Overview
This lesson explains how to graph absolute value functions and apply transformations such as shifts, stretches, and reflections, including analysis through three worked examples.
Parent Absolute Value Function
- The parent function is y = |x|, forming a sharp V-shaped graph.
- To graph y = |x|, plot points by calculating the absolute value for positive and negative x (e.g., (-3, 3), (0, 0), (3, 3)).
- The vertex, where the graph bends, is at the origin (0, 0).
Transformations of Absolute Value Functions
- The general form is y = a|x - h| + k.
- "h" shifts the graph horizontally, but the direction is opposite to the sign (x - h: right h; x + h: left h).
- "k" shifts the graph vertically in the direction of its sign (k > 0: up; k < 0: down).
- "a" vertically stretches (|a| > 1, narrower), compresses (0 < |a| < 1, wider), or reflects (a < 0, flips over x-axis) the graph.
Example 1: y = |x + 1| - 3
- Vertex: (-1, -3) (left 1, down 3 from the origin).
- "a" is 1, so the graph opens upward, maintaining the parent shape.
- Symmetrical about the line x = -1.
- Domain: all real numbers; Range: y ≥ -3.
Example 2: y = -2|x - 3| + 4
- Vertex: (3, 4) (right 3, up 4).
- “a” is -2, so graph opens downward and is vertically stretched (narrower).
- Domain: all real numbers; Range: y ≤ 4.
Example 3: y = (1/3)|x + 1| - 2
- Vertex: (-1, -2) (left 1, down 2).
- “a” is 1/3, so the graph opens upward and is vertically compressed (wider).
- Domain: all real numbers; Range: y ≥ -2.
Key Terms & Definitions
- Absolute Value — the distance a number is from zero on the number line, always non-negative.
- Vertex — the point where the graph changes direction (minimum or maximum).
- Domain — set of all possible x-values.
- Range — set of all possible y-values.
- Transformation — operations that shift, stretch, compress, or reflect a graph.
- Axis of Symmetry — vertical line running through the vertex, dividing the graph into two mirror-image halves.
Action Items / Next Steps
- Practice graphing absolute value functions with different values of a, h, and k.
- Review homework or textbook problems on graph transformations.