An affine transform performs a linear mapping from 2D coordinates to other 2D coordinates that preserves the "straightness" and "parallelness" of lines.
Such a coordinate transformation can be represented by a 3 row by 3 column matrix with an implied last row of [ 0 0 1 ]. This matrix transforms source coordinates (x,y) into destination coordinates (x',y') by considering them to be a column vector and multiplying the coordinate vector by the matrix according to the following process:
[ x ] [ a b tx ] [ x ] [ a * x + b * y + tx ]
[ y ] = [ c d ty ] [ y ] = [ c * x + d * y + ty ]
[ 1 ] [ 0 0 1 ] [ 1 ] [ 1 ]
This class is optimized for speed and minimizes calculations based on its knowledge of the underlying matrix (as opposed to say simply performing matrix multiplication).