Adding or subtracting a number shifts a graph up or down without changing its shape or its rate of change at any point -- verified point by point, and across quadratic, absolute value, and cubic functions.
Add one number and the entire graph moves. What happens when we add or subtract a number from a parent function's equation? This is a vertical translation — the simplest transformation there is.
Add a positive number and the graph moves upward; subtract a number and it moves downward. The amount of the shift equals the number added or subtracted — and this rule applies to every function.
A vertical translation moves position only. The graph's shape stays identical, and so does its rate of change at every point — the function is exactly as steep before the shift as after it, whether the graph moves up or down. Remember this detail: it is exactly what makes differentiation and derivatives work the way they do.
Take and substitute a few values of .
| x | x² | x² + 3 |
|---|---|---|
| 0 | 0 | 3 |
| 1 | 1 | 4 |
| 3 | 9 | 12 |
Every single point moved upward by exactly 3 units — the same 3 whether the starting height was 0, 1, or 9. That is the meaning of a vertical translation: not a new shape, but every point on the old one nudged by the same fixed amount.
Neither graph changes shape. The V still looks like a V, and the cubic still bends the same way — only their position on the page is different.
The quadratic function is even: its graph on the right of the y-axis is a mirror image of the left. A vertical shift moves the whole graph up or down without touching left-right symmetry, so is still even — only its height changed, not its shape.