By: Laura Lowe

It has now become a rather standard exercise, with available technology, to construct graphs to consider the equation  and to overlay several graphs of  for different values of a, b, or c as the other two are held constant. From these graphs discussion of the patterns for the roots of  can be followed. For example, if we set  for b = -3, -2, -1, 0, 1, 2, 3, and overlay the graphs, the following picture is obtained.

We can discuss the "movement" of a parabola as b is changed.  The parabola always passes through the same point on the y-axis (the point (0,1) with this equation).  For b < -2 the parabola will intersect the x-axis in two points with positive x values (i.e. the original equation will have two real roots, both positive).  For b = -2, the parabola is tangent to the x-axis and so the original equation has one real and positive root at the point of tangency.  For -2 < b < 2, the parabola does not intersect the x-axis -- the original equation has no real roots.  Similarly for b = 2 the parabola is tangent to the x-axis (one real negative root) and for b > 2, the parabola intersects the x-axis twice to show two negative real roots for each b.    Now consider the locus of the vertices of the set of parabolas graphed from .

First letŐs consider the set of vertices of the set of parabolas of the form .

From the above graph we can see that the vertices of the graphs of the form  are (-1.5,  -1.25), (-1, 0), (-0.5,  0.75), (0, 1), (0.5, 0.75), (1, 0), and (1.5, -1.25).  The locus of these points has the equation  as seen below.

Now suppose we start with the equation  and we let b = -3, -2, -1, 0, 1, 2, 3, and overlay the graphs, the following picture is obtained.

From the above graph we can see that the vertices of the graphs of the form  are (-0.75,  -.125), (-0.5, 0.5), (-0.25,  0.875), (0, 1), (0.25, 0.875), (0.5, 0.5), and (0.75, -.125).  The locus of these points has the equation  as seen below.

In general, the locus of the vertices of the equations of , is the equation .

Click here to see a movie when a varies and c = 1.