# Graph Transformations

## Graph Transformations Revision

**Graph Transformations**

You should have seen some **graph transformations** before, such as translations and reflections – recall that reflections in the x-axis flip f(x) vertically and reflections in the y-axis flip f(x) horizontally. Here, we will also look at stretches.

There are **4** main types of graph transformation that we will cover. Each transformation has the same effect on all functions.

Make sure you are happy with the following topics before continuing.

**Type 1: y = f(x+k)**

For the transformation y=f(x+k), for k>0:

**f(x+k)**is f(x) moved**k to the left****f(x-k)**is f(x) moved**k to the right**

In this example, we have f(x) = x^2 - 4 and **y=f(x+2)**

So, subtract 2 from the x-coordinates of f(x) to get **y=f(x+2)**

**Type 2: y = f(x)+k**

For the transformation y=f(x)+k, for k>0:

**f(x)+k**is f(x) moved**k upwards****f(x)-k**is f(x) moved**k downwards**

In this example, we have f(x) = x^2 - 4 and **y=f(x)+3**

So, add 3 to the y-coordinates of f(x) to get **y=f(x)+3**

**Type 3: y = af(x)**

For the transformation y=af(x):

- If
**|a|>1**then af(x) is f(x)**stretched vertically**by a factor of a - If
**0<|a|<1**then f(x) is**squashed vertically** - If
**a<0**then is f(x) also**reflected in the x-axis**

In this example, we have f(x) = x^2 - 4 and **y=2f(x)**

This is a stretch vertically, so multiply the y-coordinates of f(x) by 2 to get **y=2f(x)**

**Type 4: y = f(ax)**

For the transformation y=f(ax):

- If
**|a|>1**then f(ax) is f(x)**squashed horizontally**by a factor of a - If
**0<|a|<1**then f(x) is**stretched horizontally** - If
**a<0**then is f(x) also**reflected in the y-axis**

In this example, we have f(x) = x^2 - 4 and **y=f(2x)**

This is a squash horizontally, so divide the x-coordinates of f(x) by 2 (or multiply by \dfrac{1}{2}) to get **y=f(2x)**

**Note:**

- For these transformations, any asymptotes need to be moved correspondingly.
- A
**squash**by a factor of a is equivalent to a**stretch**by a factor of \dfrac{1}{a} - When drawing graph transformation, only a sketch including important points is necessary.

**Combinations of Transformations**

For **combinations of transformations**, it is easy to break them up and do them one step at a time (do the bit in the brackets first). You can sketch the graph at each step to help you visualise the whole transformation.

e.g. for f(x) = x^2 - 4 and y=2f(x+2), draw the graph of

**y=f(x+2)** first, and then use this graph to draw the graph of

**y=2f(x+2)**

**Note:** These transformations can also be combined with modulus functions.

## Graph Transformations Example Questions

**Question 1:** The function f(x) is shown on the graph below.

Sketch the graph of y = f(4x)

**[2 marks]**

y = f(4x) means that the graph of f(x) is squashed horizontally by a factor of 4.

Hence, the graph will look like:

**Question 2:** The function f(x) is shown on the graph below.

Sketch the graph of y = - \dfrac{1}{2} f(x)

**[3 marks]**

Firstly, since the coefficient before f(x) is negative, we need to reflect f(x) in the x-axis.

The coefficient of \dfrac{1}{2} before -f(x) means that the graph of -f(x) is squashed vertically by a factor of 2.

Hence, the graph will look like:

**Question 3:** The function f(x) is shown on the graph below.

Sketch the graph of y = 3f(x) - 1

**[3 marks]**

Split the transformation up into 2 parts – firstly sketch y=3f(x) which is a stretch vertically by a scale factor of 3 (multiply the y-coordinates by 3:

Then, do the second transformation – y=3f(x)-1 means that we need to move the graph down by 1 (subtract 1 from the y-coordinates):

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