From Jim WilsonÕs Assignment page: Consider the locus of the vertices of the set of parabolas graphed from . Show that the locus is the parabola .
By using a graphing utility, it appears that the vertices of
these parabolas do lie along another parabolic curve.
We can support this conjecture by tracing the vertex of the
parabola , using geometerÕs
sketchpad. The red parabola in the
image below was created by tracing the vertex and appears to be .
We can now explore the situation algebraically.
vertex of any parabola, , lies on the parabolaÕs line of symmetry, . So each vertex
for a parabola has as itÕs
abscissa. The ordinate of each
In general, the vertices of , have abscissa . The ordinate
of a given vertex is
So for any quadratic function , the locus of the vertex as b varies will be given by .
For calculus students: What is the locus of the point on a parabola whose tangent line has slope one? Is there an algebraic generalization? A rough sketch of a solution is here.