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 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
| -5 | = 5
| 5 | = 5
| 0 | = 0
|a - b| is the distance between points at a and b on the number line.
|5 - 15| = | - 10| = 10
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.
\( \sqrt {(x - a)^2} = |x - a| \)
\( | x | ? k \iff x \le - k \text{ or } x \ge k \)
Graph of Absolute Value of a Function
Function f(x) used is a quadratic function of the form
\[ f(x) = a x^2 + b x + c \]
The exploration is carried out by changing the parameters a, b and c included in \( f(x)\) above.
Interactive Tutorial
click on the button above "draw" to start.
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 \( f(x)\).
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.
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).
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).
Keep the values of a and b as in 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 \( f(x)\)?
Select different values for a, b and c and explore.
\( f(x) = x - 1 , h(x) = |f(x)| = |x - 1| \)
For x - 1 ? 0 or x ? 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 < 0 or x < 1 , h(x) = |x - 1| = - (x - 1) = - f(x).
Therefore for x < 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.
\( f(x) = x^2 - 4 , h(x) = |f(x)| = |x^2 - 4| \)
For x2 - 4 ? 0 or -2 ? x ? 2 , h(x) = |(x2 - 4)| = - (x2 - 4) = - f(x) . Therefore, on the interval -2 ? x ? 2, the graph of h is a reflection of the graph of f. Outside this interval, the two graphs coincide.
\( f(x) = \sin(x) , h(x) = |\sin(x)| \)
Over intervals -2? to -? , 0 to ? , 2? to 3?, ... sin(x) ? 0 and h(x) = sin(x) = f(x) therefore the graphs of h and f coincide.
For intervals - ? to 0 , ? to 2? ... sin(x) < 0 , h(x) = | sin(x) | = - sin(x) and the graph of h is a reflection of the graph of f on the x axis.
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.