Single transformations
Recall the four basic transformations. A translation slides every point by a vector; a reflection flips over a line; a rotation turns about a centre through an angle; an enlargement scales from a centre. Some useful coordinate rules about the origin:
- Reflection in the $x$-axis: $(x,y)\to(x,-y)$;
- Reflection in the $y$-axis: $(x,y)\to(-x,y)$;
- Rotation $90^\circ$ clockwise about $O$: $(x,y)\to(y,-x)$;
- Rotation $180^\circ$ about $O$: $(x,y)\to(-x,-y)$.
Combined transformations
A combined transformation carries out one transformation and then another on the result. If transformation $A$ is followed by $B$, we apply $A$ first, then $B$.
Key formula
$90^\circ\text{ clockwise about }O:\ (x,y)\to(y,-x),\qquad 180^\circ\text{ about }O:\ (x,y)\to(-x,-y)$
Remember
Order matters. In general, doing $A$ then $B$ does not give the same image as $B$ then $A$.
Example
Reflect $(4,1)$ in the $x$-axis to get $(4,-1)$, then translate by $\begin{pmatrix}2\\3\end{pmatrix}$ to get $(4+2,\,-1+3)=(6,2)$.
Useful facts
Two successive reflections in parallel lines are equivalent to a single translation. Two successive reflections in intersecting lines are equivalent to a single rotation about the point of intersection.