Why when it comes to translating functions (x-b) negative?

Question

Quick question: when in the formula:(x-b), if it was (x-2) moves to the right but when it is (x-b)+c, you have (x-2)+3 then it moves up 3. I do not understand why it is a minus sign instead of a plus sign so instead be: (x+b)+c then (x+2)+3 would move to the right 2 and up 3

Answer 1

Comment made into an answer:

Replacing $x$ with $x-3$ moves the graph $3$ to the right (direction of increasing $x$).

Replacing $y$ with $y-3$ moves the graph $3$ up (direction of increasing $y$). So it works the same in both dimensions if you do the replacements the same way.

The movement is in the opposite direction to what you might think because when you replace $x$ by $x-3$, the value of $x$ that works in the new equation with $x-3$ must be $3$ $bigger$ than it used to be to make the new formula with $x-3$ come out the same as the old formula with $x$.