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What is the effect on the graph of \( f(x)=\sqrt[3]{x} \) when \( f(x) \) is replaced by \( -2 f(x) \) ? (1 point) The graph is reflected across the \( x \)-axis and has been stretched horizontally. The graph is reflected across the \( x \)-axis and has been stretched vertically. The graph is reflected across the \( y \)-axis and has been compressed vertically. The graph is reflected across the \( y \)-axis and has been compressed horizontally.

Ask by Reed Cook. in the United States
Jan 23,2025

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Answer

The graph is reflected across the \( x \)-axis and stretched vertically.

Solution

When you replace \( f(x) = \sqrt[3]{x} \) with \( -2f(x) \), the transformation involves two main changes: 1. **Reflection across the \( x \)-axis:** The negative sign (\(-\)) flips the graph vertically. 2. **Vertical Stretch:** The factor of 2 (\( \times 2 \)) stretches the graph away from the \( x \)-axis by a factor of 2. Thus, the graph of \( f(x) = \sqrt[3]{x} \) is both reflected across the \( x \)-axis and stretched vertically when replaced by \( -2f(x) \). **Answer:** The graph is reflected across the \( x \)-axis and has been stretched vertically.

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When the function \( f(x) = \sqrt[3]{x} \) is replaced by \( -2 f(x) \), you are effectively reflecting the graph across the \( x \)-axis due to the negative sign and stretching it vertically by a factor of 2 because of the multiplication by 2. This means all the \( y \)-coordinates of the points on the original graph are flipped over the \( x \)-axis and then stretched away from the \( x \)-axis. So, the correct answer is: The graph is reflected across the \( x \)-axis and has been stretched vertically.

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