Modulus Functions - Sketching Graphs (CIE A Level Maths: Pure 3)

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Modulus Functions - Sketching Graphs

Modulus functions

  • The modulus function makes any ‘input’ positive
  • |x| = x   if x ≥ 0   |f(x)| = f(x)   if f(x) ≥ 0
  • |x| = -x if x < 0   |f(x)| = -f(x)   if f(x) < 0
    • For example: |5| = 5 and |-5| = 5
  • Sometimes called absolute value

How do I sketch the graph of the modulus function: y = a x + p| + q?

  • The graph will look like a “ꓦ” if a > 0 or a “ꓥ” if a < 0
  • There will be a vertex at the point (-p, q)
  • There could be 0, 1 or 2 roots
    • This depends on the location of the vertex and the orientation of the graph (ꓦ or ꓥ)
  • Compare this to the completed square form of a quadratic a(x + p)² + q

1-2-4-modulus-functions--sketching-graphs-notes-diagram-0

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

 STEP 1        Pencil in the graph of y = f(x)

STEP 2        Reflect anything below the x-axis, in the x-axis, to get y = |f(x)|

 Modulus Functions - Sketching Graphs Notes Diagram 2, A Level & AS Level Pure Maths Revision Notes

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

STEP 1          Sketch the graph of y = f(x) only for x ≥ 0

STEP 2          Reflect this in the y-axis

1-2-4-modulus-functions--sketching-graphs-notes-diagram-2-1-

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

  • There is a 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 y-axis
    • It can go below the y-axis

Modulus Functions - Sketching Graphs Notes Diagram 4, A Level & AS Level Pure Maths Revision Notes

  • For the exam you will only be asked to do this when f(x) is linear
    • Your graphs will all look like a “ꓦ” or a “ꓥ”
    • You can also think of these graphs as transformations of the graph y = |x|

Worked example

Modulus functions - Sketching Graphs Example Diagram, A Level & AS Level Pure Maths Revision Notes

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Paul

Author: Paul

Expertise: Maths

Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.