To find the x-intercepts of the original function \( y = 12x - 4 \) and the vertically dilated function \( y = 6(12x - 4) \), we can proceed as follows:
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Find the x-intercept of the original function:
The x-intercept occurs when \( y = 0 \): \[ 0 = 12x - 4 \] Adding 4 to both sides: \[ 4 = 12x \] Dividing by 12: \[ x = \frac{4}{12} = \frac{1}{3} \] So, the x-intercept of the original function is \( \left(\frac{1}{3}, 0\right) \).
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Determine the vertically dilated function:
The vertically dilated function with a factor of \( k = 6 \) is: \[ y = 6(12x - 4) = 72x - 24 \]
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Find the x-intercept of the dilated function:
Setting \( y = 0 \) for the dilated function: \[ 0 = 72x - 24 \] Adding 24 to both sides: \[ 24 = 72x \] Dividing by 72: \[ x = \frac{24}{72} = \frac{1}{3} \] So, the x-intercept of the dilated function is also \( \left(\frac{1}{3}, 0\right) \).
Conclusion:
The x-intercepts of both functions are:
- The original function has an x-intercept of \( \left(\frac{1}{3}, 0\right) \).
- The dilated function has an x-intercept of \( \left(\frac{1}{3}, 0\right) \).
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