A composition of transformations is a series of transformations done on an image. When written out, you have to complete the transformation all the way to the left first, and then keep going from right to left in order. This composition of transformations would be written out "r x-axis*T{-3,1}*R-90". The first transformation is a rotation 90 degrees clockwise (or -90 degrees). The rule for that is to change the coordinates from (x,y) to (y,-x). Point A is (4,5) and using the rule, it would change to (5,-4). This shows that the rotation was done correctly because A' is (5-4). The next transformation is a translation {-3,1}. The rule would be {x-3, y+1}. Point A' is (5,-4), and using the rule would change it to {5-3, -4+1}. This would simplify into (2,-3) and those are the coordinates of point A'' so we know that the translation was correct. The next transformation is a reflection over the x-axis. Using the rule, we would change (x,y) to (x,-y). Point A'' is (2,-3) and once we apply the rule, these coordinates would change to (2,3). These are the coordinates of A''', so we know that we did the reflection correctly. Figure A'''B'''C'''D''' is the final figure after all the transformations have been completed.

Example of Composition of Transformations

This image shows how you have to do the transformation all the way to the right first, and then keep going from right to left in order. In this example, you have to complete the transformation of {3,4} first and then the reflection over the x-axis.