Expert: Josh - 1/3/2007

Question
Hello,
I can't figure out how to do this problem.
A parabolic mirror is placed on the floor. It measured 30 centimeters tall and 60 centimeters wide.  Find an equation of the parabola with its vertex at the origin.

Any help would be greatly appreciated. Thanks!
Lauren

Answer
Hi Lauren,

First thing we should do is draw a diagram.

A.....B.....C
.
.
......D

The parabolic curve passes through point A, D and C. Point D is supposed to be directly below point B.

Height h=30 refers to length BD. Width w=60 refers to length AC. By construction, AB=BC=w/2.

The general equation for a parabola may be written as
(y-yo)=c*(x-xo)^2........[#1], where c represents some positive quantity since the parabola is upright "U-shape" (If the parabola is tipped up-side-down, then "c" would be negative).

This quadratic expression represents a convex curve, expanding the right hand side gives a second degree polynomial in terms of x.

The vertex (lowest point) of the parabola is given by (xo,yo). From (y-yo)=c*(x-xo)^2, it is clear that squaring (x-xo) cannot produce a negative value (i.e., RHS>=0). The minimum y-value is attained when the RHS=0 as y=yo. The parabola is symmetrical about x=xo.

Having the vertex at the origin, D=(xo,yo) coincides with (0,0). Substituting point C=(x,y)=(30,30) into the equation [#1], we get (30-0)=c*(30-0)^2, c=30/900=1/30.

So, the equation is y=c*x^2, where c=1/30