Transformations of Functions

Transformations of Functions

A transformation of a function is a way of mapping that function to another. 

The most commonly asked question in an exam on this topic is: Which geometrical transformation maps the graph y = f(x) to y = g(x)? 

There are three types of transformation: translations, stretches and reflections. This revision sheet will take you through all three in a step-by-step manner, including some exam tips for how to obtain full marks on any transformation question.

An important element to remember when doing any form of transformation is 

Vertical

Outside

Normal

Horizontal

Inside

Reversed

Translations

A translation is a type of transformation which moves a function from one position to another. These are always written in vector form and are recognised through the process of addition/subtraction.

y = x^2 maps to y = (x^2) + 2

Here, the geometrical transformation would be described as a:

Translation   0

                        2   where the 0 and 2 are enclosed in a single bracket. 

This is because, according to VON HIR, the action we are doing to the function of x (x^2 in this case) is outside of the function we began with, meaning it is a vertical translation (in the y-direction) and is taken as it is. 

If we mapped y = x^2 to y = (x+2)^2

the transformation would then become:

Translation   -2

                         0

VON HIR dictates that anything INSIDE the function we began with is in the x-direction (horizontal) and becomes reversed (i.e. positive becomes negative) 

y = x^2 maps to y = ((x+2)^2) + 2 would be described as a:

Translation   -2

                          2

To generalise: if y = f(x) maps to y = f(x-a) + b

the geometrical transformation used to describe it would be a

Translation   a

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