IBMaths: AI HLDPRevision Notes2. Functions2.5 Transformations of Graphs2.5.4 Composite Transformations of Graphs

Composite Transformations of Graphs (DP IB Maths: AI HL)

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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 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

Exam Tip

  • 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-imageSketch 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

Author: Dan

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.