Tiffany N. KeysÕ Assignment 3:

Locus of the Vertex

INVESTIGATION:  Consider the locus of the vertices of the set of parabolas graphed from y = x2 + bx + 1. Show that the locus is the parabola y = -x2 +1.   Generalize.

In order to generalize that the locus of vertices of the set of parabolas graphed from y = ax2 + bx + c, you can explore the equation with different values of a, b, or c as the other two are held constant.

For a = -3, -2, -1, 0, 1, 2, 3, the following overlay graphs are obtained.

Observations:

á     Each parabola always passes through (0, 1) on the y-axis.

á     The vertex of each parabola changes as the value of a is changed in the equation.

á     The concavity of each parabola also changes when a different value of a is substituted in each equation.

For b = -3, -2, -1, 0, 1, 2, 3, the following overlay of graphs are obtained:

Observations:

á     Like the graphs above, each parabola passes through (0, 1) on the y-axis.

á     The vertex of each parabola also appears to change in each equation as the different values of b are substituted in.

á      The concavity of each parabola appeared to remain the same.

Lastly, where c = -3, -2, -1, 0, 1, 2, 3, the following overlay of graphs are obtained:

Observations:

á      Each parabola does not pass through the same point as the value c is changed.

á     Again, the vertex of the parabola changes in each equation.

á     The concavity of the parabola increased as the value of c was increased.

We can generalize that because all of the graphs passed through the point (0, 1) on the y-axis that y = -x2 + 1 is the locus of vertices of y = ax2 + bx + c, despite the fact that when the value of c was changed.