# Math Help - Affine transformation

1. ## Affine transformation

Having a little trouble with this question:

f is an affine transformation. The transformation f is a reflection in the line
y=-x-2.

By first translating an appropriate point to the origin find f in the form

f(x)=Ax+a

2. Originally Posted by offahengaway and chips
Having a little trouble with this question:

f is an affine transformation. The transformation f is a reflection in the line
y=-x-2.

By first translating an appropriate point to the origin find f in the form

f(x)=Ax+a
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 $A(x,y)= \begin{bmatrix}0 & -1 \\ -1 & 0\end{bmatrix}\begin{bmatrix}x \\y\end{bmatrix}= \begin{bmatrix}-y \\ -x\end{bmatrix}$. 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). $A(0,2)= \begin{bmatrix}0 & -1 \\ -1 & 0\end{bmatrix}\begin{bmatrix}0 \\ 2 \end{bmatrix}= \begin{bmatrix}-2 \\ 0\end{bmatrix}$

So A(x,y)+ a is $\begin{bmatrix}0 & -1 \\ -1 & 0\end{bmatrix}\begin{bmatrix} x \\ y\end{bmatrix}+ \begin{bmatrix}-2 \\ -2\end{bmatrix}$.