If we set
for b = -3,-2,-1,0,1,2,3, and overlay the graphs, the following picture is obtained.
Now we can find the locus of each parabla.
Now let's plot the locus point only of each parabola.
Notice that the points form what looks like a concave down parabola with the locus at (0,1). We can see from the picture that the roots of the new parabola are x=-1 and x=1. Let's try to find the equation of the new parabola.
First set y equal to the roots.
But because this is a concave down parabola the x^2 must be negative, so let's see what happens when we graph y=-x^2+1.
Here is what we get when everything is graphed together.
Return to Homepage