# Absolute Value Functions

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

## Properties of the Absolute Value Function

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

## 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. $$| x | ≥ k \iff x \le - k \text{ or } x \ge k$$

## 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

 a = 1 -10+10 b = 0 -10+10 c = 1 -10+10
>

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 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. $$|2-19| - |9 - 20|$$
Part 2
Substitute and Evaluate the expressions
1. | x - 10 | + | 2 x | , x = - 7
2. | - x | , x = 8
3. $$\left |- \dfrac{|x - 2|}{|-x + 9|} \right |$$ , x = -20
4. $$| - 3 x - x^2 |$$ , x = -3
5. $$x^{| x - 7|}$$ , x = - 2
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. $$\sqrt{sin^2(x)}$$
Part 4
Solve the Equations
1. $$|x - 2| = 4$$
2. $$|x + 5| = 0$$
3. $$|10 - x| = - 5$$
Part 5
Solve the Inequalities
1. $$|x - 7| \gt 4$$
2. $$|x + 5| \lt 9$$
3. $$|x + 8| \lt - 5$$
4. $$|x + 8| \gt - 2$$
Part 6
Sketch the following pairs of functions in the same system of axes and explains the similarities and differences of the two graphs.
1. $$f(x) = x - 1 , h(x) = |f(x)|$$
2. $$f(x) = x^2 - 4 , h(x) = |f(x)|$$
3. $$f(x) = \sin(x) , h(x) = |\sin(x)|$$

Solutions to the Above Exercises
Part 1
Evaluate the expressions

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. 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 | = 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
Solve the Equations
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.
Part 5
Solve the Inequalities
1. $$|x - 7| \gt 4$$ , solution set: x < 3 ∪ x > 11
2. $$|x + 5| \lt 9$$ , solution set: -14 < x < 4
3. $$|x + 8| \lt - 5$$ , solution set: {∅}
4. $$|x + 8| \gt - 2$$ , solution set: {all real numbers}
Part 6
1. $$f(x) = x - 1 , h(x) = |f(x)| = |x - 1|$$
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.

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

3. $$f(x) = \sin(x) , h(x) = |\sin(x)|$$
Over intervals -2π to -π , 0 to π , 2π to 3π, ... sin(x) ≥ 0 and h(x) = sin(x) = f(x) therefore the graphs of h and f coincide.
For intervals - π to 0 , π to 2π ... 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.