[Math] How to write exponential function for curve that pass 4 or more points

Question

Given 3 points, we could write exponential function for a curve that would pass those points in $y = ax^n + b$ form

So I wonder that, could we write a function in exponential form that would pass 4 points, which is not quadratic or cubic bezier

Or are there any formula that could construct a function for curve to pass any given points?

Edit : Sorry I was misunderstand the word exponential function. What I really mean is rational exponent function

I don't know what it called but it not polynomial. Something like $y = ax^{1.5} + b$ but can pass any 4 or more points

Answer 1

As the equation $y=ax^n+b$ has three independent parameters, you can constrain it to pass through three given points, but not more.

If you add more terms or terms with more coefficients, you can increase the flexibility. For example, the linear combination of four independent functions.

Possibilities are infinite.