Describe the Transformations of Absolute Value Functions - Algebra-2

1. Fundamental Concepts

Definition of Transformations: Transformations of absolute value functions refer to changes in the graph of the basic absolute value function through operations such as translation, stretching, compression, and reflection. These transformations do not alter the "V" shape of the graph but modify its position, size, or direction.
Basic Function Reference: The parent function for all absolute value functions is , with a vertex at , opening upwards, and a "V" shape with slopes of 1 (right of the vertex) and (left of the vertex).

2. Key Concepts

General Form of Transformed Absolute Value Functions: The standard form of a transformed absolute value function is , where each parameter controls a specific transformation:

Parameter	Transformation Effect
a	- If : Vertical stretch (narrows the "V" shape). - If : Vertical compression (widens the "V" shape). - If : Reflection over the x-axis (opens downward).
b	- If : Horizontal compression (narrows the "V" shape horizontally). - If : Horizontal stretch (widens the "V" shape horizontally). - If : Reflection over the y-axis (flips left-right).
h	- If : Horizontal translation to the right by h units. - If : Horizontal translation to the left by units (note: the form is , so means ).
k	- If : Vertical translation upward by k units. - If : Vertical translation downward by units.

3. Examples

Easy Level

Question: Describe the transformation of from the parent function .

Answer: The graph of is obtained by shifting the graph of 2 units upward (due to ). The vertex moves from to ; no stretching, compression, or reflection occurs.

Medium Level

Question: Describe the transformation of from the parent function .

Answer:

: Vertical stretch by a factor of 3 (narrows the "V" shape).
: Horizontal translation 1 unit to the right.
: Vertical translation 4 units downward. Overall, the parent function is stretched vertically by 3, shifted right 1 unit, and shifted down 4 units. The vertex moves from to .

Hard Level

Question: Describe the transformation of from the parent function . Answer:

Rewrite the function to identify parameters: , so , , , .
: Vertical stretch by a factor of 2 and reflection over the x-axis (opens downward).
: Horizontal stretch by a factor of 2 (since ) and reflection over the y-axis (due to ).
: Horizontal translation 3 units to the left.
: Vertical translation 1 unit upward. The vertex moves from to .

4. Problem-Solving Techniques

Step 1: Identify the Parent Function: Start with the basic form and its vertex .
Step 2: Rewrite the Target Function in Standard Form: Ensure the function is in to clearly identify a, b, h, and k. For example, rewrite as .
Step 3: Analyze Each Parameter Sequentially:
- First, handle translations (h for horizontal, k for vertical) to determine the new vertex position.
- Then, apply stretching/compression ( for vertical, for horizontal) to adjust the "V" shape size.
- Finally, check for reflections ( over x-axis, over y-axis) to determine the opening direction or left-right flip.
Step 4: Verify with Vertex Tracking: The vertex of the transformed function is ; confirm that the transformations correctly map the parent vertex to .