It is known that a class of fractal functions $f\colon I(\subset\mathbf R)\to\mathbf R$ can be defined using an IFS (Iterated Function System) code. Moreover, these functions can be constructed to interpolate the data set $Y=\{(x_i,y_i)\}^n_{i=1\subset\mathbf R^2$}. Here, affine transformations (in $\mathbf R^2$) of such functions (defined as affine mappings of their graphs) are examined. Particularly, it is shown that fractal interpolating functions are affine invariant only upon the class of affine mappings whose linear part is given by a lower-triangular matrix. Also, it is proved that the fractal interpolation scheme is a linear operator and can be written in the \emph{Lagrange form}.