Investigation of Quadratic Equations


Kristy Hawkins

The standard form of a quadratic equation is

To begin investigating this equations, I will set two of the variables a, b, and c as constant while varying the other one. This will show us what happens to the graph as the one variable changes. Let us first hold a and c as constants and vary b. What will happen?

What happens to the parabola as b changes from -4 all the way to 4? They all seem to have the same shape and the same y-intercept at (0,1). But what is changing? The position of the parabola is changing as well as the roots. If we look closely we can see that from (-2,2) there are no roots, while at -2 and 2, there is a double root (This can also be seen by looking at the equation at these points and noticing that they are perfect squares). Along with this observation, we see that there are 2 roots when b=-4, -3, 3, and 4. This would lead us to assume that as b approaches infinity in both directions from 2, there will always be two real roots.

How can we describe the way that these parabolas' shift as b varies? Let's look at the locus of the vertices of these parabolas. If we can find an equation that will describe this locus, then we will have a way to talk about how the parabola shifts.

We can show that this locus is

This shouldn't be too hard. We know that the y-intercept must be at 1 because every parabola that we have looked at passes through that point. This locus must also pass through (-1,0) and (1,0) which will be it's roots. We also know that this new parabola must have a negative orientation. Putting all of these facts together gives us our formula, and our illustration below.


Now, what would these graphs look like in the xb-plane?

Consider the equation

If we decide to graph this relation in the xb plane we will get the following graph.

If we take any particular value of b, say b=3, and overlay this equation on the graph we add a line parallel to the x-axis. If it intersects the curve in the xb plane the intersection points correspond to the roots of the original equation for that value of b. Why?

So here we hve a different view of our roots than before, which can be very helpful when investigating our quadratic equations.

Try this one yourself. Fix b and c while varying c.