What are graph transformations?

• You should have come across?transformations?of shapes –?reflections,?rotations,?translations?and?enlargements
• These are often performed on 2D coordinate grids (and can be applied in 3D too)
• Therefore it is possible to?transform?graphs?as they are also drawn on 2D coordinate grids

• Graph transformations are described differently to transforming shapes
• In general, a graph can be considered as having the equation?y = f(x)?...
  • ... where?f(x)?will be some expression involving?x

   

• At IGCSE?graph?transformations?cover:
  • linear?functions?f(x) = mx + c
  • quadratic?functions?f(x) = ax²?+ bx +c
  • f(x) = sin x
  • f(x) = cos x

   

• In fact many exam questions do not state the actual function!
  • A graph is provided with it being referred to just as?y = f(x)
  • It will be impossible to tell what?f(x)?is from the graph
  • But some key points will be highlighted

   

Graph Translations f(x+a) and f(x)+a

• f(x + a)?translates?the graph?-a?units in the?positive?x?direction (‘right’)
  • f(x – a)?translates?the graph?a?units in the?negative?x?direction (‘left’)
  • Notice the?opposite?signs!

   

• f(x) + a?translates?the graph?a?units in the?positive?y?direction (‘up’)
  • f(x) – a?translates?the graph?a?units in the?negative?y?direction (‘down’)

   

• There is a logic to these but the easiest way is to memorise and recognise them
• With?y = f(x) ± a?the “±a” happens?after?the function
  • So the ‘output’?of the function changesie.?y-coordinates change

   

• With?y = f(x ± a)?the?“±a” happens?before?the function
  • So the ‘input’?of the function changesie.?x-coordinates change

   

• These second translations may seem as though they are in the wrong direction
  • In the case of “+a” the ‘input’?will be?greaterso the graph happens “sooner”

    ie. graph moves?left

  • In the case of “-a” the ‘input’?will be?smaller 

    so the graph happens “later”ie. graph moves?right

   

Graph Stretches f(ax) and af(x)

• f(ax)?is a?horizontal?stretch?by a scale factor of?1/a
  • May also be referred to as a?stretch?in (or parallel to) the?x-direction
  • This looks like a “squash” but that is not a technical term – do not use it!

   

• af(x)?is a?vertical stretch?by a scale factor of?a
  • May also be referred to as a?stretch?in (or parallel to) the?y-direction

   

• There is a logic to these as well but memory and recognition is easier!
• With?y = af(x)?the “a” happens?after?the function
  • So the ‘output’?of the function changesie.?y-coordinates change

   

• With?y = f(ax)?the?“a” happens?before?the function
  • So the ‘input’?of the function changesie.?x-coordinates change

   

• This stretch may seem it has the wrong scale factor (1/a)
  • However the ‘input’?is being?multiplied?by?a?– so?x?will need to be?divided?by?a?in order for the function to receive the?same?input

   

Special cases – graph reflections f(-x) and -f(x)

• f(-x)?and?-f(x)?are of the form of?stretches?(a = -1) but lead to special cases that are easiest dealt with separately
• f(-x)?is a?reflection?in the?y-axis
• -f(x)?is a?reflection?in the?x-axis

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• Yes, there is a logic to these too but as before it is easier to memorise them
• With?y = -f(x)?the “-” happens?after?the function
  • So the ‘output’?of the function changes ie??y-coordinates change sign
  • y-coordinates changing sign means a?reflection?in the?x-axis 

    (Think about it or draw a diagram if you’re not yet convinced!)

   

• With?y = f(-x)?the?“a” happens?before?the function
  • So the ‘input’?of the function changes ie??x-coordinates change sign
  • x-coordinates changing sign means a?reflection?in the?y-axis(Go on, draw a diagram!)

   

Recognising graph transformations?

• So far all examples have involved sketching a graph for a given transformation
• However some questions ask for a description of the transformation having provided both the original and transformed graphs