Graph Transformations

Graph Transformations

A LevelAQAEdexcelOCRAQA 2022Edexcel 2022OCR 2022

Graph Transformations

You should have seen some graph transformations before, such as translations and reflections – recall that reflections in the x-axis flip f(x) vertically and reflections in the y-axis flip f(x) horizontally. Here, we will also look at stretches.

There are 4 main types of graph transformation that we will cover. Each transformation has the same effect on all functions.

Make sure you are happy with the following topics before continuing.

A Level AQA Edexcel OCR

Type 1: y = f(x+k)

 

For the transformation y=f(x+k), for k>0:

  • f(x+k) is f(x) moved k to the left
  • f(x-k) is f(x) moved k to the right

 

In this example, we have f(x) = x^2 - 4 and y=f(x+2)

So, subtract 2 from the x-coordinates of f(x) to get y=f(x+2)

A LevelAQAEdexcelOCR

Type 2: y = f(x)+k

 

For the transformation y=f(x)+k, for k>0:

  • f(x)+k is f(x) moved k upwards
  • f(x)-k is f(x) moved k downwards

 

In this example, we have f(x) = x^2 - 4 and y=f(x)+3

So, add 3 to the y-coordinates of f(x) to get y=f(x)+3

A LevelAQAEdexcelOCR

Type 3: y = af(x)

 

For the transformation y=af(x):

  • If |a|>1 then af(x) is f(x) stretched vertically by a factor of a
  • If 0<|a|<1 then f(x) is squashed vertically
  • If a<0 then is f(x) also reflected in the x-axis

 

In this example, we have f(x) = x^2 - 4 and y=2f(x)

This is a stretch vertically, so multiply the y-coordinates of f(x) by 2 to get y=2f(x)

A LevelAQAEdexcelOCR

Type 4: y = f(ax)

 

For the transformation y=f(ax):

  • If |a|>1 then f(ax) is f(x) squashed horizontally by a factor of a
  • If 0<|a|<1 then f(x) is stretched horizontally
  • If a<0 then is f(x) also reflected in the y-axis

 

In this example, we have f(x) = x^2 - 4 and y=f(2x)

This is a squash horizontally, so divide the x-coordinates of f(x) by 2 (or multiply by \dfrac{1}{2}) to get y=f(2x)

A LevelAQAEdexcelOCR

Note:

  • For these transformations, any asymptotes need to be moved correspondingly.
  • A squash by a factor of a is equivalent to a stretch by a factor of \dfrac{1}{a}
  • When drawing graph transformation, only a sketch including important points is necessary.

Combinations of Transformations

For combinations of transformations, it is easy to break them up and do them one step at a time (do the bit in the brackets first). You can sketch the graph at each step to help you visualise the whole transformation.

e.g. for f(x) = x^2 - 4 and y=2f(x+2), draw the graph of
y=f(x+2) first, and then use this graph to draw the graph of
y=2f(x+2)

 

Note: These transformations can also be combined with modulus functions.

A LevelAQAEdexcelOCR

Example Questions

y = f(4x) means that the graph of f(x) is squashed horizontally by a factor of 4.

Hence, the graph will look like:

 

Firstly, since the coefficient before f(x) is negative, we need to reflect f(x) in the x-axis.

The coefficient of \dfrac{1}{2} before -f(x) means that the graph of -f(x) is squashed vertically by a factor of 2.

Hence, the graph will look like:

 

 

Split the transformation up into 2 parts – firstly sketch y=3f(x) which is a stretch vertically by a scale factor of 3 (multiply the y-coordinates by 3:

 

 

Then, do the second transformation – y=3f(x)-1 means that we need to move the graph down by 1 (subtract 1 from the y-coordinates):

 

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