Definition and properties

Definition of Absolute Value

The absolute value of a number is its distance from zero on the number line. It is always a non-negative value. The absolute value of a real number x is denoted as |x|. For example, |3| = 3 and |-3| = 3. In mathematical terms, we define absolute value as follows:

|x| = { x, if x ≥ 0
-x, if x < 0 }

Examples of Absolute Value

Let us evaluate some absolute values to understand this concept better:

1. For x = 5, |5| = 5.
2. For x = -2, |-2| = -(-2) = 2.
3. For x = 0, |0| = 0.

Properties of Absolute Value

Absolute value functions have several important properties that are useful for solving equations and inequalities.

1. Non-negativity

The absolute value of any real number is always greater than or equal to zero:

|x| ≥ 0 for all x ∈ ℝ.

2. Symmetry

The absolute value function is symmetric about the y-axis. This means that:

|-x| = |x| for all x ∈ ℝ.

3. Triangle Inequality

The triangle inequality states that for any two real numbers a and b:

|a + b| ≤ |a| + |b|.

4. Product Property

The product of the absolute values is equal to the absolute value of the product:

|a × b| = |a| × |b|.

5. Quotient Property

The quotient of the absolute values is equal to the absolute value of the quotient:

|a / b| = |a| / |b|, for b ≠ 0.

6. Zero Property

The only time the absolute value of a number is zero is when the number itself is zero:

|x| = 0 if and only if x = 0.

Examples of Absolute Value Properties

Example 1: Non-negativity

Let x = -4. Then, |x| = |-4| = 4, which is greater than zero. This illustrates the non-negativity property.

Example 2: Triangle Inequality

Let a = 3 and b = -5. Then:

|a + b| = |3 - 5| = |-2| = 2

|a| + |b| = |3| + |-5| = 3 + 5 = 8

Since 2 ≤ 8, the triangle inequality holds.

Example 3: Product Property

Let a = -2 and b = 3. Then:

|a × b| = |-2 × 3| = |-6| = 6

|a| × |b| = |-2| × |3| = 2 × 3 = 6

Thus, |a × b| = |a| × |b| holds true.

Example 4: Quotient Property

Let a = -8 and b = 4. Then:

|a / b| = |-8 / 4| = |-2| = 2

|a| / |b| = |-8| / |4| = 8 / 4 = 2

The quotient property also holds true.

Graphing Absolute Value Functions

The graph of an absolute value function is V-shaped. The general form of an absolute value function is:

f(x) = |x - h| + k

In this equation, (h, k) is the vertex of the V-shape. The graph opens upwards if the coefficient of the absolute value is positive.

Example: Graphing an Absolute Value Function

Consider the function f(x) = |x - 2| + 1. To graph this function, follow these steps:

1. Identify the vertex. Here, h = 2 and k = 1. Thus, the vertex is (2, 1).
2. Determine additional points. Choose values for x:

x = 0: f(0) = |0 - 2| + 1 = 2 + 1 = 3
x = 1: f(1) = |1 - 2| + 1 = 1 + 1 = 2
x = 2: f(2) = |2 - 2| + 1 = 0 + 1 = 1
x = 3: f(3) = |3 - 2| + 1 = 1 + 1 = 2
x = 4: f(4) = |4 - 2| + 1 = 2 + 1 = 3

3. Plot the points on a graph:

(0, 3), (1, 2), (2, 1), (3, 2), (4, 3)

4. Draw the V-shape, connecting the points. The graph will have a vertex at (2, 1) and will open upwards.

Common Mistakes

Summary

• The absolute value of a number is its distance from zero and is always non-negative.
• Key properties include non-negativity, symmetry, triangle inequality, product property, quotient property, and zero property.
• The graph of an absolute value function is V-shaped and can be plotted using the vertex and additional points.

Check your understanding

1. What is the absolute value of -7?
2. State the triangle inequality property using a and b.
3. How would you graph the function f(x) = |x + 3| - 2?
4. Explain the significance of the vertex in the graph of an absolute value function.