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.
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?
roots of the parabola are at x=1 and x=-1.
that the parabola opens up along the negative x-axis, so
the final equation will be,
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
Review: The graph has roots when it crosses
Looking at the graph, we see that:
are no roots when -2<b<2.
There is exactly one root
when b=2 and b=-2.
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
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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