A family of curves is a collection of curves which share many of the same properties. For example, although there are infinitely many parabolas, all of them share the same basic shape. All parabolas consist of points which are equally distant from a fixed point, called the focus, and a fixed line, called the directrix. All have a vertex that is halfway between the focus and directrix. In fact, all parabolas can be seen as "descending" from a common "parent," the parabola whose algebraic representation is y=x². This simplest of all parabolas has its vertex at the origin, its focus at the point (0,1/4) and its directrix with equation y=-1/4. Any other parabola can be "generated" from the y=x² "parent" by one or more mathematical transformations. As an example, the parabola with equation (y-3)=(x-4) ² can be created by translating the y...