To find the x-intercepts of the original function \( y = 12x - 4 \), we set \( y \) equal to 0 and solve for \( x \):
\[ 0 = 12x - 4 \ 12x = 4 \ x = \frac{4}{12} = \frac{1}{3} \]
So, the x-intercept of the original function is \( \left(\frac{1}{3}, 0\right) \).
Now, for the vertically dilated function with a factor \( k = 6 \), the function becomes:
\[ y = 6(12x - 4) = 72x - 24 \]
Next, we find the x-intercept of this dilated function by setting \( y \) equal to 0:
\[ 0 = 72x - 24 \ 72x = 24 \ x = \frac{24}{72} = \frac{1}{3} \]
Thus, the x-intercept of the dilated function is also \( \left(\frac{1}{3}, 0\right) \).
Putting this all together, both the original function \( y = 12x - 4 \) and the vertically dilated function \( y = 72x - 24 \) have the same x-intercept, which is \( \left(\frac{1}{3}, 0\right) \).
None of the listed response options in your prompt seem to be correct based on this analysis. The correct x-intercept for both functions is \( \left(\frac{1}{3}, 0\right) \).