In the previous exploration, we examined the
graph y=ax^2+bx+c and explored how the equation was affected by
changing the values of *a*, *b*, and *c*. To review
the last exploration, click
here. Now we will do further exploration
into the graph when *b* varies and *a* and *c*
are held constant.

__Part
1: Locus of Vertices__

Let us examine the graph when *b* varies
and *a* and *c* are held constant at a=1 and c=1.

Now, let us look at the vertex for each of the different parabolas and plot those on a graph. Also notice that each of the parabolas goes through the point (0,1). The point (0,1) is the y-intercept of each of the curves.

When we connect all the vertices from each
of the parabolas we get the following graph, which is called the
**locus of vertices**.

The locus of vertices forms a concave down parabola, but what is the equation of this "black" parabola?

*negative* x-axis, so
the final equation will be,

__Part
2: Exploring Roots__

Next, let us shift our thinking to the roots of the same equation we have been exploring.

For which values of *b* does the graph
have roots?

*Review: The graph has roots when it crosses
the x-axis.*

Looking at the graph, we see that:

Next, we will graph the following equation in the xb plane to further explore roots:

The equation has the following solutions:

- No solutions when -2<b<2
- One solution when b=2 and b=-2
- Two solutions when b<-2 and b>2

Wow, the solutions are the same as we observed before. The solutions are easy to see when we draw vertical lines on the graph for different values of b. When b=2, we expect that there will be one root at x=-1.

When we graph the equation on the xb-plane, we get exactly the solution we expected, at x=-1, there is exactly one root.

Now we could look at when b=-3. From our previous observations, we expect to have two roots.

The vertical line b=-3 crosses the graph twice, giving us two roots, just as we expected.