Background

    A catenary is the curve formed by a flexible cable of uniform density hanging from two points under its own weigh.  Cables of suspension bridges and attached to telephone poles hang in this shape.  If the lowest point of the catenary is at , then the equation of the catenary is   

        .  

Exploration 1.

Do the above four catenaries look similar ?  Are they the same ?
Plot them on the same graph to see what is happening.  

    Hence we see that the "shape" of the curve is preserved when it is changed to , indeed the curve is merely magnified by the scale factor when or reduced by the scale factor when  .  Therefore, in order to "preserve the shape",  use the formula  

    ,

for a "catenary" which passes through the origin.  

(c) John H. Mathews 2004