Modulus Functions (DP IB Analysis & Approaches (AA)): Revision Note

Modulus functions & graphs

What is the modulus function?

• The modulus function (or absolute value function) is defined by 

  • which gives the positive distance (or size) of a real number from zero

  • e.g.

    • 

    • 

    • 

• It can also be defined as

  • 

  • or

• Its largest domain is the set of all real values

  • and its range is the set of all real non-negative values

How do I sketch the modulus function?

• The graph of is the line for

  • and for

    • giving it a V-shape

    • with its vertex at the origin

• The function is continuous

  • but not differentiable at

    • as no gradient exists there

How do I sketch y = a|x + p| + q?

• The graph of is a transformation of the graph as follows:

  • First, apply a vertical stretch of scale factor to

    • 

  • Secondly, apply a translation of

    • 

• This transforms the vertex on to

  • the new vertex at on

• Note that

  • means a shape

    • the bigger the steeper the

  • means a shape

    • the more negative the the steeper the

How do I rearrange a graph into the form y = a|x + p| + q?

• Two useful modulus relations for rearranging are

  • 

  • 

• e.g. can be rearranged as follows:

  • 

    • using

  • 

    • by factorisation

  • 

    • using

  • and giving

    •