To determine the x-intercepts of the original function \( y = 12x - 4 \) and the vertically dilated function with a factor \( k = 6 \), we first need to find the x-intercept of the original function.
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Original Function:
Set \( y = 0 \) to find the x-intercept: \[ 0 = 12x - 4 \] Solving for \( x \): \[ 12x = 4 \ x = \frac{4}{12} = \frac{1}{3} \] Therefore, the x-intercept of the original function is \( \left(\frac{1}{3}, 0\right) \). -
Vertically Dilated Function:
The vertically dilated function with a factor of \( k = 6 \) can be written as: \[ y = 6(12x - 4) = 72x - 24 \] To find the x-intercept, set \( y = 0 \): \[ 0 = 72x - 24 \] Solving for \( x \): \[ 72x = 24 \ x = \frac{24}{72} = \frac{1}{3} \] Thus, the x-intercept of the dilated function is also \( \left(\frac{1}{3}, 0\right) \).
Conclusion:
Both the original function \( y = 12x - 4 \) and the dilated function \( y = 72x - 24 \) have the same x-intercept at \( \left(\frac{1}{3}, 0\right) \).
The options you provided do not seem to reflect this outcome. However, the accurate conclusion based on calculations is that both functions have an x-intercept of \( \left(\frac{1}{3}, 0\right) \).
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