Transformations of Graphs

What are transformations of graphs?

• A transformation is simply a change of some sort
• When it comes to graphs you need to know about two different sorts of transformation:
  • Reflections (using either the x-axis or y-axis as a mirror line)
  • Translations (moving the whole curve 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 recognise and apply them?

If you are given the graph of a function, say y = f(x) then:

• REFLECTIONS look like this:
  • In x-axis: y = -f(x) (minus outside the function)
  • In y-axis: y = f(-x) (minus inside the function)
• TRANSLATIONS:
  • + a units in the x direction: y = f(x - a)
  • + b units in the y direction: y = f(x) + b
• BEWARE the highlighted change of sign!

Tips
Translation by the vector (-3 2) means a translation of -3 in the x direction and +2 in the y direction.Applied to y = f(x) this would give y = f(x + 3) + 2
(Note: the change of sign in the x direction as pointed out in 2. above)

Example : The graph of y = f(x) is shown on the graph below.
On the same graph sketch y = f(x + 3) + 2 

This is a translation by - 3 in the x direction (ie 3 to the left, note the change of sign again) and + 2 in they direction (ie 2 up).
So we copy the given (slightly unusual) graph in its new position :