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

Modulus Functions & Graphs

What is the modulus function?

• The modulus function is defined by 

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  • Equivalently it can be defined 

• Its domain is the set of all real values

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

• The modulus function gives the distance between 0 and x

  • This is also called the absolute value of x

What are the key features of the modulus graph: y = |x|?

• The graph has a y-intercept at (0, 0)

• The graph has one root at (0, 0)

• The graph has a vertex at (0, 0)

• The graph is symmetrical about the y-axis

• At the origin

  • The function is continuous

  • The function is not differentiable

What are the key features of the modulus graph: y = a|x + p| + q?

• Every modulus graph which is formed by linear transformations can be written in this form using key features of the modulus function

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    • For example: 

  • 

    • For example: 

• The graph has a y-intercept when x = 0

• The graph can have 0, 1 or 2 roots

  • If a and q have the same sign then there will be 0 roots

  • If q = 0 then there will be 1 root at (-p, 0)

  • If a and q have different signs then there will be 2 roots at 

• The graph has a vertex at (-p, q)

• The graph is symmetrical about the line x = -p

• The value of a determines the shape and the steepness of the graph

  • If a is positive the graph looks like 

  • If a is negative the graph looks like 

  • The larger the value of |a| the steeper the lines

• At the vertex

  • The function is continuous

  • The function is not differentiable