Graph translations

Graph translations involve shifting the graph of a function either horizontally or vertically without altering its shape. The translation changes the position of the graph on the coordinate plane, but the basic form of the graph remains unchanged.

Vertical Translations

A vertical translation shifts the graph up or down. The function:

\[y = f(x) + k\]

moves the graph of \(y = f(x)\) k units up if \(k > 0\), and k units down if \(k < 0\).

Horizontal Translations

A horizontal translation shifts the graph left or right. The function:

\[y = f(x + h)\]

moves the graph of \(y = f(x)\) h units to the right if \(h < 0\), and h units to the left if \(h > 0\).

Combined Translations

You can combine horizontal and vertical translations. The function:

\[y = f(x + h) + k\]

translates the graph of \(y = f(x)\) by h units horizontally and k units vertically.