Graph, Domain and Range of Absolute Value Functions

This is a step by step tutorial on how to graph functions with absolute value. Properties of the graph of these functions such as domain, range, x and y intercepts are also discussed. Free graph paper is available.


Example 1: f is a function given by

f (x) = |x - 2|

  1. Find the x and y intercepts of the graph of f.
  2. Find the domain and range of f.
  3. Sketch the graph of f.

Solution to Example 1

  • a - The y intercept is given by
    (0 , f(0)) = (0 ,|-2|) = (0 , 2)
  • The x coordinate of the x intercepts is equal to the solution of the equation
    |x - 2| = 0
    which is
    x = 2
  • The x intercepts is at the point (2 , 0)
  • b - The domain of f is the set of all real numbers
    Since |x - 2| is either positive or zero for x = 2; the range of f is given by the interval [0 , +infinity).
  • c - To sketch the graph of f(x) = |x - 2|, we first sketch the graph of y = x - 2 and then take the absolute value of y.
    The graph of y = x - 2 is a line with x intercept (2 , 0) and y intercept (0 , -2). (see graph below)
    graph of y = x - 2

  • We next use the definition of the absolute value to graph f(x) = |x - 2| = | y |.
    If y >= 0 then | y | = y , if y <0 then | y | = -y.
  • For values of x for which y is positive, the graph of | y | is the same as that of y = x - 2. For values of x for which y is negative, the graph of | y | is a reflection on the x axis of the graph of y. The graph of y = x - 2 above has y negative on the interval (-infinity , 2) and it is this part of the graph that has to be reflected on the x axis. (see graph below).
    graph of f(x) = |x - 2|

  • Check that the range is given by the interval [0 , +infinity), the domain is the set of all real numbers, the y intercept is at (0 , 2) and the x intercept at (2, 0).

Example 2: f is a function given by

f (x) = |(x - 2)2 - 4|

  1. Find the x and y intercepts of the graph of f.
  2. Find the domain and range of f.
  3. Sketch the graph of f.

Solution to Example 2

  • a - The y intercept is given by
    (0 , f(0)) = (0 ,(-2)2 - 4) = (0 , 0)
  • The x coordinates of the x intercepts are equal to the solutions of the equation
    |(x - 2)2 - 4| = 0
    which is solved
    (x - 2)2 = 4
    Which gives the solutions
    x = 0 and x = 4
  • The x intercepts is at the point (0 , 0) and (4 , 0)
  • b - The domain of f is the set of all real numbers
    Since |(x - 2)2 - 4| is either positive or zero for x = 4 and x = 0; the range of f is given by the interval [0 , +infinity).
  • c - To sketch the graph of f(x) = |(x - 2)2 - 4|, we first sketch the graph of y = (x - 2)2 - 4 and then take the absolute value of y.
    The graph of y = (x - 2)2 - 4 is a parabola with vertex at (2,-4), x intercepts (0 , 0) and (4 , 0) and a y intercept (0 , 0). (see graph below)
    graph of y = (x - 2)<sup>2</sup> - 4

  • The graph of f is given by reflecting on the x axis part of the graph of y = (x - 2)2 - 4 for which y is negative. (see graph below).
    graph of y = |(x - 2)<sup>2</sup> - 4|

More references and links to graphing, graphs and absolute value functions.
Graphing Functions
Graphs of Basic Functions.
Absolute Value Functions.
Definition of the Absolute Value.
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