Composite Transformations of Graphs (DP IB Applications & Interpretation (AI)) : Revision Note

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Composite Transformations of Graphs

What transformations do I need to know?

  • space y equals f left parenthesis x plus k right parenthesis is horizontal translation by vector stretchy left parenthesis table row cell negative k end cell row 0 end table stretchy right parenthesis

    • If k is positive then the graph moves left

    • If k is negative then the graph moves right

  • space y equals f left parenthesis x right parenthesis plus k is vertical translation by vector stretchy left parenthesis table row 0 row k end table stretchy right parenthesis

    • If k is positive then the graph moves up

    • If k is negative then the graph moves down

  • space y equals f left parenthesis k x right parenthesis is a horizontal stretch by scale factor 1 over k 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

  • space y equals k f left parenthesis x right parenthesis is a vertical stretch by scale factor k 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

  • space y equals f left parenthesis negative x right parenthesis is a horizontal reflection about the y-axis

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

  • space y equals negative f left parenthesis x right parenthesis 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 Vthen V2 then they can be applied in the following orders:

    •  Horizontal then vertical:

      • H1 H2 VV2

    • Vertical then horizontal:

      • VVH1 H2

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

      • H1 VH2 V2

      • H1 V1 V2 H2

      • V1 HVH2

      • VH1 HV2

Examiner Tips and Tricks

  • In an exam you are more likely to get the correct solution if you deal with one transformation at a time and sketch the graph after each transformation

Worked Example

The diagram below shows the graph of space y equals f left parenthesis x right parenthesis.

we-image

Sketch the graph of space y equals 1 half f stretchy left parenthesis x over 2 stretchy right parenthesis.

2-5-4-ib-aa-sl-comp-transformation-a-we-solution

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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 stretchy left parenthesis table row 0 row b end table stretchy right parenthesis 

      • Stretch: space y equals a f left parenthesis x right parenthesis

      • Then translation: space y equals stretchy left square bracket a f left parenthesis x stretchy right parenthesis stretchy right square bracket plus b

      • Final equation: space y equals a f left parenthesis x right parenthesis plus b

    • A translation of stretchy left parenthesis table row 0 row b end table stretchy right parenthesis followed by a vertical stretch by scale factor a

      • Translation: space y equals f left parenthesis x right parenthesis plus b

      • Then stretch: space y equals a stretchy left square bracket f left parenthesis x right parenthesis plus b stretchy right square bracket

      • Final equation: space y equals a f left parenthesis x right parenthesis plus a b

  • 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 space y equals a f left parenthesis x right parenthesis plus b

      • 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 stretchy left parenthesis table row 0 row b end table stretchy right parenthesis

Worked Example

The diagram below shows the graph of space y equals f left parenthesis x right parenthesis.

we-image

Sketch the graph of space y equals 3 f left parenthesis x right parenthesis minus 2.

2-5-4-ib-aa-sl-comp-transformation-b-we-solution

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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 1 over a followed by a translation of open parentheses table row cell negative b end cell row 0 end table close parentheses

      • Stretch: space y equals f left parenthesis a x right parenthesis

      • Then translation: space y equals f left parenthesis a left parenthesis x plus b right parenthesis right parenthesis

      • Final equation: space y equals f left parenthesis a x plus a b right parenthesis

    • A translation of open parentheses table row cell negative b end cell row 0 end table close parentheses followed by a horizontal stretch by scale factor 1 over a

      • Translation: space y equals f left parenthesis x plus b right parenthesis

      • Then stretch: space y equals f open parentheses open parentheses a x close parentheses plus b close parentheses

      • Final equation: space y equals f open parentheses a x plus b close parentheses

  • If you are asked to determine the order

    • First write the equation in the form space y equals f left parenthesis a x plus b right parenthesis

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

      • First translate by open parentheses table row cell negative b end cell row 0 end table close parentheses

      • Then stretch by scale factor 1 over a

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

Worked Example

The diagram below shows the graph of space y equals f left parenthesis x right parenthesis.

we-image

Sketch the graph of space y equals f left parenthesis 2 x minus 1 right parenthesis.

2-6-4-ib-aa--ai-hl-comp-horizontal-trans-we-solution
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Dan Finlay

Author: Dan Finlay

Expertise: Maths Lead

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.

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