I have become aware that in some high school curricula they are teaching the following method to "transform a point on a function":
Say $f(x) = 2x^2 + 2$ is the original function and $g(x) = 3f(x-2)$ is the transformed function. A common homework problem is of the form "given the two functions, where does the point $(1,4)$ end up after it is transformed by $g(x)$?"
Students are required to first map $1 \rightarrow 1-2 = -1$, then to map $f(1) \rightarrow 3*f(1-2) = 3f(-1) = 12$. The "correct" answer, then, is that $(1,2)$ maps to $(-1,12)$.
I have a significant issue with this, in that the mapping is moving both $x$ and $y$. $y$ ends up as I would expect - transformed under the function. But the function doesn't transform $x$, which should be independent. What are they trying to get at with these types of questions?
We have $$g(x)=3f(x-2)=3(2(x-2)^2+2)=3(2(x^2-4x+4)+2)=3(2x^2-8x+8+2)$$ $$=3(2x^2-8x+10)=6x^2-24x+30.$$ Therefore, $g(-1)=6-24+30=12,$ as required. The point is on the new function.