Modulus Functions | DP IB Maths: AA HL Revision Notes 2021

Modulus Functions & Graphs

The modulus function is defined by $f (x) = |x|$
- $|x| = \sqrt{x^{2}}$
- Equivalently it can be defined $|x| = \{\begin{cases} x & x \geq 0 \\ - x & x < 0 \end{cases}$
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

Every modulus graph which is formed by linear transformations can be written in this form using key features of the modulus function
- $|a x| = |a| |x|$
  - For example: $|2 x + 1| = 2 |x + \frac{1}{2}|$
- $|p - x| = |x - p|$
  - For example: $|4 - x| = |x - 4|$
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 $(- p \pm \frac{q}{a}, 0)$
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 $\lor$
- If a is negative the graph looks like $\land$
- The larger the value of |a| the steeper the lines
At the vertex
- The function is continuous
- The function is not differentiable