Looking for the correct terminology for this type of problem

Question

I have a sequence of real valued tuples $(x_1,y_1),...,(x_k,y_k)$ and I am looking for ideas of modelling a function $f$ such that the error $f(x_i)-y_i$ is small.

1. Is there a theory for this type of problem?
2. How do I decide what type of function to choose?
3. How do I define the error for the whole sequence?
4. How do I fit the parameters of $f$ to th data $(x_i, y_i)$?

I could plot $x$ against $y$ to see if the graph has a certain shape.

Answer 1

Try "spline function", "approximation by function", "least squares approximation"