Translating Graphs

Adding a constant outside the function shifts the whole graph $\text{up}$ by $a$ (or down if $a$ is negative). This behaves exactly as you would expect.

Adding a constant $\text{inside}$ the brackets shifts the graph horizontally — but in the opposite direction to the sign. $f(x+a)$ moves $\text{left}$ by $a$, while $f(x-a)$ moves $\text{right}$ by $a$.

A combined translation can be written as a column vector. Moving right by $a$ and up by $b$ is the vector $\left(\begin{array}{c}a\\ b\end{array}\right)$, taking $y=f(x)$ to $y=f(x-a)+b$.