it shows diagrams as well for reference. It is a great reference to use while learning about parabolas.
http://www.purplemath.com/modules/parabola.htm

Graphically:

This picture shows where the parts of a parabola are when it is graphed.
(http://www.teacherschoice.com.au/images/parabola_types.gif)

                  

This picture shows what way the parabola will face according to the equation.
(http://home.windstream.net/okrebs/Ch6-35.gif)

The shape of a parabola is a U, but it has many components to go along with it. The vertex of a parabola is (h,k). It is important to rememer that x always goes with h and y anways goes with k. P is the direction and distance that the vertex is from the focus. P also determines if the graph goes up, down, left, or right, depending on whether is is x^2 or y^2 and positive or negative. The axis of symmetry cuts the parabola in half. It is also perpendicular to the directrix. The directrix is found outside the parabola. It is p units away from the vertex, just as the focus is. The distance away from the vertex to the focus can also determine how wide or narrow the parabola is. The farther the focus is from the vertex, the wider it gets. The variable that is squared in a parabola deterimines the direction.
               
3. Real World Application

This is a Parabolic Heater. A parabola is what makes this heater function. (http://content.costco.com/Images/Content/Product/284457.jpg)

            

This video explains how parabolas are used in everyday things. Here it explains how the Parabolic Heater works. (http://www.youtube.com/watch?v=fV9YuF__fM4)

 

A Parabolic Heater is an example of something that uses a parabola to work. The heat source is located at the focus. It then bounces off the back to be re-directed back to the person. It bounces off in parallel lines. "The circular shape of the heater provides more energy efficiency than other electric space heaters. The parabolic design converts nearly 80 percent of electric energy into radiant heat."

( http://www.ehow.com/facts_7163048_parabolic-heater_.html )

Parabola's are found everywhere. They are found in things like architecture and even nature. The shape of the parabola is what makes some things work, like the Parabolic Heater. Certain things that use parabolas to work must be constructed precisely and designed accurately in order for them to work. It is amazing how parabolas make certain things work smoothly.

4. References

• http://www.teacherschoice.com.au/images/parabola_types.gif
• http://home.windstream.net/okrebs/Ch6-35.gif
• http://www.youtube.com/watch?v=fV9YuF__fM4
• http://content.costco.com/Images/Content/Product/284457.jpg
• http://www.ehow.com/facts_7163048_parabolic-heater_.html
• http://www.purplemath.com/modules/parabola.htm
• http://www.lessonpaths.com/learn/i/unit-m-conic-section-applets/parabola-drawn-from-definition-geogebra-dynamic-worksheet