A-level Mathematics/C3/Combining Transformations of Graphs

Translation of curvesEdit

Many curves are translations of the basic curve of the same degree. For example   is a translation of   by 1 unit up on the   axis,  , and one unit to the left on the   axis,  , so the full translation is  . It is useful to remember the translations of curves, so that you can easily sketch graphs from the basic shape.

Vertical translationEdit

The constant term moves the entire graph vertically by that much. For example,   will move the graph upwards by 3 units, and   will move the graph down by 3 units.

Horizontal translationEdit

When a number is subtracted from   before it is squared, the entire graph will move in the opposite direction by the same amount. For example,   will move the graph to the right by 3 units, and   will move the graph to the left by 3 units. It may be confusing that the graph moves in opposite directions, but with the translations,   has to be smaller or larger to give the same result that it used to, so positive numbers move it in the negative direction, and vice versa. You can imagine that the numbers on the   axis are changed by the amount of the translation, so in  , the origin   is now  .

General translationsEdit

In general,   is a translation of   by  .