Conic Sections: Parabola

Questions:

• 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?

The mathematical definition of this conic section (parabola) is that the parabola curve is at equal distance from the focus and directrix. This value from the curve to the focus and the curve to the directrix is the value of p which allows for one to determine which way the parabola is going. The value of p in the parabola allows us to determine whether it is going up/down or left/right, and if it is a wide or skinny curve. If p is positive and if it is a bigger number, then the parabola will have to be drawn as opening up wide (up/right). On the other hand, if p is negative and is a small number you can conclude that the parabola will go either left or down and it will be skinny. The mathematical definition of this conic section, the parabola, also deals with the focus and the directrix which plays part in which way the parabola may be going. For example, if you determine the parabola as going up due to the value of p you are able to figure out where the focus and directrix should go because the focus is always inside of the parabola and the directrix is always right below the vertex. 

• 2. How does the focus (or foci) affect the shape of the conic section?  (If you choose ellipses, you should include information about eccentricity in your response; if you choose parabolas, "p" should be a big focus... haha, get it? "p" is the distance from vertex to focus and it should be a big focus. Ok, moving on...)

In parabolas the focus doesn't play much of a role in affecting the shape of the parabola, instead we know that the focus in a parabola must always be on the inside of it. Whether the parabola is going up,dow,left, or right, the focus of the parabola must be on the inside of the parabola, before the vertex. The major factor in the shape of the parabola is the value of p. P is the one that determines the whether the parabola will be going up/down, or left/right. If the p value were negative, then your options for the graph would either be left or down depending on whether x or y is squared. Also, depending on which term is squared then if p ended up being positive then your options would be up or right. The value of P in a parabola also has an effect on depending whether it is wide or skinny. The smaller or negative value it has then it will be skinny, but if it is bigger and positive you know that it will be opening up wide. 

• How do the properties of this conic section apply in real life?  (While using the exact examples I gave you may be acceptable, I will be looking for some research, creativity, and thought.  There is a lot out there!)

The properties of this conic section, the parabola, applies in my everyday life when I go out to basketball practice and shoot a ball. When shooting, I know that the value of p must be negative since it is going downward and that the parabola must contain an x squared term since it is going up/down. Also, the focus of the parabola made when shooting the ball will be under the vertex which is inside the parabola. This focus then allows us to understand that the distance from the focus to a point on the parabola will be equal to the distance from a point on the graph to the directrix. 

Definition A parabola is a curve where any point is at an equal distance from: a fixed point (the focus), and a fixed straight line (the directrix)

citations: 

Parabola- http://www.mathsisfun.com/geometry/parabola.html
Parabola Image- http://people.richland.edu/james/lecture/m116/conics/parabola.gif
shooting ball image- http://library.thinkquest.org/12006/images/parabola.gif
last image of parabola- http://www.sparknotes.com/math/precalc/conicsections/section2.rhtml