Overview
This lecture explains how absolute value functions can be transformed through vertical and horizontal translations, and introduces related key terms.
Transforming Absolute Value Functions
- Absolute value functions have the form f(x) = |x| and can be altered by translations, stretches, compressions, and reflections.
- Transformations change the graph's position or shape on the coordinate plane.
Vertical Translation
- Adding k to f(x) gives f(x) + k, shifting the graph up if k > 0 and down if k < 0.
- Example: g(x) = |x| + 1 shifts the graph up by 1 unit; h(x) = |x| − 1 shifts the graph down by 1 unit.
- The vertex moves from (0,0) to (0,k).
Horizontal Translation
- Replacing x with (x − h) in f(x) gives f(x − h), shifting the graph right by h if h > 0 and left by h if h < 0.
- Example: g(x) = |x + 1| shifts the graph left by 1 unit; h(x) = |x − 1| shifts the graph right by 1 unit.
- The vertex moves from (0,0) to (h,0).
Combined Translation (Vertex Form)
- The function g(x) = |x − h| + k shifts the graph h units horizontally and k units vertically.
- The new vertex is at (h, k).
Key Terms & Definitions
- Translation — Moving a graph up, down, left, or right without changing its shape.
- Stretch — Making the graph narrower.
- Compression — Making the graph wider.
- Reflection — Flipping the graph over an axis.
Action Items / Next Steps
- Practice identifying and graphing absolute value function translations.
- Review definitions of translation, stretch, compression, and reflection.