Question

How do I determine if a function transformation represents a stretch or compression?

Answer 1

To determine whether a function transformation indicates a stretch or a compression, you'll need to analyze the changes made to the function in relation to its standard form. Here are the primary ways transformations are expressed and how to classify them:

1. **Vertical transformations (stretch/compression)**:
- If a function \( f(x) \) is transformed to \( af(x) \), where \( a \) is a constant:
- **If \( |a| > 1 \)**: The transformation is a **vertical stretch**. The graph will be stretched farther away from the x-axis.
- **If \( 0 < |a| < 1 \)**: The transformation is a **vertical compression**. The graph will be compressed closer to the x-axis.

2. **Horizontal transformations (stretch/compression)**:
- If a function \( f(x) \) is transformed to \( f(bx) \), where \( b \) is a constant:
- **If \( |b| > 1 \)**: The transformation is a **horizontal compression**. The graph will be compressed closer together along the x-axis.
- **If \( 0 < |b| < 1 \)**: The transformation is a **horizontal stretch**. The graph will be stretched farther apart along the x-axis.

### Summary

- **Vertical Stretch**: \( a > 1 \) in \( y = af(x) \)
- **Vertical Compression**: \( 0 < a < 1 \) in \( y = af(x) \)
- **Horizontal Compression**: \( b > 1 \) in \( y = f(bx) \)
- **Horizontal Stretch**: \( 0 < b < 1 \) in \( y = f(bx) \)

### Example

For instance, consider the function \( f(x) = x^2 \):

1. **Vertical Stretch**: \( f(x) = 3x^2 \) (since \( a = 3 > 1 \))
2. **Vertical Compression**: \( f(x) = 0.5x^2 \) (since \( a = 0.5 < 1 \))
3. **Horizontal Compression**: \( f(x) = f(2x) = (2x)^2 = 4x^2 \) (since \( b = 2 > 1 \))
4. **Horizontal Stretch**: \( f(x) = f(0.5x) = (0.5x)^2 = 0.25x^2 \) (since \( b = 0.5 < 1 \))

Using these guidelines, you can classify any transformation as a stretch or compression.