Function Transformations

Transformation is one of the most important concepts for mathematical functions. As suggested by the name of the term, a transformation of a function is the output we obtain after putting the returned value of a function through a specific transformation technique such as a shift or a scaling. In the most general sense, we can think of this process as a composite function: putting the output of one function through another function. However, there are specific types of functions that are commonly used as transformations due to their particular characteristics and usefulness, and we will go through them in the following subsections, starting with shifting.

Since a transformation is most easily understood in the context of the changes that it applies to the plot of a function, we will also frame our following discussions accordingly.

Shifts

A shift of a function happens when the plot of a function is shifted by a specific amount along the x axis and...