Parabola
1. Mathematical Definition: "Any point on a parabola is at an equal distance from a fixed point (the focus), and a fixed straight line (the directrix)" (Math is fun)."
2. Conic Section:
A) Algebraically
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| http://www.algebra-online.com/tutorials-4/algebra-formulas/articles_imgs/7976/formul92.jpg |
| http://www.clausentech.com/lchs/dclausen/algebra2/lecture_notes/conics/vertical_parabola.jpg |
B) Graphically
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| http://www.clausentech.com/lchs/dclausen/algebra2/lecture_notes/conics/horizontal_parabola.jpg |
The parabola comes with several key features: vertex, directrix, focus, and the axis of symmetry. The vertex is the point at which the parabola starts. The directrix is a line outside the parabola, perpendicular to the axis of symmetry. The axis of symmetry is a line that breaks the graph into two equal halves. The focus is the point inside the parabola. Lastly the "p" is the distance between the directrix and the focus and the from the focus to the vertex. All of which contribute to graphing a parabola accurately.
Each of those parts can be derived from a general equation like 2y^2+x+24y+75=0, taking the first step which would be to turn it into standard form. This being done so by completing the square, remembering to there needs to be something squared on one side and a coefficient on the other. Once the equation is in standard form the key parts can be discovered. The vertex is (h,k), the "p" is the coefficient set equal to 4p, and from looking at the "p" and the equation you can tell whether it opens up, right, down or left. If "p" is positive and the equation is x^2 goes up, if "p" is positive and equation is y^2 goes right, if "p" is negative and equation is x^2 opens down, if "p" is negative and the equation is y^2 the graph opens left. With "p" being small (closer to vertex) it makes the parabola skinny, if its bigger (farther from vertex) the parabola is fatter. Then you find both the directrix and focus by counting away from the vertex by the value of "p". Then there's the axis of symmetry which is perpendicular to the directrix and passes through the vertex. Lastly, the eccentricity of a parabola should be equal to one. To visually see this worked out you can seek HERE for more assistance.
3. Real World Application:
A Satellite dish is the perfect example of a real life application of a parabola. The actual dish capturing the actual figure of a parabola. Thus creating an effect that causes waves that come in toward the dish to reflect off of it. And as the waves are reflected they bounce their way to the sphere (receiver) which in this case is the focus point of the parabola. With that it allows for their to be better signal and efficient cable.
The satellite itself is the parabola with a focus as the main point. That is the spot at which all the waves end up going no matter where the waves hit on the dish. This could serve to represent the "p" show the distance between the directrix and vertex is equal to the distance from the focus and vertex. How its an equal distance from any given point, explaining as to why the waves all end up hitting the focus. Another thing would be that in small dishes the focus would have to be closer to the vertex and in a large dish it would be farther.
4. References:
http://bit.ly/zbruh4
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