To analyze the effect of the function \( g(x) = 8\sqrt{x} \) in comparison to the parent function \( f(x) = \sqrt{x} \), we need to consider how the modifications to the function \( g(x) \) affect its graph.
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Parent Function \( f(x) = \sqrt{x} \):
- This is the standard square root function.
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Function \( g(x) = 8\sqrt{x} \):
- The presence of the constant \( 8 \) in front of the square root function indicates that there is a vertical scaling (stretching in this case), as multiplying the output of the function \( \sqrt{x} \) by \( 8 \) increases the height of the graph without changing its width.
Vertical Stretch
- A vertical stretch occurs when the output (y-value) of the function is multiplied by a factor greater than 1. In this case, since we have \( 8\sqrt{x} \), the graph of \( g(x) \) is stretched vertically by a factor of \( 8 \) compared to \( f(x) \).
Summary
- The graph of \( g(x) = 8\sqrt{x} \) is a vertical stretch of the parent function \( f(x) = \sqrt{x} \) by a factor of \( 8 \).
Conclusion
The correct response is: vertical stretch by a factor of 8.
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