Skill: Apply vertical and horizontal translations

A translation moves a graph without changing its shape.

• Vertical translation:

$$f(x)+k$$

shifts the graph up $k$ units if $(k>0)$, or down $(|k|)$ units if $(k<0)$. Each point $((x,y))$ becomes $((x,y+k))$.

• Horizontal translation:

$$f(x-h)$$

shifts the graph right $h$ units if $(h>0)$, or left $(|h|)$ units if $(h<0)$. Each point $((x,y))$ becomes $((x+h,y))$.

Notice that the sign inside the function works in the opposite direction: $(x-3)$ shifts right $3$, while $(x+3)$ shifts left $3$.

Worked example

Suppose

$$f(x)=\sqrt{x}$$

and define

$$g(x)=\sqrt{x-3}+2.$$

Identify the translations and the new domain and range.

Step 1: Identify the horizontal change.

The input is $(x-3)$, so the graph shifts right $3$ units.

$$(x,y)\longrightarrow (x+3,y)$$

Step 2: Identify the vertical change.

The $(+2)$ is outside the square root, so the graph shifts up $2$ units.

$$(x,y)\longrightarrow (x,y+2)$$

Combining both translations gives

$$(x,y)\longrightarrow (x+3,y+2).$$

For example, the point $((0,0))$ on $(f(x)=\sqrt{x})$ moves to

$$(0+3,0+2)=(3,2).$$

Therefore, $(g(x)=\sqrt{x-3}+2)$ is the graph of $(f(x)=\sqrt{x})$ shifted right $3$ and up $2$.

The original domain is $(x\ge 0)$, so shifting right $3$ gives

$$\boxed{x\ge 3}.$$

The original range is $(y\ge 0)$, so shifting up $2$ gives

$$\boxed{y\ge 2}.$$

The shape of the graph remains unchanged; only its position, domain, and range are shifted.