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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.

Every point of the graph of will be moved up or down by units, depending on the sign of
Vertical Translation of Linear Function
Likewise, a function's graph is horizontally translated by adding a number to — or subtracting from — the rule's input.

Every point on the graph of will be moved to the left or to the right by units, depending on the sign of
Horizontal Translation of Linear Function
The table below summarizes the different types of translations that can be done to a function.
Translations of
Vertical Translations Translation up units,

Translation down units,

Horizontal Translations Translation to the right units,

Translation to the left units,

Why

Why the Graph is Translated?
Consider adding a number to a function rule.
If is positive, this operation increases the value of the output for every moving the graph upward. Similarly, if is negative, then the graph is moved downward, because the output of the function is decreased.
Animated Proof of Vertical Translation of Linear Function
Now consider subtracting a number from the input.
If is positive, then the value of the input is reduced. Therefore, greater values are needed to obtain the original output, leading to a translation to the right. In contrast, when 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.
Animated Proof of Horizontal Translation of Linear Function