Jernita Randolph

Loci of Parabolas

Here we will
consider the equation

and to overlay several graphs
of

for different values of a, b, or c as the other two are
held constant.

LetŐs start by
observing the general graph of

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 values of b > 0, the vertex lies to the left of the graph. When b = 0, the vertex is on the y-axis
(the black graph). We know that we have real roots of the equation at the
points where the parabola intersects or crosses the x-axis.

For graphs that do not intersect or cross the x-axis, there no
real roots. For example, when b = 1, the graph does not cross the x-axis, therefore it does not have
any real roots.

Also notice that when b is positive, the real roots occur on the
negative side of the y-axis.

Now let's observe the equation for b < 0.

For
values of b < 0, we can see that the vertices of the equation all lie to the
right of the y-axis in other words they have a positive x-value.

Now let's take a closer look at the actual vertices of the
parabolas. Where would the locus of the vertices be?

What would it look like?

By looking at the above graphs, we can see that the vertices are
continually going further below the x-axis and further away from the y-axis. This
implies that a concave downward parabola is being formed, therefore the **a** term is negative. We can see that all
the vertices are steadily approaching the point (0, 1), so the **c** for the equation must be 1. And since the vertex is on the
y-axis, we know that b = 0. Therefore the equation must be

.

LetŐs demonstrate this by graphing the locus of vertices equation
along with our other equations.

We can see that the locus of vertices parabola passes through the
vertices of the of all the other parabolas. We can see
that all the **c**'s are the same and that the locus of vertices has the opposite **a** of the other equations. We also notice
that b does in fact equal 0. But how do we know that it will work for all
equations?

Let's plug in some different **a**'s and **c**'s to check to make sure. How
about a = 2 and c = 3.

The graph for
the locus still holds.

LetŐs
investigate for values of **a** and **c** < 0.

Once again the locus of vertices passes through all the vertices
of the parabola. Therefore we may assume that the locus of vertices equation
would be equal to the opposite of **a** ± **c**.

Therefore if **a** was positive and **c** was positive, then the locus of vertices would be

and if **a** was negative and **c** was negative then

.