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?
The roots of the parabola are at x=1 and x=-1.
y=(x-1)(x+1)
Notice, that the parabola opens up along the 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:
There are no roots when -2<b<2.
There is exactly one root when b=2 and b=-2.
There are two roots when b<-2 and b>2.
Next, we will graph the following equation in the xb plane to further explore roots:
The equation has the following solutions:
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.
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