EMAT 6680, Fall 2003
The function f(x) = ax2 + bx + c is a quadratic and gives the graph of a parabola. Changing the value of b creates a family of curves that all pass through the point (0, c). For instance, the graph of f(x) = x2 + bx + 1 gives the family of curves that all pass through (0, 1).
If we construct the path through the vertices for the quadratics in this family of curves, another parabolic shape is formed.
This is the locus of points through the vertices and in this particular instance has the equation f(x) = -x2 + 1.
Generalizing, in the family of curves generated by varying the values of b in f(x) = x2 + bx + c, the locus of the vertices has the equation f(x)= -x2 + c.