An arbitrary polynomial of degree $n$ is determined by $n+1$ points. In some sense, one can think of the polynomial as having $n+1$ parameters - the $n+1$ coefficients associated to it. Is there a generalization of this to arbitrary families of functions with $n$ parameters?
For example, for $a,b \in \mathbb{R}$, the functions of the form $g(a, b, x) = e^{x/a} + b$ can be thought of as a family of functions with two parameters. Given any member of this family, we can determine the function (by determining $a,b$) by given any two points $(x_1,y_1), (x_2,y_2)$. (First determine $a$ with $e^{x_2/a} - e^{x_1/a} = y_2-y_1$, since $e^{x_2/a} - e^{x_1/a}$ is monotonic in $a$ , and then determine $b$).
Is it generally true that for any such family with $n$ parameters, we can determine the parameters of a member of said family from $n$ points? What if we restrict to smooth, non-decreasing or strictly increasing functions? What other conditions would be needed?