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Concept

Translation of a Function

A translation of a function is a transformation that moves a function graph in some direction, without any rotation, shrinking, or stretching. A function's graph is vertically translated by adding a number to — or subtracting from — the function rule.

$g (x) = f (x) \pm k$

Every point of the graph of

y = f (x)

will be moved up or down by

k

units, depending on the sign of

k .

Likewise, a function's graph is horizontally translated by adding a number to — or subtracting from — the rule's input.

$g (x) = f (x \pm h)$

Every point on the graph of

y = f (x)

will be moved to the left or to the right by

h

units, depending on the sign of

h .

Horizontal Translation of Linear Function

The table below summarizes the different types of translations that can be done to a function.

Translations of $f (x)$
Vertical Translations	Translation up $k$ units, $k > 0$ $y = f (x) + k$
Vertical Translations	Translation down $k$ units, $k > 0$ $y = f (x) - k$
Horizontal Translations	Translation to the right $h$ units, $h > 0$ $y = f (x - h)$
Horizontal Translations	Translation to the left $h$ units, $h > 0$ $y = f (x + h)$

Why

Why the Graph is Translated?

Consider adding a number

k

to a function rule.

g (x) = f (x) + k

k

is positive, this operation increases the value of the output for every

x,

moving the graph upward. Similarly, if

k

is negative, then the graph is moved downward, because the output of the function is decreased.

Now consider subtracting a number

h

from the input.

g (x) = f (x - h)

h

is positive, then the value of the input is reduced. Therefore, greater

x -

values are needed to obtain the original output, leading to a translation to the right. In contrast, when

h

is negative, the input value is increased. This means that smaller inputs are needed to obtain the original output. This leads to a translation to the left.

Exercises

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