Absolute Value Functions: Definitions, Graphs & Exercises

Definitions, Properties, Equations, and Graphs

The absolute value function is explored through definitions, examples and exercises with solutions at the bottom of the page. The graphing of absolute value functions \( |f(x)| \) is discussed through interactive examples.

Definition of Absolute Value

Consider the number line and a point at a distance \( x \) from the origin zero of the number line.

The absolute value of \( x \) written as \( |x| \) is the distance from zero to \( x \). Since \( |x| \) gives a distance, it is always positive or equal to zero. Hence:

\( |a - b| \) is the distance between points at \( a \) and \( b \) on the number line. For example: \( |5 - 15| = |-10| = 10 \).


Formal Definition:

\[ |x| = \left\{ \begin{array}{ll} x & x\geq 0 \\ -x & x\lt 0 \\ \end{array} \right. \]

Which can be used to write:

\[ |x - a| = \left\{ \begin{array}{ll} x - a & x\geq a \\ -(x - a) = a - x & x\lt a \\ \end{array} \right. \]

Absolute Value and Square Root

\( \sqrt{x^2} = |x| \) ; because the square root is positive or equal zero.

\( \sqrt{(x - a)^2} = |x - a| \)

Properties of the Absolute Value Function

  1. \( |x| \ge 0 \)
  2. \( \sqrt{x^2} = |x| \)
  3. \( |x| = |-x| \)
  4. \( |ab| = |a||b| \)
  5. \( \left| \dfrac{a}{b} \right| = \dfrac{|a|}{|b|} \), for \( b \ne 0 \)

Equations and Inequalities with Absolute Value

For \( k \ge 0 \):

  1. \( |x| = k \iff x = k \text{ or } x = -k \)
  2. \( |x| \le k \iff -k \le x \le k \)
  3. \( |x| \ge k \iff x \le -k \text{ or } x \ge k \)

Graph of Absolute Value of a Function

Function \( f(x) \) used is a quadratic function of the form:

\[ f(x) = ax^2 + bx + c \]

The exploration is carried out by changing the parameters \( a \), \( b \) and \( c \) included in \( f(x) \) using the interactive panel below.

Interactive Tutorial

a =
-10 +10
b =
-10 +10
c =
-10 +10

Exploration Guide:

  1. Click on the button above "Draw" to start.
  2. Use the sliders to set parameter \(a\) to zero, parameter \(b\) to zero and parameter \(c\) to a positive value; \( f(x) \) is a constant function. Compare the graph of \( f(x) \) in blue and that of \( h(x)= |f(x)| \) in red. Change \( c \) to a negative value and compare the graphs again. Use the definition of the absolute value functions to explain how can the graph of \( |f(x)| \) be obtained from the graph of \( f(x) \).
  3. Keep the value of \( a \) equal to zero, select non zero values for \( b \) to obtain a linear function. How can the graph of \( h(x) = |f(x)| \) be obtained from that of \( f(x) \)?
    Hint: use the definition of the absolute value functions and reflection of a graph on the x-axis.
  4. Set \( b \) and \( c \) to zero and select a positive value for \( a \) to obtain a quadratic function. Why are the two graphs the same? (Hint: use the definition of the absolute value functions).
  5. Set \( b \) and \( c \) to zero and select a negative value for \( a \) to obtain a quadratic function. Why are the two graphs reflection of each other? (Hint: use the definition of the absolute value functions and reflection of a graph on the x-axis).
  6. Keep the values of \( a \) and \( b \) as in step 5 above and change gradually \( c \) from zero to some positive values. How can the graph of \( h(x)=|f(x)| \) be obtained from that of \( f(x) \)?
  7. Select different values for \( a, b \) and \( c \) and explore.

Exercises

Part 1: Evaluate the expressions

  1. \( | - 9 | \)
  2. \( | 9 | \)
  3. \( | 2 - 6 | \)
  4. \( |- \dfrac{5}{2}| \)
  5. \( | 11 - 4^2 | \)
  6. \( 2^{| 2 - 7|} \)
  7. \( |2-19| - |9 - 20| \)
View Solutions to Part 1
  1. \( | - 9 | = 9 \)
  2. \( | 9 | = 9 \)
  3. \( | 2 - 6 | = 4 \)
  4. \( |- \dfrac{5}{2}| = \dfrac{5}{2} \)
  5. \( | 11 - 4^2 | = 5 \)
  6. \( 2^{| 2 - 7|} = 32 \)
  7. \( |2-19| - |9 - 20| = 6 \)

Part 2: Substitute and Evaluate the expressions

  1. \( | x - 10 | + | 2 x | \) , for \( x = - 7 \)
  2. \( | - x | \) , for \( x = 8 \)
  3. \( \left |- \dfrac{|x - 2|}{|-x + 9|} \right | \) , for \( x = -20 \)
  4. \( | - 3 x - x^2 | \) , for \( x = -3 \)
  5. \( x^{| x - 7|} \) , for \( x = - 2 \)
View Solutions to Part 2
  1. For \( x = - 7 \), \( | x - 10 | + | 2 x | = 31 \)
  2. For \( x = 8 \), \( | - x | = 8 \)
  3. For \( x = - 20 \), \( \left |- \dfrac{|x - 2|}{|-x + 9|} \right | = \dfrac{22}{29} \)
  4. For \( x = - 3 \), \( | - 3 x - x^2 | = 0 \)
  5. For \( x = - 2 \), \( x^{| x - 7|} = - 512 \)

Part 3: Simplify the expressions

  1. \( | a - b | \) for \( a = b \)
  2. \( | a - b | \) for \( a > b \)
  3. \( | a - b | \) for \( b > a \)
  4. \( \sqrt {3^2} \)
  5. \( \sqrt {(- 3)^2} \)
  6. \( \sqrt { (12 - 20)^2 } \)
  7. \( | x^2 | \)
  8. \( | (-x)^2 | \)
  9. \( | (x + 1)^2 | \)
  10. \( \sqrt{(x - y + z)^2} \)
  11. \( \sqrt{\sin^2(x)} \)
View Solutions to Part 3
  1. \( | a - b | = 0 \)
  2. \( | a - b | = a - b \)
  3. \( | a - b | = b - a \)
  4. \( \sqrt {3^2} = |3| = 3 \)
  5. \( \sqrt {(- 3)^2} = |-3| = 3 \)
  6. \( \sqrt { (12 - 20)^2 } = 8 \)
  7. \( | x^2 | = x^2 \)
  8. \( | (-x)^2 | = x^2 \)
  9. \( | (x + 1)^2 | = (x + 1)^2 \)
  10. \( \sqrt{(x - y + z)^2} = |x - y + z| \)
  11. \( \sqrt{\sin^2(x)} = |\sin(x)| \)

Part 4 & 5: Solve the Equations and Inequalities

  1. \( |x - 2| = 4 \)
  2. \( |x + 5| = 0 \)
  3. \( |10 - x| = - 5 \)
  4. \( |x - 7| \gt 4 \)
  5. \( |x + 5| \lt 9 \)
  6. \( |x + 8| \lt - 5 \)
  7. \( |x + 8| \gt - 2 \)
View Solutions to Equations & Inequalities
  1. \( |x - 2| = 4 \) , two solutions: \( x = - 2 \) and \( x = 6 \)
  2. \( |x + 5| = 0 \) , one solution: \( x = - 5 \)
  3. \( |10 - x| = - 5 \) , no real solutions.
  4. \( |x - 7| \gt 4 \) , solution set: \( x \lt 3 \cup x \gt 11 \)
  5. \( |x + 5| \lt 9 \) , solution set: \( -14 \lt x \lt 4 \)
  6. \( |x + 8| \lt - 5 \) , solution set: \( \{\emptyset\} \)
  7. \( |x + 8| \gt - 2 \) , solution set: {all real numbers}

Part 6: Graphs

Sketch the following pairs of functions in the same system of axes and explains the similarities and differences of the two graphs.

  1. \( f(x) = x - 1 \quad , \quad h(x) = |f(x)| \)
  2. \( f(x) = x^2 - 4 \quad , \quad h(x) = |f(x)| \)
  3. \( f(x) = \sin(x) \quad , \quad h(x) = |\sin(x)| \)
View Solutions to Graphs

  1. \( f(x) = x - 1 \quad , \quad h(x) = |f(x)| = |x - 1| \)

    For \( x - 1 \ge 0 \) or \( x \ge 1 \), \( h(x) = | x - 1 | = x - 1 \) and we therefore have \( h(x) = f(x) \) which we can be clearly seen from the graphs.

    For \( x - 1 \lt 0 \) or \( x \lt 1 \) , \( h(x) = |x - 1| = - (x - 1) = - f(x) \).

    Therefore for \( x \lt 1 \), \( h(x) = - f(x) \) which means that the graph of \( h \) is a reflection of the graph of \( f \) on the x axis and this is can be clearly seen on the graphs.

    graph of f(x) = x - 1 and h(x) = |x - 1|

  2. \( f(x) = x^2 - 4 \quad , \quad h(x) = |f(x)| = |x^2 - 4| \)

    For \( x^2 - 4 \le 0 \) or \( -2 \le x \le 2 \), \( h(x) = |(x^2 - 4)| = - (x^2 - 4) = - f(x) \). Therefore, on the interval \( -2 \le x \le 2 \), the graph of \( h \) is a reflection of the graph of \( f \). Outside this interval, the two graphs coincide.

    graph of f(x) = x^2 - 4 and h(x) = |x^2 - 4|

  3. \( f(x) = \sin(x) \quad , \quad h(x) = |\sin(x)| \)

    Over intervals \( -2\pi \) to \( -\pi \), \( 0 \) to \( \pi \), \( 2\pi \) to \( 3\pi \), ... \( \sin(x) \ge 0 \) and \( h(x) = \sin(x) = f(x) \) therefore the graphs of \( h \) and \( f \) coincide.

    For intervals \( -\pi \) to \( 0 \), \( \pi \) to \( 2\pi \) ... \( \sin(x) \lt 0 \), \( h(x) = | \sin(x) | = - \sin(x) \) and the graph of \( h \) is a reflection of the graph of \( f \) on the x axis.

    graph of f(x) = sin(x) and h(x) = |sin(x)|

Conclusion: In all three exercises above, the graph of \( h \) is on or above the x axis because \( h(x) = |f(x)| \) and the absolute value is positive or equal to 0.

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