Which Way Did It Move?
Drag the corner of the dashed V to a new spot. The graph moves exactly where you put it — and the equation's sign points the opposite way from the move, every single time.
g(x) = |x − 3| − 2
Across, it moved 3 to the right and the equation prints "x − 3" — backwards from the move. Up and down, it moved 2 down and the "− 2" outside the bars matches exactly.
- Moved across
- 3 right
- Moved up/down
- 2 down
Drag the warm corner anywhere on the grid, or use the arrow keys.
Ready to try some? Function transformation worksheets
Where it usually goes wrong
f(x − 3) is read left to right like a sentence, so "minus" is taken to mean "move left" — the same direction a number line moves for ordinary subtraction. It moves right. The minus sign is not shifting a point on a line; it is asking "where do I need x to be so that x − 3 becomes the OLD input", and the answer to that question is always three steps past where you would guess.
Try this
- Drag the corner three squares to the right and read the equation. It prints "x − 3", the minus sign pointing the opposite way from the move you just made.
- Now drag it three squares to the left instead. The equation prints "x + 3" — a plus sign for a move in the negative direction.
- Drag it straight up or down instead. The number outside the bars matches the move exactly, with no sign flip at all — only the sideways move is backwards.
Where this goes next
Every family of graphs gets shifted this same backwards way — parabolas, square roots, absolute values, trig curves — so the sign is worth being suspicious of for good, rather than re-deriving it each time a new shape shows up.