To find the x-intercepts of the original function \( y = 12x - 4 \) and the vertically dilated function with a factor \( k = 6 \), we need to analyze both functions.
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Original Function \( y = 12x - 4 \): The x-intercept occurs when \( y = 0 \). \[ 0 = 12x - 4 \ 12x = 4 \ x = \frac{4}{12} = \frac{1}{3} \] So, the x-intercept for the original function is \( \left(\frac{1}{3}, 0\right) \).
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Vertically Dilated Function: A vertical dilation by a factor of \( k \) means multiplying the entire function by \( k \). Thus, the dilated function becomes: \[ y = 6(12x - 4) = 72x - 24 \] We find the x-intercept for the dilated function: \[ 0 = 72x - 24 \ 72x = 24 \ x = \frac{24}{72} = \frac{1}{3} \] So, the x-intercept for the dilated function is also \( \left(\frac{1}{3}, 0\right) \).
Summary of x-intercepts:
- Original function: \( \left(\frac{1}{3}, 0\right) \)
- Dilated function: \( \left(\frac{1}{3}, 0\right) \)
Conclusion: The correct response is: "The original function has an x-intercept of \( \left(\frac{1}{3}, 0\right) \), and the dilated function has an x-intercept of \( \left(\frac{1}{3}, 0\right) \)."
However, none of the provided responses match this conclusion, indicating an error in the options given.
1 answer