Question
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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Mind Expander
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.