To analyze the transformation from \( b(x) = 3 - \sqrt{12x} \) to \( m(x) = \frac{3}{4} \sqrt{x} \), we need to break down the transformation step by step.
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Understand the original function: The function \( b(x) = 3 - \sqrt{12x} \) consists of a square root function multiplied by \( \sqrt{12} \), where \( 12 \) indicates a horizontal compression because the argument of the square root is multiplied by a factor of 12.
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Identify the basic transformation: The term \( \sqrt{12x} \) can be rewritten as \( \sqrt{12} \sqrt{x} \), which illustrates that \( \sqrt{x} \) is being compressed horizontally because multiplying \( x \) by 12 compresses the graph towards the y-axis.
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Transformation of the \( y \)-value: The transformation includes changing the vertical shift from 3 into a different function. The reflection over the x-axis occurs here because of the negative sign in front of \( \sqrt{12x} \) in the original function.
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Final steps towards the transformation: The transformation leads you to \( m(x) = \frac{3}{4} \sqrt{x} \). Upon comparing, you can observe that there is indeed a vertical compression too, as the output is multiplied by \( \frac{3}{4} \).
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Conclusion: The correct description for the transformation involves a horizontal compression by a factor of \( \frac{1}{3} \) and a reflection over the x-axis.
The correct response matches: Horizontal compression by a factor of \( \frac{1}{3} \) and a reflection over the x-axis.