Parabolas
Definition: The set of all points that are equidistant from a given point, known as the focus, and a given line, known as the directrix.
Properties:
Algebraically: The parabola equation is (x-h)^2 = 4p(y-k) or (y-k)^2 = 4p(x-h). The square can only be on the x or the y and not on both. The 4p must be on the term that is not squared. From the equation, you can tell which direction the parabola is facing and what the parabola will look like. Sometimes the equation for a parabola won't be given to you in standard form, so remember that a parabola will always only have one squared term.
Graphically: The general shape of a parabola is a "U" shape. If the x term is squared, the the parabola will either face up or down, the p if it is positive, then the parabola will face up while if p was negative, it would face down. If the y term is squared, then it will face either left or right. If the p is positive, it will face to the left while if it was negative, it will face to the left. The graph will have a focus and a directrix that go above and below, respectively, in a distance based on p. The foci determines the overall shape of the parabola, if the focus is close to the vertex, then the parabola will be "skinny" while if the focus is farther away from the vertex, then the parabola will be "fatter". This is because the more away the focus is, the less the parabola is focused on the center.
How to find the key parts of a parabola: First, make sure that the equation is put into standard form by using completing the square with two variables. To find the vertex, you look at h and k, they are put into the coordinates h,k. The signs will be opposite of what they are in the equation. The value of p is found by taking the coefficient in front of the term that is not squared and set it equal to 4p, you solve for p and that p will be the distance from the vertex to focus and from vertex to directrix. The axis of symmetry is usually the number that stays the same in your ordered pairs. It will either be x = # or y = #.
Real World Application: There are several uses of parabolas, they are found in car headlights, satellite dishes, parabolic skis, the arch of a bridge, etc (http://www3.ul.ie/~rynnet/swconics/UP.htm). The parabola can be used in the dishes used to for cable for television. The shape of the parabola bounces the beams and waves from the sides of the parabola into the focus of the parabola. The reason for this is because the shape, it reflects the beam from whatever location from inside the parabola into the focus point. This gives good and strong signals.
References:
https://blogger.googleusercontent.com/img/proxy/AVvXsEi6yrC-h3whYf8XeD_L2knefMEbiEJ-WvAoAgT-Pm33bfWaW_kIf2gICSUs5l_1CoWZn6fx2Kd8DpAKvw9D8_qXXTxH5oM9KRe3RzKQ6Cc94gICMRmj_k0PZTH-6i9ZGlQMF9Xljrm_uBdddnbNX8hn8YXQB1T8W9M1lE5b8Z2oDvQyf4wSqoIXAqlWrCQ=
http://www3.ul.ie/~rynnet/swconics/UP.htm
http://www.youtube.com/watch?v=Djnwlj6OG9k
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