# Transformations of Graphs

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

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) = x2−5x+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  – (x2−5x+4) = – x2 +5x-4

To work out the equation of  f(x) = x2−5x+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 =  x2 +5x+4

Suppose we wish to work out the equation of  f(x) = x2−3x+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) = x2−3x+2, we need f(x+3) for the equation of the transformed graph
f(x+3) = (x+3)2−3(x+3)+2 = x2+6x+9−3x−9+2 = x2+3x+2