Investigating the pattern of roots

as the linear coefficient b changes

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: