# Definition:One-Parameter Family of Curves

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## Definition

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

For each value of $c$, we have that $\map f {x, y, c} = 0$ 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 $F$.

## Sources

- 1956: E.L. Ince:
*Integration of Ordinary Differential Equations*(7th ed.) ... (previous) ... (next): Chapter $\text {I}$: Equations of the First Order and Degree: $1$. Definitions: $(1.1)$ - 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $1$: The Nature of Differential Equations: $\S 2$: General Remarks on Solutions