# Definition:Family of Curves/One-Parameter

## 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.

### Parameter

The value $c$ is the parameter of $F$.

## Examples

### Concentric Circles

The equation:

$x^2 + y^2 = r^2$

is a one-parameter family of concentric circles whose centers are at the origin of a Cartesian plane and whose radii are the values of the parameter $r$.

### Circles of Equal Radius with Centers along $x$-Axis

Consider the equation:

$(1): \quad \paren {x - h}^2 + y^2 = a^2$

where $a$ is constant.

$(1)$ defines a one-parameter family of circles of constant radius $a$ whose centers are on the $x$-axis of a Cartesian plane at $\tuple {h, 0}$ determined by values of the parameter $h$.