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Parabola
Parabola
Graphing Parabola
Writing Equations of Parabola according to given information
How to find focus, axis of symmetry, and directrix:
The standard form is (x - h)2 = 4p (y - k), where the focus is (h, k + p) and the directrix is y = k - p. If the parabola is rotated so that its vertex is (h,k) and its axis of symmetry is parallel to the x-axis, it has an equation of (y - k)2 = 4p (x - h), where the focus is (h + p, k) and the directrix is x = h - p.
Real World Example
A radio telescope has a parabolic dish with a diameter of 100 meters. The collected radio signals are reflected to one collection point, called the "focal" point, being the focus of the parabola. If the focal length is 45 meters, find the depth of the dish, rounded to one decimal place.
To simplify my computations, I'll put the vertex of my parabola (that is, the base of the dish) at the origin, so (h,k) = (0, 0). Since the focal length is 45, then p = 45 and the equation is:
4py = x2
4(45)y = x2
180y = x2
This parabola extends forever in either direction, but I only care about the part of the curve that models the dish. Since the dish has a diameter of a hundred meters, then I only care about the part of the curve from x = –50 to x = +50.
The height of the edge of the dish (and thus the depth of the dish) will be the y-value of the equation at the "ends" of the modelling curve. The height of the parabola will be the same at either x-value, since they're each the same distance from the vertex, so it doesn't matter which value I use. I prefer positive values, so I'll plug x = 50 into my modelling equation:
180y = (50)2
180y = 2500
y = 250/18
...or about 13.9 meters.
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