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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.


g(x)=f(x) ± 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.
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.


g(x)=f(x± 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

Translation down k units, k>0

y=f(x)-k

Horizontal Translations Translation to the right h units, h>0

y=f(x-h)

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 If 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.
Animated Proof of Vertical Translation of Linear Function
Now consider subtracting a number h from the input. g(x) = f(x - h) If 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.
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 x- or y-axis. A reflection in the x-axis is achieved by changing the sign of the y-coordinate of every point on the graph. y = - f(x) The y-coordinate of all x-intercepts is 0. Thus, changing the sign of the function value at x-intercepts makes no difference — any x-intercepts are preserved when a graph is reflected in the x-axis.
Reflecting a Function in the x-axis
A reflection in the y-axis is instead achieved by changing the sign of every input value. y = f(- x) When x = 0, which is at the y-intercept, this reflection does not affect the input value. Therefore, the y-intercept is preserved by reflections in the y-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 f(x)
Reflections In the x-axis
y=- f(x)
In the y-axis
y=f(- x)
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 a. y = a * f(x) The vertical distance between the graph and the x-axis will then change by the factor a at every point on the graph. If a > 1, this will lead to the graph being stretched vertically. Similarly, a < 1 leads to the graph being shrunk vertically. Note that x-intercepts have the function value 0. 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 f(x)
Vertical Stretch or Shrink Vertical stretch, a>1
y= af(x)
Vertical shrink, 0< a< 1
y= af(x)
Concept

Horizontal Stretch and Shrink of a Function

By multiplying the input of a function by a positive number b, its graph can be horizontally stretched or shrunk. y = f(b * x) If b > 1, every input value will be changed as though it was farther away from the y-axis than it really is. This leads to the graph being shrunk horizontally — every part of the graph is moved closer to the y-axis. Conversely, b < 1 leads to a horizontal stretch. The horizontal distance between the graph and the y-axis is changed by a factor of 1b.
Stretching and Shrinking of a Graph Horizontally
Note that y-intercepts have the x-value 0, which is why they are not affected by this transformation. The general form of this transformation is shown in the table.
Transformations of f(x)
Horizontal Stretch or Shrink Horizontal stretch, 0< b<1
y=f( bx)
Horizontal shrink, b>1
y=f( bx)
Exercises

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