What can be known about the formulas for energy only from the fact that it is conserved?

The question is to figure out how the energy can be derived knowing just one thing:

There is a quantity called Energy that is conserved over time.

The goal is to get an equation that somehow implies the basic formulas for kinetic energy and potential energy, $\frac{1}{2}mv^2$ and $mgh$. So how can we get this with just a rudimentary knowledge of physics and algebra, and the first principle (stated above)?