This is one of the most annoying things to teach in any functions-based class: Supposing $a, b > 0$, why does the transformation $y=f(x-a)+b$ move the graph of $f$ to the right by $a$ and up by $b$, even though the signs are different? Why does the thing that's happening horizontally work “backwards” from the thing that's happening vertically? Prompted by a nice activity from Matt Enlow, I finally have an explanation that feels persuasive. I road-tested this with my “Functions Modeling ...
