The rules for absolute value functions on a graph dictate its shape, symmetry, and key characteristics. An absolute value function transforms any input into its non-negative value.
Key Rules and Characteristics
Here's a breakdown of the rules governing absolute value functions on a graph:
- Definition: The absolute value of a number represents its distance from the origin (zero) on the number line. This is a core concept directly impacting the graph.
- Symmetry: The graph of an absolute value function is symmetric about the y-axis. This means if you fold the graph along the y-axis, the two halves will perfectly overlap. Mathematically, this is represented by f(x) = f(-x), meaning the function is an even function.
- V-Shape: The basic absolute value function, f(x) = |x|, forms a "V" shape. The point of the "V" (the vertex) is at the origin (0, 0).
- Right Angle at the Origin: The basic absolute value graph f(x) = |x|, creates a right angle at its vertex (the origin). However, transformations (stretches, compressions) can alter this angle.
- Non-Negativity: The absolute value function always returns a non-negative value. Therefore, the graph will always be above or on the x-axis. The range of the basic absolute value function is y ≥ 0.
Impact of Transformations
While the basic absolute value function follows the above rules, transformations can affect its position and shape. These transformations include:
- Vertical Shifts: f(x) = |x| + c. Shifts the graph up (if c > 0) or down (if c < 0).
- Horizontal Shifts: f(x) = |x - c|. Shifts the graph right (if c > 0) or left (if c < 0).
- Vertical Stretches/Compressions: f(x) = a|x|. Stretches the graph vertically (if |a| > 1) or compresses it (if 0 < |a| < 1). If a is negative, the graph is reflected across the x-axis.
- Reflections: f(x) = -|x|. Reflects the graph across the x-axis, making the "V" shape point downwards.
Example
Consider the function f(x) = |x - 2| + 1.
- This graph is the basic absolute value function shifted 2 units to the right and 1 unit up.
- The vertex of the "V" shape is at (2, 1).
- The graph maintains its symmetry, but now around the vertical line x = 2.
- The range is y ≥ 1.
Summary
In essence, graphing absolute value functions involves understanding the fundamental "V" shape, its symmetry around the y-axis (or a vertical line after a horizontal shift), and how transformations impact the vertex and overall position of the graph. Remembering that absolute value represents distance from zero is key to understanding its non-negative output and characteristic shape.
