Translating Functions | Moosa Academy

Translating Functions — Moosa Academy

A translation slides a graph without changing its shape. Two versions exist — vertical and horizontal — and although they look alike on the page, one of them has a sign convention that catches almost everyone out the first time.

Concept Vertical translation

g(x) = f(x) + k

Add $k$ to the whole function, outside everything else. Every output rises by $k$ , so the entire graph lifts.

f(x) = x^2

becomes

g(x) = x^2 + 4

— up 4 units.

f(x) = x^2

becomes

g(x) = x^2 - 4

— down 4 units.

The sign matches the direction: positive lifts, negative lowers. This is the intuitive case.

Concept Horizontal translation

g(x) = f(x - h)

Replace

x

with

(x - h)

inside the function.

h > 0

shifts right;

h < 0

shifts left.

The sign inside the bracket looks reversed: $f(x - 4)$ moves the graph to the right. The reason is that the function now produces its original output when $x - h = 0$ , that is when $x = h$ — so whatever used to happen at zero now happens at $h$ .

Note The two rules side by side

Transformation	Equation	Sign against direction
Up	f(x) + k, k > 0	Matches
Down	f(x) + k, k < 0	Matches
Right	f(x − h), h > 0	Reversed
Left	f(x − h), h < 0	Reversed

The underlying rule is short: a horizontal translation touches the independent variable only, a vertical one touches the whole equation.

Example Combining both

Translate $f(x) = |x|$ two units right and three units up.

Right 2 means

h = 2

, so the bracket becomes

(x - 2)

Up 3 means

k = 3

, added outside.

g(x) = |x - 2| + 3

The vertex has moved from

(0,\ 0)

(2,\ 3)

⟹ g(x) = |x − 2| + 3, vertex (2, 3)

In general $g(x) = f(x - h) + k$ moves the marked point of any parent function to $(h,\ k)$ . The order of the two shifts does not matter.

Example Other parent functions

f(x) = \sqrt{x}

moved 3 right and 2 down:

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

f(x) = \dfrac{1}{x}

moved 4 left and 5 up:

g(x) = \dfrac{1}{x + 4} + 5

⟹ the same two rules apply to every parent function

Note that a translation can move a domain restriction or an asymptote with it. The second graph has its vertical asymptote at $x = -4$ rather than $x = 0$ .

Summary

A vertical translation adds k outside the function; the sign matches the direction.
A horizontal translation replaces x with (x − h); the sign is reversed.
Combined, g(x) = f(x − h) + k moves the key point of the graph to (h, k).
Translations change position only — never the shape of the curve.