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# Transformations of Graphs

All functions can be transformed, which means they are altered in a certain way. These alterations can be described through geometrical transformations.

## What are the types of graphical transformations?

There are three main transformations of graphs that you need to be aware of at A-Level: Translations, Stretches and Reflections.

### Translations

Translations are a type of graphical transformation where the function is moved. To explain a translation, you use a vector in the form , where the top part of the vector shows how the function has been translated horizontally and the bottom part of the vector shows the function has moved vertically.

The direction of the translation depends on whether each variable is positive or negative.

 variable Positive negative 0 x moves right moves left does not move y moves up moves down does not move

If you find this difficult to remember, just think about how coordinates work, it's a similar principle. For instance, a negative x coordinate would be on the left side of the graph. Equally, a positive y coordinate would be the top part of the graph.

A function can be expressed as. It is, then, translated by . Sketch the new translated function.

The original function can be seen graphically:

Original graph - StudySmarter Originals

The vector tells you the function will be translated to the right by 5 and down by 3.

If you sketch this, the new function should look like this:

Transformed graph - StudySmarter Originals

As it is a sketch, it is important to label key points on the graph like turning points

#### Expressing translated functions

You might be asked to write the new translated function using the vector. Translating the function by , the translated function can be written as. Notice how a becomes negative in the function but b stays the same.

The function is translated by . What is the new translated function?

According to the vector, the function is translated 4 to the right and 3 down.

### Stretch

A function can be stretched both horizontally and vertically.

• Vertical stretch: where a is the scale factor the function is stretched by. For instance, if function is stretched vertically by 5, the function becomes.
• Horizontal stretch: where a is the scale factor the function is stretched by. As you can see the reciprocal of a is used in the function equation. For instance, if the function is stretched horizontally by 2, the function becomes.

Function has a turning point at (2, 9) and (10, -6). The function is stretched so that the new function can be expressed as. What would be the new turning point?

As the scale factor is within the brackets, the function is being stretched horizontally. Furthermore, as the function is being stretched horizontally, the actual scale factor of the stretch of.

Therefore, we multiply each of the x-coordinate by 4. The (2, 9) becomes (8, 9) and (10, -6) becomes (40, -6). As you can see, the y-coordinate is unaffected as it is not being stretched.

### Reflections

Reflections are where the whole functions are inverted in a line of reflection.

All horizontal in the x-axis and vertical reflections can be expressed as a function.

• A function that has been reflected in the x-axis can be written as. For turning points of the function, the x-coordinate stays the same but y-coordinates becomes inverted. For instance, if a function has a turning point of (4, -2) and is reflected in the x-axis, the turning point would become (4, 2).

• A function that has been reflected in the y-axis can be written as . Where the function has been reflected in the y-axis, the y-coordinate of the turning point becomes inverted which the x-coordinate stays the same.

 Reflection in the... y-coordinate x-coordinate x-axis becomes inverted stays the same y-axis stays the same becomes inverted

## Combining graphic transformations

At A-level, you need to be able to work with a combination of graphical transformations within a question. To do so, you need to know the order of operations for graph transformations. Here is a the list of graph transformations, in the correct order:

1. Stretch

2. Reflection

3. Translations

## Transformations of Graphs - Key takeaways

• There are three main transformations of graphs: stretches, reflections and translations.
• Translations are a type of graphical transformation where the function is moved. To explain a translation, you use a vector in the form where the top part of the vector shows how the function has been translated horizontally, and the bottom part of the vector shows the function has moved vertically.
• A function can be stretched both horizontally and vertically.
• Reflections are where the whole functions are inverted in a line of reflection.

## Frequently Asked Questions about Transformations of Graphs

How to describe the transformation of a graph: the geometrical alteration of a graph is called the transformation of a graph.

There are mainly three types of graph transformations.

1) Translation

2) Reflection

3) Stretch

Graph transformation is found when the existing graph is modified to produce a variation of the proceeding graph.

## Final Transformations of Graphs Quiz

Question

What is the correct sequence in graph transformations?

1. Stretch

2. Reflections

3. Translations

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Question

All functions can be transformed. True or false?

True

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Question

What direction is the graph expected to make during translation for a negative x value?

movement is towards the left.

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Question

What direction is the graph expected to make during a translation for a positive y value?

movement is upwards.

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Question

What direction is the graph expected to make during a translation for an x value of 0?

No movement is made on the horizontal direction.

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Question

The turning point of a function is (5,3). What would the coordinate be upon reflection on the y-axis?

(5,-3)

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