Definition:One-Parameter Family of Curves

Definition
Consider the implicit function $\map f {x, y, c} = 0$ in the cartesian $\tuple {x, y}$-plane where $c$ is a constant.

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

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

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