In the exam you may be asked to apply transformations to graphs. This revision bite deals with translations, stretches and reflections.
Graphs and translations
Here is the graph y = x2
Below are the graphs of:
- y = x2 + 2
- y = (x - 3)2
- y = (x - 3)2 + 2
You can see that the y = (x - 3)2 + 2 (in red) is the same shape as y = x2, but it has been moved 2 up the y axis and 3 along the x axis. This is called a translation, and we show it with the vector:
The graph of y = x² + 2 is exactly the same shape as y = x², but it has been translated by 2 units in the y-direction (translation with vector 
)
In general, if we start with y = f(x), then:-
y = f(x) + a translates f(x) 'a' units in the y-direction (translation 
)
The graph of y = (x - 3)² is exactly the same shape as y = x², but it has been translated by 3 units in the x-direction (translation with vector
)
In general, if we start with y = f(x), then:-
y = f(x + a) translates f(x) '-a' units in the x-direction (translation 
)
You can see that y = (x - 3)² + 2 is a combination of the two translations. In this case the vector of translation is 
These transformations will work in the same way for any other functions. Find out more by playing this activity.
- Question
The graph of y = x3 is drawn in the diagram. On the same diagram sketch the graph of y = (x + 2)3
- Answer
Same shape as y = x3 with translation
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