Apply vertical and horizontal translations

A vertical translation changes $f(x)$ to $f(x)+k$ , shifting every point $(x,y)$ to $(x,y+k)$ , while a horizontal translation changes it to $f(x-h)$ , shifting points to $(x+h,y)$ ; the sign in the input expression therefore produces the opposite apparent direction. The understanding applies across equations, graphs, and tables, preserving the function’s shape while shifting its domain and range by the corresponding amounts, without extending to rotations, stretches, or more advanced transformation theory.

Detailed Explanation: 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.

Learn by doing: Apply vertical and horizontal translations

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Function Transformations - Mapping Notation - Action to Double Transformation

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