Translating Functions
A translation slides a graph without changing its shape. Vertical shifts add outside the function and match direction; horizontal shifts change x inside the function and reverse the sign.
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A translation slides a graph without changing its shape. Vertical shifts add outside the function and match direction; horizontal shifts change x inside the function and reverse the sign.
A translation slides a graph without changing its shape. Two versions exist — vertical and horizontal — and although they look alike on the page, one of them has a sign convention that catches almost everyone out the first time.
Add to the whole function, outside everything else. Every output rises by , so the entire graph lifts.
The sign matches the direction: positive lifts, negative lowers. This is the intuitive case.
The sign inside the bracket looks reversed: moves the graph to the right. The reason is that the function now produces its original output when , that is when — so whatever used to happen at zero now happens at .
| Transformation | Equation | Sign against direction |
|---|---|---|
| Up | f(x) + k, k > 0 | Matches |
| Down | f(x) + k, k < 0 | Matches |
| Right | f(x − h), h > 0 | Reversed |
| Left | f(x − h), h < 0 | Reversed |
The underlying rule is short: a horizontal translation touches the independent variable only, a vertical one touches the whole equation.
Translate two units right and three units up.
In general moves the marked point of any parent function to . The order of the two shifts does not matter.
Note that a translation can move a domain restriction or an asymptote with it. The second graph has its vertical asymptote at rather than .