IBMaths: AA HLDPRevision Notes2. Functions2.9 Further Functions & Graphs2.9.2 Modulus Transformations

Modulus Transformations (DP IB Maths: AA HL)

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Modulus Transformations

How do I sketch the graph of the modulus of a function: y = |f(x)|?

  • STEP 1: Keep the parts of the graph of y = f(x) that are on or above the x-axis
  • STEP 2: Any parts of the graph below the x-axis get reflected in the x-axis

How do I sketch the graph of a function of a modulus: y = f(|x|)?

  • STEP 1: Keep the graph of y = f(x) only for x ≥ 0
  • STEP 2: Reflect this in the y-axis

What is the difference between y = |f(x)| and y = f(|x|)?

  • The graph of y = |f(x)| never goes below the x-axis
    • It does not have to have any lines of symmetry
  • The graph of y = f(|x|) is always symmetrical about the x-axis
    • It can go below the x-axis

When multiple transformations are involved how do I determine the order?

  • The transformations outside the function follow the same order as the order of operations
    • y equals open vertical bar a f left parenthesis x right parenthesis plus b close vertical bar
      • Deal with the a then the b then the modulus
    • y equals a open vertical bar f left parenthesis x right parenthesis close vertical bar plus b
      • Deal with the modulus then the a then the b
  • The transformations inside the function are in the reverse order to the order of operations
    • y equals f left parenthesis open vertical bar a x plus b close vertical bar right parenthesis 
      • Deal with the modulus then the b then the a
    • y equals f left parenthesis a open vertical bar x close vertical bar plus b right parenthesis
      • Deal with the b then the a then the modulus

Exam Tip

  • When sketching one of these transformations in an exam question make sure that the graphs do not look smooth at the points where the original graph have been reflected
    • For y equals open vertical bar f left parenthesis x right parenthesis close vertical bar the graph should look "sharp" at the points where it has been reflected on the x-axis
    • For y equals f open parentheses open vertical bar x close vertical bar close parentheses the graph should look "sharp" at the point where it has been reflected on the y-axis

Worked example

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

2-9-1-we-diagram

(a) Sketch the graph of space y equals open vertical bar f open parentheses x close parentheses close vertical bar.

2-9-1-ib-aa-hl-modulus-trans-a-we-solution

(b) Sketch the graph of space y equals f left parenthesis open vertical bar x close vertical bar right parenthesis  .

2-9-1-ib-aa-hl-modulus-trans-b-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.