Let $E \colon \mathbb{R}^2 \to \mathbb{R}^2$ be a rotation about some point $(x_0,y_0)$ and let $\theta$ the angle of rotation for $E$ A formula for $E$ can be determined as follows.

First, translate the point $(x_0,y_0)$ to $(0,0)$ The map of this translation is $T(x,y)=(x-x_0,y-y_0)$

Next, rotate by $\theta$ about the origin. The map of this rotation is $R(x,y)=(x\cos\theta-y\sin\theta,x\sin\theta+y\cos\theta)$

Finally, translate the point $(0,0)$ back to $(x_0,y_0)$ The map of this translation is $T^{-1}(x,y)=(x+x_0,y+y_0)$

The fact that $E=T^{-1} \circ R \circ T$ can be used to obtain a formula for $E$

Let $E \colon \mathbb{R}^2 \to \mathbb{R}^2$ be a reflection about some line $y=mx+b$ Let $\theta=2\arctan m$ A formula for $E$ can be determined as follows.

First, translate the $y$ intercept $(0,b)$ to $(0,0)$ The map of this translation is $B(x,y)=(x,y-b)$

Next, reflect about the line $y=mx$ The map of this reflection is $F(x,y)=(x\cos\theta+y\sin\theta,x\sin\theta-y\cos\theta)$

Finally, translate the point $(0,0)$ back to $(0,b)$ The map of this translation is $B^{-1}(x,y)=(x,y+b)$

The fact that $E=B^{-1} \circ F \circ B$ can be used to obtain a formula for $E$

Let $E \colon \mathbb{R}^2 \to \mathbb{R}^2$ be a reflection about some line $x=c$ A formula for $E$ can be determined as follows.

First, translate the $x$ intercept $(c,0)$ to $(0,0)$ The map of this translation is $C(x,y)=(x-c,y)$

Next, reflect about the line $x=0$ The map of this reflection is $M(x,y)=(-x,y)$

Finally, translate the point $(0,0)$ back to $(c,0)$ The map of this translation is $C^{-1}(x,y)=(x+c,y)$

The fact that $E=C^{-1} \circ M \circ C$ can be used to obtain a formula for $E$