To find the x-intercepts of the original function \( y = 12x - 4 \) and the vertically dilated function with a factor of \( k = 6 \), we first need to determine the x-intercept of the original function.
1. Finding the x-intercept of the original function:
The x-intercept occurs where \( y = 0 \).
Set the equation to zero:
\[ 0 = 12x - 4 \]
Now solve for \( x \):
\[ 12x = 4 \] \[ x = \frac{4}{12} = \frac{1}{3} \]
So, the x-intercept of the original function is \( \left(\frac{1}{3}, 0\right) \).
2. Finding the x-intercept of the vertically dilated function:
The vertically dilated function is given by:
\[ y = k(12x - 4) = 6(12x - 4) = 72x - 24 \]
Now, we find the x-intercept of the dilated function by setting \( y = 0 \):
\[ 0 = 72x - 24 \]
Solving for \( x \):
\[ 72x = 24 \] \[ x = \frac{24}{72} = \frac{1}{3} \]
So, the x-intercept of the dilated function is also \( \left(\frac{1}{3}, 0\right) \).
Conclusion:
The x-intercepts for both functions are the same:
- The original function has an x-intercept of \( \left(\frac{1}{3}, 0\right) \).
- The dilated function also has an x-intercept of \( \left(\frac{1}{3}, 0\right) \).
None of the provided response options correctly reflect these results, as they mention (8,0) or other values that are not accurate.