Vertical Stretch/Compression

Algebra-2

1. Fundamental Concepts

Definition: A vertical stretch or compression of a function is a transformation that changes the shape of the graph by multiplying the output values by a constant.
Vertical Stretch: If the constant \(c > 1\), the graph is stretched vertically.
Vertical Compression: If the constant \(0 < c < 1\), the graph is compressed vertically.

2. Key Concepts

Basic Rule: $$f(x) \rightarrow cf(x)$$

Degree Preservation: The degree of the polynomial remains unchanged after a vertical stretch or compression.

Application: Used to model real-world phenomena such as sound waves and light intensity.

3. Examples

Example 1 (Basic)

Problem: Given the function \(f(x) = x^2\), find the equation of the function after a vertical stretch by a factor of 3.

Step-by-Step Solution:

Multiply the output of the function by 3: \(3f(x)\)
The new function is \(3x^2\).

Validation: Substitute \(x = 1\) → Original: \(1^2 = 1\); Stretched: \(3 \cdot 1^2 = 3\) ✓

Example 2 (Intermediate)

Problem: Given the function \(g(x) = 2x^3 - 4x + 1\), find the equation of the function after a vertical compression by a factor of \(\frac{1}{2}\).

Step-by-Step Solution:

Multiply the output of the function by \(\frac{1}{2}\): \(\frac{1}{2}g(x)\)
The new function is \(\frac{1}{2}(2x^3 - 4x + 1)\).
Simplify: \(x^3 - 2x + \frac{1}{2}\).

Validation: Substitute \(x = 1\) → Original: \(2(1)^3 - 4(1) + 1 = -1\); Compressed: \(1^3 - 2(1) + \frac{1}{2} = -\frac{1}{2}\) ✓

4. Problem-Solving Techniques

Visual Strategy: Graph the original and transformed functions side by side to visualize the change.
Error-Proofing: Always check the transformation by substituting a simple value for \(x\).
Concept Reinforcement: Understand that the transformation affects only the y-values, not the x-values.