# Compound transformations of graphs (A-level Maths)

This is Part 3 of a three-part tutorial on graph transformations, and covers compound transformations. If you haven’t already worked through Part 1 and Part 2 then I recommend you look at those before trying to make sense of this one!

This tutorial goes beyond GCSE/IGCSE; in fact, compound transformations are part of the second-year content of the A-level specification.

## A quick recap

### Basic rules for graph transformations

The table below summarises the rules you need to remember.

Remember, the GCSE spec doesn’t include stretches, though Edexcel IGCSE and A-level both do.

### Skills covered so far

In Part 1 we practised the following skills:

- Given an original graph and the description of a transformation, finding the equation of a transformed graph and sketching it
- Given the equations of an original and a transformed graph, identifying the transformation that has taken place

In Part 2 we looked at sketching transformations of a graph with unknown equation, and how the different transformations affect the x- and y-coordinates of points on the graph.

In this instalment we’re going to do the same again, but with compound transformations.

## Compound transformations

A compound transformation is what you have when a graph undergoes two or more transformations in succession, for example, it might be stretched in the x-direction with scale factor 2 and then translated with vector .

You’ve already encountered one type of compound transformations when we’ve looked at the two components of a vector transformation: a translation with vector could also be described as a translation with vector followed by a translation with vector (in either order).

### Key rules for compound transformations

The key things to remember with compound transformations are:

- Transformations in the y-direction go “as expected” and transformations in x-direction go “backwards”; this applies to the
of compound transformations as well as the individual transformations themselves (see VON HIR, below).*order* - Horizontal and vertical transformations do not affect each other – remember, horizontal transformations only affect the x-coordinates and vertical ones only the y-coordinates – so you only need to worry about the order
the vertical transformations and*within*the horizontal transformations.*within*

**Note: **Since changing the sign is the same as multiplying by -1, reflections and stretches share equal priority.

You might find it helpful to remember the acronym **VON HIR**:

Vertical, Outside Normal; Horizontal, Inside, Reverse.

### Identifying and describing a sequence of transformations

#### Examples

**y = xÂ˛****â†’ y = 3 sin x + 4** or **y = 4 + 3 sin x**

The transformations are applied to the **Outside **of the original function so theyâ€™re **Vertical**.

The entire sin x has been multiplied by 3 and then had 4 added to it; the order is **Normal **so that equates to

A stretch in the y-direction with SF 3 and then a translation with vector

y = sin x

â†’ y = sin (2x â€“ 50Â°)

The transformations are applied to the **Inside **of the original function (changing/expanding the x) so theyâ€™re **Horizontal**.

Working in **Reverse **to â€śundress the xâ€ť, we add 50Â° and then divide by 2, so this equates to

A translation with vector and ** then **a stretch in the x-direction with SF Â˝.

Putting both of these together will give a format that you might encounter in a trig modelling question:

y = sin x

â†’ y = 4 + 3 sin (2x â€“ 50Â°)

It doesnâ€™t matter whether we list the vertical or the horizontal transformations first, but it makes sense to combine the two vectors, so that gives us

A stretch in the y-direction with SF 3** then **a translation with vector

**a stretch in the x-direction with SF Â˝.**

*then*#### Your turn 1

Describe each series of transformations. For questions 4-7 there are (at least) two possible interpretations; can you find both?

### Finding the equation of the resulting graph

Work out what equation youâ€™d end up with if you applied each of the sets of transformations to the start equation.

Start by separating out the vertical and horizontal transformations, then apply each set separately.

#### Example

Starting with , what would be the result of a translation by followed by a stretch SF 2 in the x-direction?

**Vertical: **

Translation => add 20 to the whole thing

**Horizontal:**

Translation then stretch SF 2 in the x-direction

Remember, REVERSE order for horizontal transformations,

so first divide the x by 2

then subtract 10 from the

So you end up with

#### Your turn 2

Find the result of each set of transformations:

- Stretch SF 3 in the y-direction then translation
- Reflection in the x-axis then stretch SF 2 in the y-direction
- Stretch SF 4 in the x-direction then translation
- Translation then reflection in the y-axis
- Translation , then stretch SF 4 in the y-direction, then stretch SF 2 in the x-direction.

### Transforming a graph with unknown equation

Below is the graph of y = f(x), the same as we used in Part 2 of this series.

As you can see, four coordinate points are marked on the graph. Your task is to (a) sketch each of the transformed graphs listed under “Your turn 3” below, labelling the new position of each of the four marked points, and (b) describe the sequence of transformations in each case.

Don’t forget VON HIR, and that

**A transformation in the x-direction will change**, and*only*the x-coordinates**A transformation in the y-direction will change**.*only*the y-coordinates

#### Your turn 3

Now try sketching these graphs and describing the sequence of transformations in each case:

- y = f(2x) + 1
- y = 2f(x + 1)
- y = 2f(x) + 1
- y = f(2x + 1)
- y = -f(x) + 1
- y = f(1 – x) [
*Hint: think of it as -x + 1*] - y = 2f(-x) + 1
- y = -f(1 – 2x)

That about covers it – you should now be able to deal with just about anything that the topic of compound transformations throws at you!

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#### Your turn 1: answers

- Stretch in y-direction SF 3 then translation
- Stretch in y-direction SF 2 then translation
- Translation then stretch in x-direction SF .
- Stretch in y-direction SF 9 THEN translation

OR Stretch in x-direction SF then translation - Think of it as Stretch in y-direction SF and reflection in x-axis (either way round), THEN translation

OR Stretch in x-direction SF 2 and reflection in x-axis (either way round), THEN translation - Stretch in the x-direction SF (could go anywhere in sequence) ; reflection in the x-axis then translation

Could also be expressed as Stretch in y-direction SF 3 and reflection in x-axis (either way round), THEN translation

OR

Stretch in the x-direction SF (OR stretch in y-direction SF 3), THEN translation , THEN reflection in x-axis - H: translation then reflection in y-axis; V: translation then stretch in y-direction SF 5 (can combine translations as long as combined translation precedes both of the other transformations)

OR H: reflection in y-axis then translation ; V: stretch in y-direction SF 5 then translation (can combine translations as long as the combined translation comes after both other transformations)

Click here to return to questions

#### Your turn 2: answers

- H:

so we get - H:

V:

so we get

Click here to return to questions