To find the new coordinates of the point \((4, \frac{1}{4})\) on the graph of the function \(f(x) = \frac{1}{x}\) after it is vertically stretched by a factor of 8, we follow these steps:
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Determine the original \(y\)-value: We start with the point \((4, \frac{1}{4})\). Here, the \(x\)-coordinate is 4, and the corresponding \(y\)-coordinate is \(f(4) = \frac{1}{4}\).
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Apply the vertical stretch: A vertical stretch by a factor of 8 means we multiply the \(y\)-value by 8. Therefore, the new \(y\)-value will be: \[ y_{\text{new}} = 8 \cdot y_{\text{original}} = 8 \cdot \frac{1}{4} = 2. \]
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Form the new coordinates: The new coordinates after the vertical stretch will keep the same \(x\)-coordinate but have the updated \(y\)-coordinate. Thus, the new coordinates will be: \[ (4, 2). \]
In conclusion, the new coordinates of the point after the vertical stretch by a factor of 8 are \(\boxed{(4, 2)}\).