 Affine covariance of natural cubic spline interpolation

If $S$ is the natural cubic spline interpolating data $y_i$ at fixed distinct knots $x_i$, then $aS(x)+b+cx$ is the natural cubic spline interpolating $a{y}_i+b+c{x}_i$.

The transformed function is piecewise cubic, preserves continuity of the first two derivatives, and has endpoint second derivatives $a{S}^{\prime \prime }=0$. It takes exactly the transformed nodal values. Uniqueness of the natural spline therefore proves the claim. This permits affine changes in data or the addition of a linear trend without solving a new spline system.