The most common equation for a parabola is described by the traditional function format:
f(x) = (ax)squared + bx + c
or
y = (ax)squared + bx + c
The "a" value must be a non-zero real number and the "b" and "c" values can be any real number, including zero. If the "a" value is positive, the curve will be U-shaped; if the "a" value is negative, the solution with describe an inverted U- shape.
The alternate parabolic equation is of great help when sight recognition of the parabola's vertex, axis of symmetry, locations of the focus and directrix are "a" priorities:
(x-h)squared = 4p(y-k)
The Hubble telescope has made phenomenal use of parabolic-shaped mirrors to locate and photograph interstellar images. The large, parabolic dishes that serve as radio telescopes track and collect data from space probes. The largest in current use is the 1,001-foot dish located in Arecibo, Puerto Rico. Suspension bridges are another large real-world application of parabolas. Renowned for their beauty, these structures may appear fragile, but due to the balance of tension forces, as prescribed by the parabolic design of the bridge, the combination of tough materials, like steel, and meticulous placement of the elements, creates bridges that are strong and attractive.
Because a parabolic-shaped item is, by definition, a U-shaped object, this curve is particularly suitable for small three-dimensional reflectors. Parabolic reflectors are found in flashlights, solar ovens, radio and optical telescopes, the small satellite receiver mounted on homes for television reception and even for amplifying on-field sounds during a football game. For the football-field application, it would be more accurate to describe the sound-enhancing function of the parabola as a receiver. In either case, the parabolic portion of any tool, appliance or other equipment works by reflecting waves traveling parallel to the axis of symmetry, amplifying and directing said waves into a more efficient, usable operation.