Definition:One-Parameter Family of Curves

Definition
Consider the implicit function $$f \left({x, y, c}\right) = 0$$ in the $$\left({x, y}\right)$$-plane where $$c$$ is a constant.

For each value of $$c$$, we have that $$f \left({x, y, c}\right) = 0$$ defines a relation between $$x$$ and $$y$$ which can be graphed in the cartesian coordinate plane.

Thus, each value of $$c$$ defines a particular curve.

The complete set of all these curves for each value of $$c$$ is called a one-parameter family of curves.