Question

What does the transformation f(x)↦
1/
5
f(x) do to the graph of f(x)?

Answer 1

The transformation \( f(x) \mapsto \frac{1}{5} f(x) \) compresses the graph of the function \( f(x) \) vertically.

Here's a breakdown of what this means:

1. **Scaling Factor**: The factor \( \frac{1}{5} \) is less than 1, which indicates vertical compression. For each \( y \)-value of the function \( f(x) \), the new \( y \)-value will be \( \frac{1}{5} \) of that \( y \)-value.

2. **Effect on Points**: If a point \( (x, y) \) is on the graph of \( f(x) \), it will transform to the point \( (x, \frac{1}{5}y) \) on the graph of \( \frac{1}{5} f(x) \). This means that every point on the graph of \( f(x) \) will be moved closer to the \( x \)-axis by a factor of \( \frac{1}{5} \).

3. **Graph Appearance**: Overall, the graph will appear "squished" or "flattened" vertically. The function values will all be smaller, but the \( x \)-intercepts (where the graph crosses the x-axis) will remain unchanged, as scaling does not affect zero values.

In summary, the transformation \( f(x) \mapsto \frac{1}{5} f(x) \) results in a vertical compression of the graph of \( f(x) \) by a factor of 5.