To find the x-intercepts of the original function \( y = 12x - 4 \) and the vertically dilated function with a factor \( k = 6 \), we start by calculating the x-intercepts for both functions.
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Original Function: \[ y = 12x - 4 \] To find the x-intercept, set \( y = 0 \): \[ 0 = 12x - 4 \] Solving for \( x \): \[ 12x = 4 \ x = \frac{4}{12} = \frac{1}{3} \] So the x-intercept of the original function is \( \left(\frac{1}{3}, 0\right) \).
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Vertically Dilated Function: The vertically dilated function is given by: \[ y = 6(12x - 4) = 72x - 24 \] Now, we find the x-intercept 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 \( \left(\frac{1}{3}, 0\right) \).
Summary:
- The original function has an x-intercept of \( \left(\frac{1}{3}, 0\right) \).
- The vertically dilated function also has an x-intercept of \( \left(\frac{1}{3}, 0\right) \).
None of the response options you provided match the findings, but the correct intercepts for both functions are indeed \( \left(\frac{1}{3}, 0\right) \).