Parabola
Geometrically, a parabola is the set of all points in the plane which are equally distant from a fixed point, called the focus, and a fixed line, called the directrix. We will say more about this geometric definition below. Algebraically, a parabola is the graph, in the x-y coordinate system, of an equation of the form y=ax2+bx+c. Such equations define functions called "quadratic" functions. The simplest of all quadratic functions is defined by y=x*2. This function's graph is the parabola with it's lowest point, the vertex, at the origin and which is symmetric to the y-axis. It passes through the set of points of the form (a,a2); in other words, the second coordinate of any point on this parabola is the square of the first coordinate. All other parabolas can be generated from the y=x2 parabola with some combination of translations, reflections, dilations, and/or rotations. For this reason, we sometimes call the y=x2 parabola the "parent" of all other parabolas. Since all other parabolas "inherit" the basic characteristics of the y=x2 parabola, the mathematician can learn most of what she needs to know about parabolas by studying the parent in depth.
Using the geometric definition, given in the first sentence of this article, let the focus have coordinates (0,p) and the directrix have equation y=-p. One point on this parabola must then be at (0,0) since that point is halfway between (0,p) and (0,-p), and this latter point is on the line y=-p, the directrix. In fact, (0,0) is the vertex of this parabola. If (x,y) is a variable point on the parabola, then the distance formula from algebra gives (x-0)2+(y-p)2=(y+p) 2. Expanding the binomials in this equation and "simplifying", we arrive at the equation x2=4py, or y=(1/(4p))x2. Now in the special case where the focus is at (0,1/4) and the directrix is y=-1/4, we have the "parent" parabola y=x2. In fact, any parabola whose vertex is at the origin may be mapped onto the y=x2 parabola and vice versa by a transformation called a dilation or size transformation. A dilation is a transformation which maps points of the form (x,y) onto points of the form (kx,ky), where k is a nonzero real number called the dilation factor or magnitude. In particular, the y=x2parabola can be mapped onto a parabola with equation y=ax2 by the dilation which maps (x,y) onto ((1/a)x,(1/a)y).
Dilations belong to a class of transformations called similarity transformations, which means that dilations map geometric figures onto other geometric figures that are similar to the original figures. In fact, the transformational definition of similarity is: Two figures are similar if and only if one can be mapped onto the other by a similarity transformation. This gives us the rather remarkable fact that all parabolas with vertex at the origin are similar to the y=x2 and, by transitivity, all parabolas centered at the origin are similar to one another. In other words, they all have the same shape in the same sense that all circles have the same shape, or all similar triangles have the same shape. This is hard for the beginning student to understand because when one graphs, for example, y=x2 and y=2x2 on the same axis system, the y=2x2 parabola looks "thinner" or is closer to the y-axis for any specific value of x than is the y=x2 parabola. But if the graphs are accurately plotted on two separate transparencies, with the scale for the y=2x2 parabola adjusted by a factor of ½, then if one transparency is placed on top of the other so that the origin and axes coincide, the portion of the y=2x2 parabola on its transparency will exactly cover the portion of the y=x2 parabola on its transparency. The story does not end here, however. As mentioned above, every parabola with vertex at the origin can be mapped onto any other parabola in the plane by translations, reflections, and rotations. These are also similarity transformations and this means that all parabolas are similar.
The mathematical properties of parabolas make them excellent models for physical objects in which a focusing component is essential. It can be shown that parallel lines drawn on the inside of any parabola are reflected from the curve of the parabola to its focus. Thus, many telescopes are designed using parabolic reflectors with the light collection instrument located precisely at the focus of the parabola. Satellite television receivers use this same focal property to gather television signals from satellites in stationary orbit around the earth. Searchlights that require concentrated beams of light use this property in reverse. The light source is located at the focus of a parabolic reflector so that when the light is turned on, it bounces off the sides of the reflector and is directed outward as beam of parallel light rays. Parabolas also model the motion of a body in free fall towards the surface of the earth. Isaac Newton's theories of gravitation and motion lead to an equation of the parabolic form y=-16t2 for a body falling towards the earth with negligible air resistance. In this case, time is measured in seconds and distance is measured in feet. Newton also showed that the path of a projectile launched from a point on the earth's surface will follow a parabolic path until it hits the ground at the end of its flight. Parabolas are also used in the design of bridges and other structures involving arches.
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