To find the x-intercepts of the functions given, we need to set \( y = 0 \) and solve for \( x \).
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Original function: \( y = \frac{1}{2}x - 4 \)
- Set \( y = 0 \): \[ 0 = \frac{1}{2}x - 4 \]
- Solve for \( x \): \[ \frac{1}{2}x = 4 \quad \Rightarrow \quad x = 4 \times 2 = 8 \]
- Thus, the x-intercept of the original function is \( (8, 0) \).
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Vertically dilated function with a factor of \( k = 6 \):
- The equation transforms to \( y = k \cdot \left(\frac{1}{2}x - 4\right) = 6 \cdot \left(\frac{1}{2}x - 4\right) \) or \( y = 3x - 24 \).
- Set \( y = 0 \): \[ 0 = 3x - 24 \]
- Solve for \( x \): \[ 3x = 24 \quad \Rightarrow \quad x = \frac{24}{3} = 8 \]
- Thus, the x-intercept of the dilated function is also \( (8, 0) \).
Therefore, both functions have the same x-intercept:
Original function: \( (8, 0) \)
Dilated function: \( (8, 0) \)
The correct response is: "The original function has an x-intercept of (8,0), and the dilated function has an x-intercept of (8,0)."
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