Vertical Translations Up and Down (AF 604)

A vertical translation happens when you add or subtract a number outside the function, after everything else has been calculated. In y = f(x) + k, every single output gets k added to it. The curve doesn't stretch, flip, or lean, because the x-values never change and the relationship between neighboring outputs never changes either. The whole graph just rides straight up or down like an elevator.

This works because k is applied last. For any x you plug in, you first get f(x), the original height, then you add k to that height. Since every output shifts by the same fixed amount, distances between points on the curve stay identical, so the shape is preserved exactly. A positive k lifts the graph; a negative k drops it.

This sits next to the other single-coefficient moves: a constant added inside the parentheses, like f(x + h), shifts the graph sideways instead, and a coefficient multiplied on the outside, like a·f(x), stretches or flips it rather than sliding it. Vertical translation is the simplest of these moves because it only resets the baseline, the height the whole graph sits at, without touching the curve's width, direction, or steepness. Seeing k by itself, added or subtracted outside the function, is the signal that you're dealing with this move.