Now, if we look at the following equation, and evaluate the roots, we can start noticing some patterns:
We can look at the roots of the graph and that is the solution to our initial equation for our values of 'a', 'b', and 'c'.

Let's explore what changes in 'a' do to our parabola.
1. Variations with 'a' ...
We will set our 'b' and 'c' to be 1. Our equation is now as follows:
And our graph looks like this:

We can generalize that variations in 'a' do not effect our yinterception, but do effect our roots and the orientation of our graph. Let's look at a very big and very small value of 'a'.

So, we can see that our a = 0 line acts as an asymptote for our equation. We can also see that as a gets very large, it thins our parabola up.
So, with a = 1, we have a basic graph. Let's leave that to be our default, and explore what happens when we vary our 'b'.
2. Variations with 'b' ...
Let's again see a quick overview of values around 0:

We can generalize from this that orientation is not effected by the value of our 'b', but we start moving away from the orgin the higher our value of 'b' is.
In fact, the graph:
... goes through the vertex of all of our graphs.
So let's find the vertex using our our given equation and its derivative:
The vertex happens when our derivative = 0, so:
Plugging that in for our yvalue, we get:
That's our vertex. We can show that equation 2.1 satisfies that very point. Plug in x, and one gets:
And so we are done. One thing that makes me uneasy is my need for calculus to derive the formula for the vertex. Is there another way to obtain it? Something to think about for the future.
2. Variations with 'c' ...
Finally, let's examine what happens when different values are inputed into our core equation, which now looks like this:
This isn't very interesting, as changes in c only move the graph up or down by that number. Check out this quicktime movie to observe this behavior.