Combinations of Transformations

When you alter a function in certain ways, the effects on the graph of the function can be described by geometrical transformations
In additional to single transformations, you need to be able to interpret the effects of multiple transformations applied to the same function

Combinations What Is, A Level & AS Level Pure Maths Revision Notes

Make sure you understand the effects of individual translations, stretches, and reflections on the graph of a function (see the previous pages)
When applying combinations of these transformations, apply them to the graph one at a time according to the following guidelines:
- First apply any horizontal transformations inside the brackets (if you have more than one transformation inside the brackets, translations must be applied before stretches and reflections)
  - y = kf(ax + b)+ c
- Then apply any vertical transformations outside the brackets (if you have more than one transformation outside the brackets, stretches and reflections must be applied before translations)
  - y = kf(ax + b)+ c

Comb Xforms Illustration inside, A Level & AS Level Pure Maths Revision Notes

Comb Xforms Illustration outside, A Level & AS Level Pure Maths Revision Notes

Any asymptotes of the function are also affected by the combined transformation (perform the transformations one at a time in the same order as above)

Be sure to apply transformations in the correct order – applying them in the wrong order can produce an incorrect transformation.
When you sketch a transformed graph, indicate the new coordinates of any points that are marked on the original graph.
Try to indicate the coordinates of points where the transformed graph intersects the coordinate axes (although if you don't have the equation of the original function this may not be possible).
If the graph has asymptotes, don't forget to sketch the asymptotes of the transformed graph as well.

Comb Xforms Example, A Level & AS Level Pure Maths Revision Notes

Combinations of Transformations (AQA A Level Maths: Pure)