When doing absolute value functions and graphing them, you can add constant in, or outside of the absolute value symbols to translate (move) the function. Please explain why putting a constant inside the absolute value symbols moves it horizontally and putting a constant outside of them makes it move vertically. If y = |x| - 2, then when x = 0 the plotted point should be (0,-2). If y = |x - 2| when x = 0 the plotted point should be (0,2). Either way, the V of the function should only move vertically. However, I am being told that if x = 0 in y = |x - 2|, the plotted point would be (2,0). This would mean the function translates to the right horizontally, how does this make sense?

If the point is described as P(x,y) than for y=|x-2| the point for x=0 can be (0,2), because 2=|0-2|.
The point (2,0) also solves the equation. 0=|2-2|
That means both points lay on the function.
In my picture you can see both functions. The green: y=|x|-2 and the blue y=|x-2|. The blue function compared to the green was shifted up and to the right.
That is not what the rule is about.
If you change the function y=|x|-2 to y=|x|-3, the function moves vertically down.
If instead you changed the function y=|x-2| to y=|x-3|, now the function would move horizontally only, and in this case to the right.
Got it?