The properties depend on what the transformations are.
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Transformations can translate, reflect, rotate and enlarge shapes on the Cartesian plane.
Dilation, rotation, reflection and translation (Go to www.mathwarehouse.com/transformations/) For more information
You can usually make valid transformations in one of the expressions until you get the other expression. A "valid transformation" in this context means one that doesn't change the value of the expression.
Not all books at all levels have Transformations in Chapter 9! Besides, that is not enough information for a sensible question.
Linear transformations can be very important in graphics. Also, linear transformations come up whenever you need to solve systems of linear equations, which arise quite often. Finally, they can be useful in further areas of mathematics such as topology.