f is an affine transformation. The transformation f is a reflection in the line
By first translating an appropriate point to the origin find f in the form
The line y= -x-2 is a line with slope -1 passing through the point (0, -2). The translation (x,y)-> (x,y+2) maps that point into (0, 0) and maps everypoint on the line into a point on y= -x. A reflection in that line is (x,y)->(-y, -x), which would be represented by the matrix equation . Finally, the translation (x,y)->(x, y-2) maps back to the original position. So we are doing A((x,y)+ (0,2))+ (0,-2)= A(x,y)+ A(0,2)+ (0,-2).