Can we represent a parabola or data on the parabolic curve as a sum of two exponential functions?

Question

I am just wondering, is it possible to represent a parabola or data on the parabolic curve as a sum of two exponential functions?

Your suggestions will be appreciated.

Thanks, Raghu

Answer 1

Since, by definition, parabolas are a specific conic section, you can represent all of them using quadratic equations, but there's no way to precisely represent them using a sum of $2$ exponential functions. However, for a "parabolic curve", if you mean something which is similar to a parabola, how about double the hyperbolic cosine function of $\cosh(x)$, i.e., $f(x) = e^x + e^{-x}$? It's the sum of $2$ exponential functions. As shown in cosh function graph, its graph looks much like a concave up parabola with an axis of symmetry at $x = 0$, plus a minimum of $1$ at $x = 0$ for $\cosh(x)$, and a minimum of $2$ for $f(x)$.