Conic Section (Ellipse)

Conic Section

(Parabola)

Q #1: What is the mathematical definition of this conic section and how does that definition play a role in the properties of the conic section and how it is shaped or formed?

A #1: A parabola consists of a focus point, and a directrix. The focus is just a point inside the parabola, and the directrix is a fixed line beneath the parabola to help draw the curve. We can see that the length between the focus and directrix are equidistant when connected to touch the parabola. The eccentricity of a graph is usually 1. This is because if it were less, the curve would close in, and if it were greater, the parabola would eventually open wider and create a hyperbola. Click here to get a visual understanding of eccentricity. 


Q #2: How does the focus (or foci) affect the shape of the conic section?
A #2: The distance of of the vertex to the focus point named as the term 'p'. The 'p' is equidistant to the length of the directrix connected to the graph. The distance of the focus and directrix to the vertex can fluctuate the size of the parabola. If the focus and directrix were farther away from the vertex, the graph would open much wider, otherwise it would be very thin. Click here to experiment with a live parabola. 

Q #3: How do the properties of this conic section apply in real life? 
A #3: Here we can see that the cables of the S.F.G.G. Bridge is very close to a parabola. It is shaped this way because the two towers holding the cables are very close to each other. Therefore, the cable must equally distribute its weight from the distance of one tower to the other. The closer it is to the tower, the higher the curve. This creates the shape of a parabola. 

Citation

http://www.mathsisfun.com/geometry/eccentricity.html

https://www.google.com/images

http://www.intmath.com/plane-analytic-geometry/parabola-interactive.php