Catenary: Shape of a Hanging Chain

What you are seeing: a uniform, perfectly flexible chain hung between two supports. Carrying only its own weight it takes the catenary $y(x) = a \cosh(x / a) - a$, where $a = T_0 / (\mu g)$ is the catenary parameter (horizontal tension $T_0$, linear mass density $\mu$, gravity $g$). A small $a$ is a deep, slack, low-tension sag; a large $a$ is a shallow, taut, nearly straight cable. The readout shows $a$, the cable length, span, sag and the peak tension (largest at the supports).

Drag either support: the chain length is fixed, so the catenary re-solves through the new endpoints. Pull the supports far enough apart that the chain cannot reach and it snaps taut to a straight line. A shallow chain looks almost parabolic, $y \approx x^2 / 2a$ (the classical suspension-bridge approximation), but the computed shape is always the exact cosh, which grows faster than that quadratic for deep sag.