A-level Mathematics/C3/Combining Transformations of Graphs

Translation of curves
Many curves are translations of the basic curve of the same degree. For example $$y=(x+1)^2+1$$ is a translation of $$y=x^2$$ by 1 unit up on the $$y$$ axis, $$\begin{pmatrix} 0 \\ 1 \end{pmatrix}$$, and one unit to the left on the $$x$$ axis, $$\begin{pmatrix} -1 \\ 0 \end{pmatrix}$$, so the full translation is $$\begin{pmatrix} -1 \\ 1 \end{pmatrix}$$. It is useful to remember the translations of curves, so that you can easily sketch graphs from the basic shape.

Vertical translation
The constant term moves the entire graph vertically by that much. For example, $$y=x^2+3$$ will move the graph upwards by 3 units, and $$y=x^2-3$$ will move the graph down by 3 units.

Horizontal translation
When a number is subtracted from $$x$$ before it is squared, the entire graph will move in the opposite direction by the same amount. For example, $$y=(x-3)^2$$ will move the graph to the right by 3 units, and $$y=(x+3)^2$$ 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, $$x$$ 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 $$x$$ axis are changed by the amount of the translation, so in $$y=(x-3)^2$$, the origin $$(0,0)$$ is now $$(3,0)$$.

General translations
In general, $$y=(x-t)^2+s$$ is a translation of $$y=x^2$$ by $$\begin{pmatrix} t \\ s \end{pmatrix}$$.