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.

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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 aaa is equivalent to a stretch by a factor of 1adfrac{1}{a}a1​
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:

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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:

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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 $3$ :