To determine the x-intercepts of the original function \( y = 12x - 4 \) and the vertically dilated function with a dilation factor \( k = 6 \), we first find the x-intercept of the original function.
Original Function:
Set \( y = 0 \): \[ 0 = 12x - 4 \]
Add 4 to both sides: \[ 4 = 12x \]
Divide by 12: \[ x = \frac{4}{12} = \frac{1}{3} \]
Thus, the x-intercept of the original function is: \[ \left(\frac{1}{3}, 0\right) \]
Vertically Dilated Function:
The dilated function can be expressed as: \[ y = k \cdot (12x - 4) = 6(12x - 4) \] Expanding this: \[ y = 72x - 24 \]
Now, set \( y = 0 \): \[ 0 = 72x - 24 \]
Add 24 to both sides: \[ 24 = 72x \]
Divide by 72: \[ x = \frac{24}{72} = \frac{1}{3} \]
Thus, the x-intercept of the vertically dilated function is also: \[ \left(\frac{1}{3}, 0\right) \]
Conclusion:
Both the original function and the vertically dilated function have the same x-intercept, which is \( \left(\frac{1}{3}, 0\right) \).
None of the provided options correspond to this result, so it seems there may be an error in the given options.
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