Graph the Transformations of Absolute Value Functions - Algebra-2

1. Fundamental Concepts

Parent Function: The basic absolute value function is . Its graph is a V-shape with the vertex at the origin , opening upwards.
Transformations: Changes made to the parent function that alter its graph, including translations (shifts), stretches, compressions, and reflections.

Transformations of the absolute value function are based on the general form , where:

a: Controls vertical stretch, compression, or reflection.
- If : Vertical stretch (narrower than the parent function).
- If : Vertical compression (wider than the parent function).
- If : Reflection over the x-axis (opens downward).
h: Controls horizontal translation (shift along the x-axis).
- If : Shift right by h units.
- If : Shift left by units (equivalent to ).
k: Controls vertical translation (shift along the y-axis).
- If : Shift up by k units.
- If : Shift down by units (equivalent to ).
Reflection over the y-axis: Represented by , which is equivalent to (since absolute value eliminates the sign, the graph remains unchanged).

Describe the transformations of to .

Solution: The parent function is shifted right by 2 units () and up by 3 units ().

Graph using transformations of .

Solution:

Identify the Vertex: For a transformed absolute value function, the vertex is the starting point for analyzing shifts. It can be directly read from the graph or the equation.
Determine a: Check if the graph opens up () or down (). Compare the "width" to the parent function: narrower means (stretch), wider means (compression).
Reverse-Engineer Equations from Graphs:
1. Locate the vertex from the graph.
2. Determine the direction of opening and stretch/compression to find a.
3. Substitute a, h, and k into .
Graph Step-by-Step: Start with the parent function , apply horizontal/vertical shifts first, then stretch/compression, and finally reflections (order ensures accuracy).