There are two curves that look a LOT alike, both occur in nature, but have very distinct mathematical form and properties. This activity compares the two curves.
1) Play with the coefficients a through e. How do they effect the curves?
2) How closely can you get the red and blue curves to match? Extension: if someone gave you a parabola, can you always find a catenary to (almost) match? Vice versa?
3) Show the picture of the 144th St Bridge in Holland, Michigan. Can you match it with either curve? Do you think it was designed to be a parabola or a catenary?

4) Research catenaries on Wikipedia. Why do you think they come up so much in architecture?
Note that the curve here is really a flattened catenary. To be a true catenary, you need a*b=1.
More GeoGebra posted at mathhombre.blogspot.com.