To find the x-intercepts of the functions \(y = 12x - 4\) and its vertically dilated version with a factor \(k = 6\), we start with the original function:
Original Function
The x-intercept occurs where \(y = 0\). \[ 0 = 12x - 4 \] Solving for \(x\): \[ 12x = 4 \ x = \frac{4}{12} \ x = \frac{1}{3} \] Thus, the x-intercept of the original function \(y = 12x - 4\) is \(\left(\frac{1}{3}, 0\right)\).
Vertically Dilated Function
The vertically dilated function with factor \(k = 6\) is given by: \[ y = 6(12x - 4) = 72x - 24 \] Again, to find the x-intercept, we set \(y = 0\): \[ 0 = 72x - 24 \] Solving for \(x\): \[ 72x = 24 \ x = \frac{24}{72} \ x = \frac{1}{3} \] Thus, the x-intercept of the dilated function \(y = 72x - 24\) is also \(\left(\frac{1}{3}, 0\right)\).
Conclusion
Both the original function \(y = 12x - 4\) and the vertically dilated function \(y = 72x - 24\) have the same x-intercept: The original function has an x-intercept of \(\left(\frac{1}{3}, 0\right)\), and the dilated function also has an x-intercept of \(\left(\frac{1}{3}, 0\right)\).
None of the provided response options correctly reflect these x-intercepts.
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