## Combined Transformation

 5.3 Combined Transformation

 How to determine the image and object of a combined transformation? Let $$P$$ be the object and $$P^\prime$$ be the image, then To determine the object when an image is given, the transformation of translation needs to be performed in the opposite direction. For example, translation $$\begin{pmatrix} 2\\ -4 \end{pmatrix}$$ becomes $$\begin{pmatrix} -2\\ 4 \end{pmatrix}$$. For the transformation of rotation, rotation in clockwise direction will become rotation in anticlockwise direction. For the transformation of enlargement, enlargement with a scale factor $$k=2$$ will become reciprocal, that is enlargement with scale factor \begin{aligned}k=\frac{1}{2}\end{aligned}.An object can perform more than one transformation and will produce an image based on the transformations involved. In general, the combination between the transformation $$A$$ and transformation $$B$$ can be written as transformation $$AB$$ or transformation $$BA$$ in the order of the desired transformation. Combined transformation $$AB$$ means transformation $$B$$ followed by transformation $$A$$.

 Example 5 The diagram below shows several triangles drawn on a Cartesian plane. It is given that transformation $$P=$$ Translation $$\begin{pmatrix}3\\1\end{pmatrix}$$ $$Q=$$ Rotation of $$90^{\circ}$$ anticlockwise at centre $$(3,4)$$ $$R=$$ Enlargement at centre $$(8,0)$$ with scale factor $$2$$ Determine the image of triangle $$A$$ under the combined transformation: (a) $$\text{P}^2$$            (b) $$\text{RQ}$$ Solution: a) Combined transformation $$\text{P}^2$$ means transformation $$\text{P}$$ performed 2 times in a row. b) Combined transformation $$\text{RQ}$$ means transformation $$\text{Q}$$ followed by transformation $$\text{R}$$.