Question

A manufacturer is designing a flashlight. For the flashlight to emit a focused beam, the bulb needs to be on the central axis of the parabolic reflector, 3 centimeters from the vertex. Write an equation that models the parabola formed when a cross section is taken through the reflector’s central axis. Assume that the vertex of the parabola is at the origin in the xy coordinate plane and the parabola can open in any direction.

Answer 1

A parabola in analytical geometry is a curve drawn that has either 2 x-intercepts or y-intercepts. You can see visually that it is a parabola if it forms an arc that passes either the x-axis or y-axis twice. Nevertheless, it would have two roots, so the equation must be quadratic.There are 2 possible equations for a parabola:

(x-h)² = +/- 4a(y-k) or
(y-k)² = +/- 4a(x-h), where

(h,k) are the coordinated of the vertex
a is the distance of the focus from the vertex

From the problem, it says that the vertex is at the origin, so the coordinates are (0,0). That means h=0 and k=0. Also, the focus is a=3. Thus, the possible equations for the reflector are

x² = +/- 12y and y² = +/- 12x

Any of those are possible because the problem mentions that it can open anywhere. Therefore, it could open downwards(-) or upwards(+) if it passes the x-axis twice; or to the left(-) or to the right(+) if its passes the y-axis twice.