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 k is positive then the graph moves left

  • If k is negative then the graph moves right

• is vertical translation by vector 

  • If k is positive then the graph moves up

  • If k is negative then the graph moves down

• is a horizontal stretch by scale factor  centred about the y-axis

  • If k > 1 then the graph gets closer to the y-axis

  • If 0 < k < 1 then the graph gets further from the y-axis

• is a vertical stretch by scale factor  centred about the x-axis

  • If k > 1 then the graph gets further from the x-axis

  • If 0 < k < 1 then the graph gets closer to the x-axis

• is a horizontal reflection about the y-axis

  • A horizontal reflection can be viewed as a special case of a horizontal stretch

• is a vertical reflection about the x-axis

  • A vertical reflection can be viewed as a special case of a vertical stretch

How do horizontal and vertical transformations affect each other?

• Horizontal and vertical transformations are independent of each other

  • The horizontal transformations involved will need to be applied in their correct order

  • The vertical transformations involved will need to be applied in their correct order

• Suppose there are two horizontal transformation H₁ then H₂ and two vertical transformations V₁ then V₂ then they can be applied in the following orders:

  •  Horizontal then vertical:

    • H₁ H₂ V₁ V₂

  • Vertical then horizontal:

    • V₁ V₂ H₁ H₂

  • Mixed up (provided that H₁ comes before H₂ and V₁ comes before V2):

    • H₁ V₁ H₂ V₂

    • H₁ V₁ V₂ H₂

    • V₁ H₁ V₂ H₂

    • V₁ H₁ H₂ V₂

Composite Vertical Transformations af(x)+b

How do I deal with multiple vertical transformations?

• Order matters when you have more than one vertical transformations

• If you are asked to find the equation then build up the equation by looking at the transformations in order

  • A vertical stretch by scale factor a followed by a translation of  

    • Stretch: 

    • Then translation: 

    • Final equation: 

  • A translation of followed by a vertical stretch by scale factor a

    • Translation: 

    • Then stretch:

    • Final equation: 

• If you are asked to determine the order

  • The order of vertical transformations follows the order of operations

  • First write the equation in the form 

    • First stretch vertically by scale factor a

    • If a is negative then the reflection and stretch can be done in any order

    • Then translate by 

Composite Horizontal Transformations f(ax+b)

How do I deal with multiple horizontal transformations?

• Order matters when you have more than one horizontal transformations

• If you are asked to find the equation then build up the equation by looking at the transformations in order

  • A horizontal stretch by scale factor followed by a translation of 

    • Stretch: 

    • Then translation: 

    • Final equation: 

  • A translation of followed by a horizontal stretch by scale factor 

    • Translation: 

    • Then stretch: 

    • Final equation: 

• If you are asked to determine the order

  • First write the equation in the form 

  • The order of horizontal transformations is the reverse of the order of operations

    • First translate by 

    • Then stretch by scale factor 

    • If a is negative then the reflection and stretch can be done in any order