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IBMaths: AI HLDPRevision Notes2. Functions2.5 Transformations of Graphs2.5.3 Stretches of Graphs

Stretches of Graphs (DP IB Maths: AI HL)

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Stretches of Graphs

What are stretches 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 stretch:
    • the graph is stretched about one of the coordinate axes by a scale factor
      • Its size changes
    • the orientation of the graph remains unchanged
  • A particular stretch is specified by a coordinate axis and a scale factor:
    • The distance between a point on the graph and the specified coordinate axis is multiplied by the constant scale factor
    • The graph is stretched in the direction which is parallel to the other coordinate axis
    • For scale factors bigger than 1
      • the points on the graph get further away from the specified coordinate axis
    • For scale factors between 0 and 1
      • the points on the graph get closer to the specified coordinate axis
      • This is also sometimes called a compression but in your exam you must use the term stretch with the appropriate scale factor

Stretches What Is

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

  • A horizontal stretch of the graph space y equals f left parenthesis x right parenthesis by a scale factor q centred about the y-axis is represented by 
    • space y equals f stretchy left parenthesis x over q stretchy right parenthesis
  • The x-coordinates change
    • They are divided by q
  • The y-coordinates stay the same
  • The coordinates left parenthesis x comma space y right parenthesis become stretchy left parenthesis q x comma space y stretchy right parenthesis
  • Horizontal asymptotes stay the same
  • Vertical asymptotes change
    • x equals k becomes x equals q k

Stretches statement_horiz_Illustration

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

  • A vertical stretch of the graph space y equals f left parenthesis x right parenthesis by a scale factor p centred about the x-axis is represented by
    • space y over p equals f left parenthesis x right parenthesis
    • This is often rearranged to space y equals p f left parenthesis x right parenthesis
  • The x-coordinates stay the same
  • The y-coordinates change
    • They are multiplied by p
  • The coordinates left parenthesis x comma space y right parenthesis become left parenthesis x comma space p y right parenthesis
  • Horizontal asymptotes change
    • space y equals k becomes space y equals p k
  • Vertical asymptotes stay the same

Stretches statement_vert_Illustration

Examiner Tip

  • To get full marks in an exam make sure you use correct mathematical terminology
    • For example: Stretch vertically by scale factor ½
    • Do not use the word "compress" in your exam

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 2 f left parenthesis x right parenthesis.

2-5-3-ib-aa-sl-stretch-graph-a-we-solution

b)
Sketch the graph of space y equals f left parenthesis 2 x right parenthesis.

2-5-3-ib-aa-sl-stretch-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.