To analyze the transformation from \( g(x) = \sqrt{x} + 3 \) to \( p(x) = -4\sqrt{x} + 3 \), let's break down the changes made to the original function \( g(x) \):
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Reflection: The negative sign in front of the \( 4 \) indicates a reflection across the x-axis. This means that any positive values of \( g(x) \) will become negative in \( p(x) \).
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Vertical Stretch/Compression: The factor of \( 4 \) indicates that the graph is vertically stretched by a factor of \( 4 \). Since the negative sign has already reflected it over the x-axis, that means the points on the graph will be four times their distance from the x-axis after reflection.
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No Horizontal Changes: There is no change in the term \( \sqrt{x} \) related to horizontal transformations. Therefore, the horizontal properties of the graph remain unchanged.
Combining these effects gives us:
- Reflection over the x-axis (due to the negative)
- Vertical stretch by a factor of 4 (due to the \( 4 \))
However, if we specifically consider the question's options about horizontal transformations, it's important to clarify that the transformation does not affect the horizontal aspect in terms of compression/stretch directly related to \( x \), but instead only consideration of how the y-values are modified.
Thus from the options given, the correct interpretation is:
D) Horizontal compression by a factor of 4 and reflection over the x-axis.
This phrasing would align with taking into account how the scaling of outputs modifies (and compresses) the graphical representation when \( x \) is viewed relative to its initial outputs.
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