The absolute value graph visually represents the absolute value function, which always returns the non-negative value of a number (its distance from zero).
Understanding the Absolute Value Function
The absolute value function is defined as:
- f(x) = |x|
- Where |x| = x if x ≥ 0
- And |x| = -x if x < 0
In simpler terms, if the input (x) is positive or zero, the output is the same as the input. If the input is negative, the output is the positive version of that number. For example, |3| = 3 and |-3| = 3.
Characteristics of the Absolute Value Graph
The graph of the basic absolute value function, f(x) = |x|, has the following key characteristics:
- Shape: It forms a "V" shape.
- Vertex: The "V" shape's point (vertex) is located at the origin (0, 0).
- Symmetry: The graph is symmetric about the y-axis. This means that if you fold the graph along the y-axis, the two halves will perfectly overlap. This is because f(x) = f(-x), making it an even function.
- Domain: All real numbers. You can input any real number into the absolute value function.
- Range: All non-negative real numbers (y ≥ 0). The output of an absolute value function is never negative.
Transformations of the Absolute Value Graph
The basic absolute value graph can be transformed through various operations:
- Vertical Shifts: Adding a constant outside the absolute value, f(x) = |x| + k, shifts the graph up (if k > 0) or down (if k < 0).
- Horizontal Shifts: Adding a constant inside the absolute value, f(x) = |x - h|, shifts the graph right (if h > 0) or left (if h < 0).
- Vertical Stretches/Compressions: Multiplying the absolute value by a constant, f(x) = a|x|, stretches the graph vertically (if |a| > 1) or compresses it vertically (if 0 < |a| < 1). If 'a' is negative, it also reflects the graph across the x-axis, turning the "V" upside down.
- Reflections: Multiplying the absolute value by -1, f(x) = -|x|, reflects the graph across the x-axis.
Example
Consider the function f(x) = |x - 2| + 1. This graph is the basic absolute value graph shifted 2 units to the right and 1 unit up. The vertex is at (2, 1).
