Locus of Parabolas
For quadratic equations in standard form y=ax2+bx+c, we can explore what happens to the parabola as we change the values of a, b, and c.
If we maintain a and c a constant of one, and use the integer values -3<b<3, we can see that the parabolas intersect at (0, 1).
Furthermore, the locus of vertices lies on the parabola y = -x2-1
Now, we can investigate what happens when we use c = 2.
Notice that we have the same pattern with the parabolas except we have shifted them all so that the intersection is now (0, 2). This includes the parabola which lies on the locus of vertices- the
equation of this parabola is y = -x2 +2.
We can generalize this for various values of c. Perhaps we can consider the values of c for -5<c<5.
In the following animation, we can see for these various values of c, the locus of all of these parabolas travels along the y-axis with the parabolas. Changing the values of c causes a vertical translation of the parabolas.