**Moving b in a Quadratic Equation**

**by**

**Rob Walsh**

A quadratic equation is a second degree polynomial with at most three terms and two roots. It's graphic representation is parabolic in shape and by definition, is the locus of points that is equidistant from a given point (known as the focus) and a line (known as the directrix.). Constructions of certain parabolas have led to the common standard equation representation y = ax^{2} + bx + c, where x and y are the coordinates for each of the locus points and a, b, and c are parameters that describe the parabola's shape and location on the Cartesian Plane. If we fix our y value to zero, then some minor exploration shows us how parameter * a* and parameter

We explore y = x^{2} + bx + 1, where a and c are held constant. Let's graph this quadratic for b = -3, -2, -1, 0, 1, 2, 3.

There are a few things of interest here:

- The parabolas share the same y-intercept. This is not too surprising, though, since this point is where our value for
, hence, the y-intercept is always our parameter*x = 0*. Since we have fixed*c*, it stands to reason that all of our parabolas will cross the y-axis at (0, 1).**c = 1** - When
and*b < -2*, our parabolas cross the x-axis in two places. this tells us that at these values of*b > 2*, our equations have two real roots. Similarly, when*b*, the parabolas touch the x-axis in exactly one place (they are tangent) which says that the equations have exactly one real root. Finally, for all values*x = -2, 2*, there are no real roots (which implies that there are complex roots to these equations), since they do not touch the x-axis at all.*-2 < b < 2* - The location of our parabola seems to follow some uniform path. If we focus on one particular point, say, the vertex of our parabola when
, we can observe how it "moves up and to the right" for awhile and then "back down" again. In general, we acquire the coordinates of a parabola's vertex as such:*b = -3*and*x = -b/2a*is solved for with this value of*y*. We produce the vertices of our parabolas here:*x*

If we plot these coordinate pairs, the vertices start to take a very obvious shape:

If we consider the locus of the vertices of the set of parabolas graphed from y = ax^{2} + bx + c, what we find is that we get a parabola that opens opposite of our set:

Let's finish off by finding out the equation of this locus. We start with the general equation y = ax^{2} + bx + c where we know that our values for * a* and c are both 1. In our locus graph, since it opens opposite our set of parabolas, we take the opposite of

x = -b/2a

0 = -b/2(-1)

0 = -b/-2

(0)(-2) = -b

0 = -b

So, we can generalize the equation of this locus to be the parabola y = -x^{2} + 1.

One final observation is that the vertex of our locus is the one common point of our set of parabolas. Therefore, we can further generalize the locus equation as such:

Given a uniform set of equations y = ax^{2} + bx + c, we determine the equation of the locus of their vertices to be y = dx^{2} + ex + fx, where ** d = -a**,