I have 3 data points for a material property (permeability of a gas through an solid elastomer) as a function of temperature. My goal is to extrapolate the property at a lower temperature.
As a slightly-educated guess, I chose a power fit because many chemical properties and reaction rates have the Arrhenius form: $f(T)=Ce^{-E/kT}$.
So I chose a power fit $f(T)=Ae^{BT}$ (although that's not quite the same as Arrhenius form) and got very good agreement to the 3 existing points, $R^2 = 0.998$, which seems to support that functional form for the dependence. But then I started wondering if that is saying anything profound, because a power fit could hit 3 data points by necessity. In the same way as a parabolic equation $f(x)=Ax^2+Bx+C$ would fit nearly any 3 points, or a linear equation $f(x)=Ax+B$ would fit any 2 points.
Hence my question, does a power fit $f(x)=Ae^{Bx}$ automatically fit 3 data points well, or is the good fit an indication that it may be a good model for the physics?