Hamilton HardisonÕs Exploration of

Assignment 3:  Locus of Vertices of Parabolas of the Form y = x2 + 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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