Investigating the pattern of
roots

as the linear
coefficient *b* changes

in the general quadratic

Write-up by

Blair T. Dietrich

EMAT 6680

This investigation will explore the pattern of
roots for the
general quadratic . While holding
constant the values of each of *a* and *c* at 1, the value
of *b*
will be allowed to vary.

Consider the
equation . Try your own
values of *b* here.

By overlaying the
graphs for *b* = -3, -2, -1, 0, 1, 2, 3, the following picture is
obtained:

We notice that all of the graphs shown have the
point (0,1)
in common. This is reasonable since *c*
is fixed at 1 and this corresponds to the y-intercept.
If we were to observe the graph of as *b* varies,
we could try to visualize the movement of the curve along a fixed path
that
would be traced out by the successive vertices of each parabola. This path seems to be symmetric about and
centered at the line *x* = 0.

In general, we can determine the vertex of a parabola by finding the axis of symmetry for the general quadratic . Specifically, for quadratics of the form , the line of symmetry is . By substitution, we can find the vertex:

So we find that the vertex will occur at the point
for
any given value of *b*.

Due to the symmetric nature of the path of vertices the locus of points is known to also contain the point .

Also, as we have observed, the point (0,1) will also be on this path.

Recall that three points will define a quadratic
equation . We know that *C* = 1
(the *y*-intercept). We can find the
values of *A* and *B*
by solving the following system of equations:

By substitution,

Since *b* varies (i.e. not constant) it is
not (in
general) equal to zero. Therefore, *B*
= 0.

Back-substituting, we can find A:

*A*
= -1

Therefore, the quadratic function that passes
through the
vertices of all parabolas of the form

is . This equation is
shown in black on the graph below: