Assignment #3

More Quadratics

by

Megan Dickerson

The Quadratic Equation is a big part of High School math curriculum. As a teacher of secondary mathematics, it is a very important formula to understand.

So I am going to examine a little further.

First, look at the graphs for various values of b:

It is evident that each of these graphs cross the y-axis at the same point (0,1). But what is true for the vertices of each of these graphs?

If you imagine a line connecting the vertices it would resemble a parabola of negative leading coefficient.

To see this more clearly examine this animation of as b moves from -3 to 3.

To investigate the locus of the Quadratic Equation I am going to first complete the square of the original so that it now reads:

The equation of a parabola is y=a(x-h)^2 +k with (h,k) being the vertex. I have turned the standard form into this form with:

If I solve for b in the top equation, then b = -2h. Substitute this into the bottom equation to get:

In the last step, I placed y=k and x=h. This equation is the locus for .

Here is an animation.