Understanding Absolute Value Functions and Transformations
The absolute value function represents a fundamental concept in mathematics, defined as the distance of a number from zero on a number line. When graphing the absolute value parent function fx = |x|, several key properties emerge that form the foundation for understanding more complex transformations.
Definition: The absolute value of a number represents its positive distance from zero, regardless of whether the original number was positive or negative.
The parent function fx = |x| has distinct characteristics that make it recognizable. Its domain includes all real numbers −∞,∞, while the range spans from zero to positive infinity [0, ∞). The vertex, positioned at 0,0, serves as a crucial point where the function changes direction. The graph exhibits symmetry around the y-axis x=0, creating a distinctive V-shape.
Highlight: Key features of the absolute value parent function:
- Domain: All real numbers
- Range: [0, ∞)
- Vertex: 0,0
- Axis of Symmetry: x=0
Understanding intervals of increase and decrease helps analyze the function's behavior. For x-values greater than zero 0,∞, the function increases with a positive slope. Conversely, for x-values less than zero −∞,0, the function decreases with a negative slope. This creates the characteristic V-shape of the absolute value graph.