To find the x-intercepts of the original function \( y = 12x - 4 \) and the vertically dilated function with a factor \( k = 6 \), we first determine the x-intercept of the original function.
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Finding the x-intercept of the original function \( y = 12x - 4 \):
- Set \( y = 0 \): \[ 0 = 12x - 4 \]
- Solve for \( x \): \[ 12x = 4 \ x = \frac{4}{12} = \frac{1}{3} \]
- Therefore, the x-intercept of the original function is \( \left(\frac{1}{3}, 0\right) \).
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Finding the x-intercept of the vertically dilated function \( y = k(12x - 4) \) where \( k = 6 \):
- The vertically dilated function is: \[ y = 6(12x - 4) = 72x - 24 \]
- Set \( y = 0 \): \[ 0 = 72x - 24 \]
- Solve for \( x \): \[ 72x = 24 \ x = \frac{24}{72} = \frac{1}{3} \]
- Therefore, the x-intercept of the vertically dilated function is also \( \left(\frac{1}{3}, 0\right) \).
From this analysis, we see that the x-intercepts of both the original function and the dilated function are the same, at \( \left(\frac{1}{3}, 0\right) \).
Reviewing the provided responses, none of them correctly state that the x-intercepts of both functions are \( \left(\frac{1}{3}, 0\right) \). Therefore, it seems that the correct response is not available in the provided options.