**Transformations of Functions : **

In this section, we 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.

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.

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

**3. Reflection through the x-axis **

**4. Reflection through the y-axis**

**5. Horizontal Expansions and Compressions**

**6. ****Vertical Expansions and Compressions**

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)

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

1. Stretches / Compressions

2. Reflections

3. Translations

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.

How is this transformation made ?

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.

After having gone through the stuff given above, we hope that the students would have understood, how to do transformations of functions.

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