The Locus of Vertices for a Family of Parabolas

**By**

Equation 1 is the standard form for the equation of a parabola.

Equation 1:

Holding *a* and *c*
constant (*a* = *c* = 1) we let *b* = *n*, for *n*
varies between these values. This animation can also be viewed in movie format,
by opening this AVI file. The graphs of Equation 1
are overlaid in Figure 1, for {*b* < 3}.

**Figure 1:
**, with

We
hypothesize from the animation and from the graphs in Figure 1 that the locus
created by the vertices of each parabola in the family of Equation 1, for

**Figure 2:
**and

**Proof:**

For

and solving for *x*, we have

This equation describes the *x*
coordinates of the locus of points. We wish to show that this locus forms a
parabola. Substituting back into the equation,

we have

and squaring the first term yields

Simplifying the first term and
multiplying the second term by

Combining like terms, we have

Multiplication of the first term by

Factoring the –*a*, we have

Squaring the negative root yields

for which we then substitute x,
resulting in the desired parabolic equation

Ò

Return to Homepage