Family of Curves
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=x2 parabola 4 units to the right and 3 units vertically upward. Thus, all parabolas do share a kind of "family" resemblance, and the same can be said of other types of curves. This is important to the mathematician because it allows her to study the parent curve and draw conclusions about the entire family of curves, without having to investigate each curve individually.
Many of the key properties of a curve will be "inherited" from its parent.
Families of curves play an important role in the study of differential equations. For example, it is known that 2x is the derivative of x2, but it is also the derivative ofx2+1 and x2+13 and x2+0.5, among infinitely many others. In fact, 2x is the derivative of any member of the family of curves with equations of the form x2+C, where C can be any constant. This means that the differential equation y=2x has a family of solutions of the form x2+C. This form is also called the general solution of the differential equation. If some specific condition is given, then it is possible to pick out the one member of the family of solutions that satisfies this condition. This member is called a particular solution of the differential equation for which the given condition is true. For instance, if it is known that the point (2,7) is on the graph, then we can say that the particular solution of y=2x which satisfies this condition is y=x2+3. Thus, there is an entire family of curves that have the property that y=2x, but only one member of that family will pass through the point (2,7). Such a family of curves may be illustrated by means of a "slope field" in which short segments of tangent lines are drawn at each point on a grid. Such a picture gives one a sense of the "flow" of the family of solutions curves for a differential equation.
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