Can these data points be used to form a polynomial?

Question

If I have the following data points: $(-7,-1)(-5,1)(-1,-3)(3,1)(6,-2)$

With the correct assumptions, is this enough information to derive a $4^{th}$ degree polynomial function?

Answer 1

Yes, there is a unique way to define a polynomial function of degree $n-1$ given its values on $n$ points. See Lagrange interpolation, for example.

https://en.wikipedia.org/wiki/Lagrange_polynomial