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Reference

Types of Transformations

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

Concept

Reflection of a Function

A reflection of a function is a transformation that flips a graph over some line. This line is called the line of reflection and is commonly either the or axis. A reflection in the axis is achieved by changing the sign of the coordinate of every point on the graph.
The coordinate of all intercepts is Thus, changing the sign of the function value at intercepts makes no difference — any intercepts are preserved when a graph is reflected in the axis.
Reflecting a Function in the x-axis
A reflection in the axis is instead achieved by changing the sign of every input value.
When which is at the intercept, this reflection does not affect the input value. Therefore, the intercept is preserved by reflections in the axis.
Reflecting a Function in the y-axis
The following table illustrates the different types of reflections that can be done to a function.
Transformations of
Reflections In the axis
In the axis

Concept

Vertical Stretch and Shrink of a Function

The graph of a function can be vertically stretched or shrunk by multiplying the function rule by a positive number
The vertical distance between the graph and the axis will then change by the factor at every point on the graph. If this will lead to the graph being stretched vertically. Similarly, leads to the graph being shrunk vertically. Note that intercepts have the function value Therefore, they are not affected by this transformation.
Stretching and Shrinking of a Graph Vertically
The general form of this transformation is shown in the table.
Transformations of
Vertical Stretch or Shrink Vertical stretch,
Vertical shrink,

Concept

Horizontal Stretch and Shrink of a Function

By multiplying the input of a function by a positive number its graph can be horizontally stretched or shrunk.
If every input value will be changed as though it was farther away from the axis than it really is. This leads to the graph being shrunk horizontally — every part of the graph is moved closer to the axis. Conversely, leads to a horizontal stretch. The horizontal distance between the graph and the axis is changed by a factor of
Stretching and Shrinking of a Graph Horizontally
Note that -intercepts have the -value which is why they are not affected by this transformation. The general form of this transformation is shown in the table.
Transformations of
Horizontal Stretch or Shrink Horizontal stretch,
Horizontal shrink,