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:
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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.
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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 \( \frac{1}{5} \).
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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.