Algebra Absolute Value Study Guide

Absolute Value Concepts

• Absolute value represents the distance of a number from zero on a number line and is always positive.

Absolute Value Function

• Forms a piecewise function yielding a V-shaped graph that opens upwards or downwards.
• The parent function is y = |x|, where f(x) is non-negative (f(x) ≥ 0) for all x values.

Extraneous Solution

• Refers to a solution that appears to be valid but does not satisfy the initial equation and must be discarded.

Graph Transformations

• Reflection Over X-Axis: The transformation of -|x| indicates a reflection of the graph across the x-axis.
• Reflection Over Y-Axis: The expression y = f(-x) reflects the graph across the y-axis.
• Horizontal Shifts:
  • y = f(x + c) results in shifting the graph left by c units.
  • y = f(x - c) shifts the graph right by c units.

Vertical Transformations

• Vertical Compression: The equation y = (1/c) * f(x) compresses the graph vertically.
• Vertical Stretch: The function y = c * f(x) stretches the graph vertically.

Horizontal Transformations

• Horizontal Stretch: The transformation y = f(1/c * x) applies a horizontal stretch to the graph.
• Horizontal Compression: The equation y = f(c * x) indicates a horizontal compression of the graph.

Additional Reflection

• The expression y = |-x| shows reflection around the y-axis, providing symmetry relative to the y-axis.