Translations of Graphs

What are transformations of graphs?

A transformation is simply a change of some sort
- Reflections (using either the x-axis or y-axis as a mirror line)
- Translations (moving the whole graph in the x and/or y direction)
You should be able to recognise these two different sorts of transformation and apply them to a given graph.

How do we translate the graph of a function?

Given a transformation in function notation, then

y = f(x + a), represents a translation −a units in the x axis
- a is inside the bracket
- Note the change of sign highlighted!
y = f(x) + b, represents a translation + b units in the y direction
- a is outside the bracket
For example, y = f(x + 3) + 2 means
- a translation of y = f(x) by −3 in x the direction and +2 in the y direction
- Note the change of sign in the x direction as pointed out above!

How do we describe translations of graphs?

Some questions give a transformed function in the form y = f(x + a) or y = f(x) + a and ask you to describe the transformation
To describe a translation fully, you must include;
1. the transformation: "translation"
2. the direction in the x-axis and in the y-axis
  - this can be given as a worded description, e.g. "3 left and 2 up" / "−3 in the x-axis and +2 in the y-axis"
  - or it can be given as a vector $(\begin{matrix} x \\ y \end{matrix})$ , e.g. "by $(\begin{matrix} - 3 \\ 2 \end{matrix})$ "
Remember that
- if the a is inside the bracket, its a translation by −a in the x-axis
- if the a is outside the bracket, its a translation by a in the y-axis

Exam Tip

y = f(x + a); "a" next to x, translates in x-axis by −a
y = f(x) + a; "a" not next to x, translates in y-axis

Worked example

The graph of $y = f (x)$ is shown on the graph below.
On the same graph sketch $y = f (x + 3) + 2$ .

This is a translation by −3 in the x direction (i.e. 3 to the left, note the change of sign again) and +2 in the y direction (i.e. 2 up)

So we copy the given graph in its new position. Translate key points- endpoints and vertices like the vertex shown below- first, then join the points with straight lines

Worked example

Describe the transformation that maps the graph of $y = f (x)$ to the graph of $y = f (x - 4) - 6$ .

The number inside the bracket (next to x) is −4 so this is a translation by +4 in the x-axis (note the change in sign again)
The number outside the bracket (not next to x!) is −6 so this is a translation by −6 in the y-axis

Translation by $(\begin{matrix} 4 \\ - 6 \end{matrix})$
or Translation 4 right and 6 down
or Translation by +4 in the x-axis and −6 in the y-axis

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