Absolute Value Function

Understanding Absolute Value Functions

An absolute value function is a function that produces positive or zero values from any input, regardless of the original sign of the input. Geometrically, absolute value can be understood as the distance of a number from the zero point on the number line.

Mathematical Definition

For any real number $x$, the absolute value function is defined as:

Components of absolute value functions:

• The symbol $|x|$ is read as absolute value of $x$.
• The function result is always non-negative ($|x| \geq 0$)
• This function is even: $|-x| = |x|$

Properties of Absolute Value Functions

Absolute value functions have several important properties that need to be understood:

Basic properties:

Triangle inequality properties:

Graphs of Absolute Value Functions

The following is a visualization of the basic absolute value function:

Value table for function $f(x) = |x|$:

Transformations of Absolute Value Functions

Absolute value functions can be transformed in various ways:

Vertical Translation

The function $f(x) = |x| + k$ shifts the graph upward (if $k > 0$) or downward (if $k < 0$).

Horizontal Translation

The function $f(x) = |x - h|$ shifts the graph to the right (if $h > 0$) or to the left (if $h < 0$).

Stretching and Compression

The function $f(x) = a|x|$ changes the slope of the graph:

• If $a > 1$: the graph becomes steeper
• If $0 < a < 1$: the graph becomes gentler
• If $a < 0$: the graph is inverted (reflection across the $x$-axis)

To make it easier to understand, let's look at the following example:

General Form of Absolute Value Functions

The general form of an absolute value function is:

where:

• $a$: stretching/compression factor and reflection
• $h$: horizontal translation
• $k$: vertical translation
• The vertex is located at $(h, k)$

Transformation table:

Absolute Value Equations and Inequalities

Solving absolute value equations:

To solve $|x| = a$ with $a \geq 0$:

Example: Solve $|x - 3| = 5$

Solving absolute value inequalities:

For $|x| < a$ with $a > 0$:

For $|x| > a$ with $a > 0$:

Exercises

1. Determine the value of $f(x) = |2x - 6|$ for $x = -1, 0, 3, 5$

2. Solve the equation $|3x + 1| = 7$

3. Solve the inequality $|x - 2| < 4$

4. Determine the vertex of the function $f(x) = 2|x - 3| + 1$

5. The distance between two cities is $150 \text{ km}$. If city $A$ is located at coordinate $-50 \text{ km}$, where is city $B$ located?

Answer Key

1. Calculating function values for various inputs:

   Substitute each value of $x$ into the function $f(x) = |2x - 6|$:

   $$f(-1) = |2(-1) - 6| = |-2 - 6| = |-8| = 8$$

   $$f(0) = |2(0) - 6| = |0 - 6| = |-6| = 6$$

   $$f(3) = |2(3) - 6| = |6 - 6| = |0| = 0$$

   $$f(5) = |2(5) - 6| = |10 - 6| = |4| = 4$$

2. Solving absolute value equations:

   For the equation $|3x + 1| = 7$, we use the definition of absolute value which produces two possibilities:

   $$3x + 1 = 7 \quad \text{or} \quad 3x + 1 = -7$$

   $$3x = 6 \quad \text{or} \quad 3x = -8$$

   $$x = 2 \quad \text{or} \quad x = -\frac{8}{3}$$

3. Solving absolute value inequalities:

   For $|x - 2| < 4$, we use the property that $|a| < b$ is equivalent to $-b < a < b$:

   $$-4 < x - 2 < 4$$

   $$-4 + 2 < x < 4 + 2$$

   $$-2 < x < 6$$

   So the solution set is $x \in (-2, 6)$.

4. Determining the vertex:

   From the function $f(x) = 2|x - 3| + 1$, we can identify the parameters:

   • $a = 2$ (stretching factor)
   • $h = 3$ (horizontal translation)
   • $k = 1$ (vertical translation)

   The vertex is located at $(h, k) = (3, 1)$.

5. Calculating position based on distance:

   Given that the distance between cities $A$ and $B$ is $150 \text{ km}$, with city $A$ at coordinate $-50 \text{ km}$. Let city $B$ be at coordinate $x_B$:

   $$|-50 - x_B| = 150$$

   $$-50 - x_B = 150 \quad \text{or} \quad -50 - x_B = -150$$

   $$x_B = -50 - 150 = -200 \quad \text{or} \quad x_B = -50 + 150 = 100$$

   So city $B$ can be located at coordinate $100 \text{ km}$ or $-200 \text{ km}$.