body{margin:0;} noscript{z-index: 10000; position: fixed; display: block; height: 100%; width: 100%; background: #fff;} noscript .fa{font-size: 40px; display:block; color: #daa520;} noscript div{margin-top: 6%; padding-left: 20px; padding-right: 20px; text-align: center;} ! You must have JavaScript enabled to use this site.

Recommended

Tests

An error ocurred, try again later!

Chapter {{ article.chapter.number }}

{{ article.number }}.

Show less Show more

Lesson Settings & Tools

	{{ 'ml-lesson-number-slides' \| message : article.intro.bblockCount }}
	{{ 'ml-lesson-number-exercises' \| message : article.intro.exerciseCount }}
	{{ 'ml-lesson-time-estimation' \| message }}

Image Credits

{{ item.file.title }}
{{ presentation }}

No file copyrights entries found

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.

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

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.

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 -

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.

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.

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 $y = - f (x)$
Reflections	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 \cdot 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, $a > 1$ $y = a f (x)$
Vertical Stretch or Shrink	Vertical shrink, $0 < a < 1$ $y = a f (x)$

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 \cdot x)

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

\frac{1}{b} .

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, $0 < b < 1$ $y = f (b x)$
Horizontal Stretch or Shrink	Horizontal shrink, $b > 1$ $y = f (b x)$

Loading content

{{ article.displayTitle }}

Why