To find the x-intercepts of the original function \( y = 12x - 4 \) and the vertically dilated function \( y = k(12x - 4) \) with a factor \( k = 6 \), we can proceed as follows:
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Find the x-intercept of the original function \( y = 12x - 4 \):
- Set \( y = 0 \): \[ 0 = 12x - 4 \]
- Solve for \( x \): \[ 12x = 4 \quad \Rightarrow \quad x = \frac{4}{12} = \frac{1}{3} \]
- The x-intercept of the original function is \( \left( \frac{1}{3}, 0 \right) \).
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Find the x-intercept of the vertically dilated function \( y = 6(12x - 4) \):
- Set \( y = 0 \): \[ 0 = 6(12x - 4) \]
- Since \( 6 \) is a multiplicative factor, we only need to solve the equation inside the parentheses: \[ 0 = 12x - 4 \]
- This is the same equation that we solved earlier: \[ 12x = 4 \quad \Rightarrow \quad x = \frac{4}{12} = \frac{1}{3} \]
- The x-intercept of the dilated function is also \( \left( \frac{1}{3}, 0 \right) \).
Both functions have the same x-intercept.
Thus, 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 \( \left( \frac{1}{3}, 0 \right) \).
(Note: None of the provided options matches the correct answer. The correct intercept for both functions is \( \left( \frac{1}{3}, 0 \right) \).)
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