In this post we will look at reflections and translations (the requirement for the UK GCSE syllabus). For GCSE students are required to sketch translations and reflections of a given function.

Experiment with the following Desmos pages to examine these transformations, f(x) + a, f(x + b), -f(x) and f(-x) where a and b are integers.

Translations

**Translations parallel to the x axis.**as illustrated above.- Or try translating this graph of
**f(x) = sin x parallel to the x axis**. **Translations parallel to the y axis**.**Graph of sin x translated parallel to y axis**

Reflections

Equations of Transformed Graphs

Note that you can work out the equation of the transformed graphs.

Suppose we wish to work out the equation of f(x) = x^{2}−5x+4 after reflection in the x axis.

To reflect in the x axis the transformation is –f(x)

So f(x) becomes –f(x) and we have – (x^{2}−5x+4) = – x^{2} +5x-4

To work out the equation of f(x) = x^{2}−5x+4 after reflection in the y axis.

To reflect in the y axis the transformation is f(–x)

So f(x) becomes f(–x) so we replace x by –x

and we have (–x)^{2}−5(–x)+4 = x^{2} +5x+4

Suppose we wish to work out the equation of f(x) = x^{2}−3x+2 after a translation of 3 units to the left parallel to the x axis as shown in the first image in this post.

f(x) = x^{2}−3x+2, we need f(x+3) for the equation of the transformed graph

f(x+3) = (x+3)^{2}−3(x+3)+2 = x^{2}+6x+9−3x−9+2 = x^{2}+3x+2