To analyze the transformations that produce the graph of \( g(x) = -|3x| \) from the parent function \( f(x) = |x| \), we can consider the following pieces:
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Horizontal Stretch by a Factor of 3: The function \( |3x| \) can be seen as a horizontal compression of \( |x| \) by a factor of \( \frac{1}{3} \) (since it "compresses" the x-values by multiplying them by 3). Therefore, this transformation does not apply.
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Horizontal Compression by a Factor of 3: This is correct. The term \( 3x \) means that every x-value in the original function is compressed toward the y-axis by a factor of 3.
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Reflection over the x-axis: The negative sign in front of the absolute value function indicates a reflection over the x-axis. So this transformation does apply.
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Reflection over the y-axis: There is no negative sign in front of \( x \) itself, so there is no reflection over the y-axis. This transformation does not apply.
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Vertical Stretch by a Factor of 3: The term \( |3x| \) does not produce a vertical stretch by a factor of 3; instead, it affects the horizontal scaling. So this transformation does not apply.
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Vertical Compression by a Factor of 3: Similarly, there is no vertical compression happening; the negative sign only reflects it over the x-axis. So this transformation does not apply.
Based on the information above, the transformations that apply are:
- Reflection over the x-axis
- Horizontal compression by a factor of 3
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