Composite Transformations of Graphs (DP IB Analysis & Approaches (AA)): Revision Note

Composite transformations of graphs

What transformations do I need to know?

• 

  • is horizontal translation by vector 

    • If is positive then the graph moves right

    • If is negative then the graph moves left

• 

  • is horizontal translation by vector 

    • If is positive then the graph moves left

    • If is negative then the graph moves right

• 

  • is vertical translation by vector 

    • If is positive then the graph moves up

    • If is negative then the graph moves down

• 

  • is a horizontal stretch by scale factor  parallel to the -axis

    • If then the graph gets closer to the -axis

    • If then the graph gets further from the -axis

• 

  • is a vertical stretch by scale factor  parallel to the -axis

    • If then the graph gets further from the -axis

    • If then the graph gets closer to the -axis

• 

  • is a horizontal reflection about the -axis

• 

  • is a vertical reflection about the -axis

Does the order of transformations matter?

• The order of applying transformations does matter

  • In general

    • different horizontal transformations need to be applied in order

    • different vertical transformations need to be applied in order

  • but horizontal and vertical transformations can swap orders

    • They are independent of each other

• e.g. if there are two horizontal transformations H₁ then H₂ and two vertical transformations V₁ then V₂

  • then the following orders are all acceptable

    •  Horizontal then vertical: H₁ H₂ V₁ V₂

    • Vertical then horizontal: V₁ V₂ H₁ H₂

    • Mixed up (but H₁ before H₂ and V₁ before V₂):

    • H₁ V₁ H₂ V₂

    • H₁ V₁ V₂ H₂

    • V₁ H₁ V₂ H₂

    • V₁ H₁ H₂ V₂

Composite vertical transformations af(x)+b

What does the transformation af(x)+b represent?

• The transformation represents, in order:

  • a vertical stretch of by scale factor

    • 

  • followed by a translation of  

    • 

    • giving

• It is not a translation of followed by a vertical stretch by scale factor

  • because that would give

    • translation: 

    • then stretch:

    • final equation:

• If is negative, then there is also a reflection in the -axis

  • either before or after the vertical stretch

Composite horizontal transformations f(ax+b)

What does the transformation f(ax+b) represent?

• The transformation represents, in order:

  • a translation of by

    • 

  • followed by a horizontal stretch by scale factor 

    • 

    • giving 

• It is not a horizontal stretch by scale factor followed by a translation of 

  • because that would give

    • stretch: 

    • then translation: 

    • final equation: 

• If is negative, then there is also a reflection in the -axis

  • either before or after the horizontal stretch