Problem: Consider the locus of the vertices of the set of parabolas graphed from

.

Show that the locus is the parabola

.

Consider , where , and the corresponding graphs:

Now using calculus to find the minimum of each of the graphs we have for the minimum x-values:

,

since

.

Now, , and so we have that for each of our graphs

The minimum y-values of:

.

So for the vertex of each of our graphs is:

.

Now consider the function . By overlaying it on our previously graphed functions we notice that it appears as though intersects every graph at its vertex.

Indeed, this is the case. Consider again our general , and set it equal to .

So we have

.

Hence, intersects each at its respective x-coordinate minimum.

Similarly, we check what we already suppose to be true, namely that

,

which is the y-minimum of each of our respective .

Whereby, we have shown that the locus of is given by the function

.