# TRANSFORMATIONS OF FUNCTIONS

In this section, you will learn how to do different types of transformations of functions like translation, stretch, compression and reflection.

To make the students to understand the different types of transformations, we have explained each kind of transformation with step by step explanation along with the corresponding figures.

### Transformations - Definition

A very simple definition for transformations is, whenever a figure is moved from one location to another location, a Transformation occurs.

If a figure is moved from one location another location, we say, it is transformation.

Our next question is, how will the transformation be?

To know that, we have to be knowing the different types of transformations.

Now, let us come to know the different types of transformations.

### Different Types of Transformations

The different types of transformations which we can do in the functions are

1. Horizontal Translation

2. Vertical Translation

3. Reflection across  the x-axis

4. Reflection across the y-axis

6. Vertical Stretch and Compression

How different types of transformations occur in terms of x-coordinate and y-coordinate have been summarized below.

Summary of Transformation

Transformation

Appearance in Function

Transformation of Point

Vertical Translation

f(x) ---> f(x) + d

(x, y) --> (x, y + d)

Horizontal Translation

f(x) ---> f(x - c)

(x, y) --> (x + c. y)

Vertical Stretch or Compression

f(x) ---> af(x)

(x, y) --> (x, ay)

Horizontal Stretch or Compression

f(x) ---> f(kx)

(x, y) --> (x/k, y)

Reflection in  x-axis

f(x) ---> -f(x)

(x, y) --> (x, -y)

Reflection in y-axis

f(x) ---> f(-x)

(x, y) --> (-x, y)

### Order of Transformations

In transformations of functions, if we have more than one transformation, we have to do the transformations one by one in the following order.

1. Stretches/Compressions

2. Reflections

3. Translations

### What is image and pre-image?

When a transformation occurs, the original figure is known as the pre-image and the new figure is known as the image.

It has been clearly shown in the below picture.

When a figure is moved from one location to another location, we say that it is a transformation.

In this point, always students have the following questions.

More clearly, on what grounds is the transformation made ?

Is there any pre-decided rule to make transformation?

Yes, there is a pre-decided rule to make each and every transformation.

The rule we apply to make transformation is depending upon the kind of transformation we make.

We have already seen the different types of transformations in functions.

For example, if we are going to make transformation of a function using reflection through the x-axis, there is a pre-decided rule for that. According to the rule, we have to make transformation.

The rule that we apply to make transformation using reflection and the rule we apply to make transformation using rotation are not same.

Hence, for each type of transformation, we may have to apply different rule.

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