Is there a formula to find a translating point of a function?
Suppose I have $y = x^2 + 2$ and after getting translated, it becomes $y' = x^2 - 9x + 5$. Let's say we don't know the translating point yet, how do you find $T(x,y)$ such that : $$ y\overset{\begin{array}{c} T(x,y) \end{array}}{\longrightarrow}y' $$
I know I can just plot the functions and determining it manually, but what about a general formula? I've searched about this, but Google only suggested translation from horizontal and vertical which doesn't seem to answer my question. I'm not sure, but since it's a point, it must be the combination of those two i.e. vertical and horizontal.
You have $f(x) = x^2+2$ and you are looking for $a,b \in \mathbb{R}$ such that $f(x-a)+b = x^2 - 9x + 5$. Multiplying out, we get $$f(x-a) + b = x^2 - 2ax + a^2 + 2 + b.$$ Comparing with the expression in the first line, we can obtain $a$ and $b$.