1) Definition: The set of all points equidistant from a given point, known as the focus and a line, known as the directrix.
2) Algebraically: The algebraic equation that is used to standardize parabolas is as follows: (x-h)^2 = 4p(y-k) or (y-k)^2 = 4p(x-h). Basically, one would modify any given equation that is determined to be one of a parabola into this format, most usually, through completing the square. In order to distinguish equations of parabolas from equations from other conic sections, you basically need to know only one thing: one variable is square, while the other isn't. This is a distinct feature of the parabola that we don't find occurring within equations for circles, ellipses, or hyperbolas. Anyways, by looking at the value of p, and at which variable is squared, you may determine what the parabola would look like graphically, as in its orientation and all.
Graphically: The first and foremost characteristic that one would find while examining the parabola, is that it looks awfully similar to the letter 'U'. Its orientation can be determined through the sign that is complementary with the value of p, and the variable that holds the square. If x is squared, then the parabola will either face up or down - up if the p is positive, and down if it is negative. If y is squared, then the parabola will either face right or left - right if the p is positive, and left if the p is negative. The point at which the parabola can be directly divided in half - that is, also considered its vertex - will be known as the vertex, and the line which divides it exactly in half, known as the line of symmetry. The focus point, which will be on the line of symmetry the distance of 'p' away from the center, will help one decide what the orientation of the graph is like - by way of being in the direction of the lines. Lastly, the focus point may also determine whether the parabola is fat or skinny - if it is close to the vertex, then it will be skinny; if it is far away from the vertex, it will be fat. AND the directrix is basically the opposite of the focus point, and is a line.
http://www.mathsisfun.com/geometry/images/parabola-formula.gif
Features Of: The vertex is most basically, (h,k), the focus, (h+-p,k) or (h,k+-p), depending on which variable has the square and what the value of p is, the line of symmetry, either h or k, again, depending on which value has the square, and p is the variable that precedes the variable without the square, divided by 4. Ah, and the directrix is the exact same thing as the focus point - save that it is determined by subtracting p if the focus is found through addition, and vice-versa - and, it is a line.
3) Real World Application: As demonstrated in the video below, there are many instances in real life in which parabolas occur. For example, when the teenager shoots the basketball into the basketball hoop, a parabola is seen - and even outlined, too. If it were practical, I suppose that basketball players would study up a bit more in math, in order to improve their shots.
4) Works Cited: http://mathworld.wolfram.com/Parabola.html
http://www.mathsisfun.com/geometry/images/parabola-formula.gif
https://www.youtube.com/watch?v=FXuTKpp-vME
