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?

Translations in the x-axis

Given an original equation in x, the graph of that equation will be translated −a units in the x direction by adding a inside a bracket next to x
- For example, in relation to y = x²;
  - y = (x − 2)² is a translation 2 units to the right/ a translation by +2 in the x-axis
  - y = (x + 3)² is a translation 3 units to the left/ a translation by −3 in the x-axis
- Another example, in relation to y = sin(x) where x is in degrees;
  - y = sin(x + 90) is a translation 90 degrees to the left/ a translation by −90 degrees in the x-axis
  - y = sin(x − 180) is a translation 180 degrees to the right/ a translation by +180 degrees in the x-axis
- Note that for changes in the x direction, the translation is in the opposite direction to the sign of a (as highlighted)!

Translations in the y-axis

Given an original equation in x, the graph of that equation will be translated +b units in the y direction by adding b outside the bracket
- For example, in relation to y = x²;
  - y = x² + 2 is a translation 2 units up/ a translation by +2 in the y-axis
  - y = x² − 3 is a translation 3 units down/ a translation by −3 in the y-axis
- Another example, in relation to y = sin(x) where x is in degrees;
  - y = sin(x) − 2 is a translation 2 units down/ a translation by −2 units in the y-axis
  - y = sin(x) + 1 is a translation 1 unit up/ a translation by +1 unit in the y-axis
- Note that for translations in the y direction, the direction is the same as the sign of b

How do we describe translations of graphs?

Some questions give a transformed a transformed graph of an equation with an original graph of an equation and ask you to describe the transformation
As with translations of shapes, 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 $(\begin{matrix} - 3 \\ 2 \end{matrix})$ "

Exam Tip

REMEMBER that:

number next to x (inside the bracket): the graph translates in the x direction but with the opposite sign

Worked example

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

ocr-7-graphs-transformations-translations-1

This is a translation by +3 in the y direction (i.e. 3 up)
So we copy the given graph in its new position. Translate key points first- x-intercepts, maximums and minimums as shown below

ocr-7-graphs-transformations-translations2

Now join your new points with a curved line. The new curve should go through the key points shown in the answer below

ocr-7-graphs-transformations-translations3

Worked example

Describe the transformation that maps the graph of $y = x^{3}$ to the graph of $y = {(x - 4)}^{3} - 6$ .

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

Translations of Graphs (OCR GCSE Maths)

Revision Note