Edexcel IGCSE Maths 复习笔记 3.8.1 Transformations of Graphs

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

• 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