# How to Translate a Function’s Graph

Two types of translation are possible for a function’s graph: shifting and scaling. You can also reflect a graph over the x-axis (or the y-axis); reflections are a special case of scaling, where x- or y-values are multiplied by a negative number.

## How to Translate a Function’s Graph with Shifts

Shifts, which change location but not shape, can be horizontal or vertical:

• Vertical translation: a shift up and down the y-axis. These shifts add or subtract a constant to every y-coordinate. For example, to move a graph up two units, add 2 to the function; to move the graph down, subtract 2.
• Horizontal translation: a shift left or right along the x-axis. These shifts add or subtract a constant to every x-coordinate. For example, to shift a graph left two units, add 2; to shift the graph right two units, subtract 2.

## How to Translate a Function’s Graph with Scaling

Scaling alters the size and shape of a function’s graph by multiplying or dividing the function by a constant. This results in shrinks or stretches that can be horizontal or vertical:

• Vertical scale: Multiply or divide every y-coordinate by a constant.
• Horizontal scale: Multiply or divide every x-coordinate by a constant.

If the absolute value of the constant is less than 1, the graph will shrink. If the absolute value of the constant is greater than 1, the graph will stretch.

## How to Translate a Function’s Graph with Reflections

Reflections are a special type of scaling where the graph is multiplied by a negative number. Multiplying the graph by -1 will reflect the graph without changing the shape, but you can multiply by another number to shrink or stretch the graph as well. For examples, see: Reflection Over The X-Axis.

## References

Graphs: Desmos.com.