I know how to use transformations of functions. The question is why do we need to learn transformations of functions? Also, how do we use them in real life, so as a real life application.
Transformations such as graphing y =(x-2)^2 + 1 using the graph of y =x^2.
I'm assuming a transformation of a function is like taking $f(x) = x^2$ as a base and then considering, for example $f(x) = ax^2 + c$. One example where this comes in handy to understand intuitively, that comes up in my job all the time, is in statistical modeling. Basically, when you want to fit a set of known data points to a function, it helps to understand how different transformations affect the output of the function, and how to interpret them.
Transforming the graph can also be used "backwards" in the case of linear transformations, to keep the graph in place and shift/scale the axes, instead. For example (courtesy wikipedia), this is what allows the following chart to display both Celsius (bottom/left) and Fahrenheit (top/right) degrees.
