Conic Sections

Parabolas

1. One of the conic sections formed in a cone is a parabola. The mathematical definition for a parabola is described as the following: the distance from the focus to any point on the graph (parabola) and from that point directly perpendicularly down to the directrix is the same. A parabola is constructed when it has a focus on the inside and the directrix on the outside.The distance from the vertex to the focus is the same  as the distance from the vertex to the directrix and this value is known as "p".  A parabola is created on a cone when it is sliced so that it is cut parallel to the slope of the cone. 

2.  How far the focus is from the vertex of the parabola or how close it is, affects its shape. We determine the distance from the focus to the vertex by the value of "P". "P" tells us how many units away the focus is from the vertex and how skinny or fat the parabola is. If the "P" value is a small number such as a fraction then it will be a narrow parabola. If the "p" value is a big number or anything greater than one then it will open wider. 

3. The properties of parabolas are seen in real life all the time. They are seen in parabolic satellites, parabolic mirrors, parabolas in bridges,the path of a ball, whisper dishes, and much more. In whisper dishes since they are paraboliod shaped so when a person speaks into the speaker which is the focus the other person will hear what is said at the end of the other dish. This is because when the whisper enters the speaker or the focus, the sound waves travel to the other focus because of the angle of the paraboliod is shaped and the focus.Parabolas are also present when using flashlights or car headlights. The properties of the parabola are used to reflect the light rays flowing from the focus of the parabola in parallel rays.   

Citations  

 Parabola image from: http://www.mathwarehouse.com/quadratic/parabola/focus-and-directrix-of-parabola.php
Conic sections video: http://www.youtube.com/watch?v=BnWEI5o35G8