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:
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 .