(Junior high/ high school level)
The mathematical formulas for the catenary and the parabola both look easy. The formula for the parabola is a quadratic polynomial, y=k x^2. Here, k is any positive number, and "^" is a sign we use to denote exponents. So x^2 means "x raised to the 2nd power".
Experiment 3: On a computer or graphing calculator, or just by hand, try plotting the function y=k x^2 for different values of k until you get a shape that looks like the shape of the Golden Gate Bridge in the picture. What effect does changing k have on the shape?
The formula for a catenary looks easy, but looks can be deceiving.
This is the hyperbolic cosine function, and we pronounce it just like it's spelled, to rhyme with "gosh". You might not have heard of it. But if you have heard of the exponential function e^x, then we can write down cosh in terms of the exponential function,
Experiment 4: On a computer or graphing calculator, try plotting the function y=k cosh(x/k) for different values of k until you get a shape that looks like the Gateway Arch from the picture. If you use a positive value for k the curve will be upside down to what the Gateway Arch is. What effect does changing k have on the shape?
We said earlier that these curves -- the parabola and the catenary -- arise from considering all the forces acting on the hanging cables. We will talk more about forces in the next section.