We know that the degree 2 equations $x^2 + y^2 =1$ and $x^2 - y^2 =1$ can be parametrized by exponential functions. How come exponential functions show up in this seemingly unrelated area? I think it has to do with the fact that exponential functions satisfy $y'=y$, while the degree 2 equations satisfy $y'=\frac{-x}{y}$ and $y'=\frac{x}{y}$ respectively.
Also, there are two extreme cases of degree two equations- parabola and straight lines. Exponential functions don't show up in those. What exactly happens as we approach those extreme cases which causes the exponential behavior to collapse?
Can degree 3 or higher curves be parametrized by exponentials? $y=\frac{1}{x}$ is also a hyperbola. Can exponentials parametrize that?