# Hamilton
HardisonÕs Exploration of

# Assignment 3: Locus of Vertices of Parabolas of the
Form y = x^{2} + bx + 1

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.

The
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
vertex is

.

## So for each value *b*, we have a
vertex . Thus, vertices
of the parabolas trace out the curve .

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.

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