Question
How does the graph of \( f(x)=\sqrt[3]{x} \) change when \( f(x) \) is replaced by \( -7 f(x) \) (1 point) The graph is reflected across the \( x \)-axis and has been stretched horizontally. The graph is reflected across the \( y \)-axis and has been compressed vertically. 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 horizontally.
Ask by Reid Harper. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The graph is reflected across the \( x \)-axis and stretched vertically.
Solution
When you replace \( f(x) = \sqrt[3]{x} \) with \( -7f(x) = -7\sqrt[3]{x} \):
1. **Reflection across the \( x \)-axis**: The negative sign (-7) reflects the graph across the \( x \)-axis.
2. **Vertical Stretch**: The factor of 7 stretches the graph vertically by a factor of 7.
So, the correct transformation is:
**The graph is reflected across the \( x \)-axis and has been stretched vertically.**
**Answer:**
The graph is reflected across the \( x \)-axis and has been stretched vertically.
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Simplify this solution The Deep Dive
When you replace \( f(x) \) with \( -7 f(x) \), you are essentially multiplying the original function by -7. This means the entire graph of \( f(x) = \sqrt[3]{x} \) is reflected across the \( x \)-axis, which flips all the positive \( y \)-values to negative. Additionally, multiplying by 7 stretches the graph vertically by a factor of 7, making it taller! So, in simpler terms, the graph takes a fun dive downwards (thanks to the reflection) and becomes much taller (due to the vertical stretch). It's a wild ride for our cube root!🥳
