If you take a string of beads or a necklace, hold it by two points at the same height, and watch the work of gravity, you will get a smooth curve, quite similar to a parabola. Indeed, it was once mistaken to be a parabola. However, it is not parabola in spite of its tricky appearance. If you happen to know the trajectory of a projectile is a parabola, then you could think about the physics behind the chain curve. What is the physics behind the curve? What makes the chain drape ? What keeps it from falling all the way down? What is the balance when you hold it between the two points? Specifically, imagine a point P along the curve, what keeps that point P in its current place?
In fact, a hanging chain follows the catenary curve, which is everywhere if you pay attention to it: up in the sky when you look at the power lines, down near the ground along the fences, or, maybe, it is right on your neck if you wear a necklace. There are many interesting aspects about the catenary curve, you can read more about its historical and physical connections at the Wikipedia Catenary page.
In this activity, you could take a picture of a hanging chain and match it to a catenary function, thus finding its mathematical representation and conducting further mathematical analysis.
where a is a constant that is determined by the gravity and the material of the chain, hs is the horizontal shift, vs if the vertical shift. Try adjusting a, hs, vs in using either the sliders or their algebraic counterparts (with arrow keys). See if you can find a mathematical function that best matches the picture.