IBMaths: AA HLDPRevision Notes2. Functions2.9 Further Functions & Graphs2.9.1 Modulus Functions

Modulus Functions (DP IB Maths: AA HL)

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Modulus Functions & Graphs

What is the modulus function?

  • The modulus function is defined by f left parenthesis x right parenthesis equals open vertical bar x close vertical bar
    • open vertical bar x close vertical bar equals square root of x squared end root
    • Equivalently it can be defined open vertical bar x close vertical bar equals open curly brackets table row x cell x greater or equal than 0 end cell row cell negative x end cell cell x less than 0 end cell end table close
  • 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

2-4-2-ib-aa-hl-modulus-function

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
    • open vertical bar a x close vertical bar equals open vertical bar a close vertical bar open vertical bar x close vertical bar
      • For example: open vertical bar 2 x plus 1 close vertical bar equals 2 open vertical bar x plus 1 half close vertical bar
    • open vertical bar p minus x close vertical bar equals open vertical bar x minus p close vertical bar
      • For example: open vertical bar 4 minus x close vertical bar equals open vertical bar x minus 4 close vertical bar
  • 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 open parentheses negative p plus-or-minus q over a comma 0 close parentheses
  • 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 logical or
    • If a is negative the graph looks like logical and
    • The larger the value of |a| the steeper the lines
  • At the vertex
    • The function is continuous
    • The function is not differentiable
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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.