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IBMaths: AI HLDPRevision Notes2. Functions2.5 Transformations of Graphs2.5.1 Translations of Graphs

Translations of Graphs (DP IB Maths: AI HL)

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Dan

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Dan

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Translations of Graphs

What are translations of graphs?

  • When you alter a function in certain ways, the effects on the graph of the function can be described by geometrical transformations
  • For a translation:
    • the graph is moved (up or down, left or right) in the xy plane
      • Its position changes
    • the shape, size, and orientation of the graph remain unchanged
  • A particular translation (how far left/right, how far up/down) is specified by a translation vector stretchy left parenthesis table row x row y end table stretchy right parenthesis:
    • x is the horizontal displacement
      • Positive moves right
      • Negative moves left
    • y is the vertical displacement
      • Positive moves up
      • Negative moves down

Translations What Is

What effects do horizontal translations have on the graphs and functions?

  • A horizontal translation of the graph space y equals f left parenthesis x right parenthesis by the vector stretchy left parenthesis table row a row 0 end table stretchy right parenthesis is represented by 
    • space y equals f left parenthesis x minus a right parenthesis
  • The x-coordinates change
    • The value a is subtracted from them
  • The y-coordinates stay the same
  • The coordinates left parenthesis x comma space y right parenthesis become left parenthesis x plus a comma space y right parenthesis
  • Horizontal asymptotes stay the same
  • Vertical asymptotes change
    • x equals k becomes x equals k plus a

Translations statement_horiz_Illustration

What effects do vertical translations have on the graphs and functions?

  • A vertical translation of the graph space y equals f left parenthesis x right parenthesis by the vector stretchy left parenthesis table row 0 row b end table stretchy right parenthesis is represented by
    • space y minus b equals f left parenthesis x right parenthesis
    • This is often rearranged to space y equals f left parenthesis x right parenthesis plus b
  • The x-coordinates stay the same
  • The y-coordinates change
    • The value b is added to them
  • The coordinates left parenthesis x comma space y right parenthesis become left parenthesis x comma space y plus b right parenthesis
  • Horizontal asymptotes change
    • space y equals k becomes space y equals k plus b
  • Vertical asymptotes stay the same

Translations statement_vert_Illustration

Examiner Tip

  • To get full marks in an exam make sure you use correct mathematical terminology
    • For example: Translate by the vector open parentheses table row 2 row cell negative 4 end cell end table close parentheses

Worked example

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

we-image

a)
Sketch the graph of space y equals f left parenthesis x plus 3 right parenthesis.

2-5-1-ib-aa-sl-translate-graph-a-we-solution

b)
Sketch the graph of space y equals f left parenthesis x right parenthesis plus 3.

2-5-1-ib-aa-sl-translate-graph-b-we-solution

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    Dan

    Author: Dan

    Expertise: Maths

    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.