Horizontal Stretch/Compression - Algebra-2

1. Fundamental Concepts

Horizontal stretch/compression is a type of function transformation that changes the width of a function’s graph along the x-axis (horizontal direction) while preserving its shape and vertical position relative to key features (e.g., vertices, intercepts). It is determined by the coefficient of the input variable x within the function, typically denoted as b in the transformed function form .

2. Key Concepts

General Form: For a parent function $y = f(x)$ , the transformed function $y = f(bx)$ represents a horizontal stretch or compression:
- If : The graph of is horizontally stretched by a factor of . This means each x-coordinate of the original function is multiplied by , making the graph wider.
- If : The graph of is horizontally compressed by a factor of . This means each x-coordinate of the original function is multiplied by , making the graph narrower.
Impact on Coordinates: For any point on the parent function , the corresponding point on is . The y-coordinates remain unchanged, while x-coordinates are scaled by .

3. Examples

Simple

Given the parent function , find the transformed function for a horizontal compression by a factor of .

Solution: A horizontal compression by means (since ). Thus, the transformed function is .

Medium

The parent function has a point on its graph. What is the corresponding point on ?

Solution: represents a horizontal stretch by a factor of 2 (since , ). For the original point , the new x-coordinate is , and the y-coordinate remains 1. Thus, the corresponding point is .

4. Problem-Solving Techniques

Identify the Coefficient b: In the transformed function , isolate b to determine if the transformation is a stretch () or compression ().
Scale X-Coordinates: For points on the parent function, calculate new x-coordinates by dividing the original x-values by b; y-coordinates stay unchanged.
Relate to the General Transformation Rule: Use the formula to confirm horizontal stretch/compression (focus on b) and distinguish it from vertical transformations (involving a).
Verify with the Vertical Line Test: Ensure the transformed function remains a function (passes the vertical line test) after horizontal scaling.