To find the x-intercepts of both the original function \( y = 12x - 4 \) and the vertically dilated function with a factor \( k = 6 \), we can break the steps down as follows:
1. Finding the x-intercept of the original function
The x-intercept occurs where \( y = 0 \). For the original function: \[ 0 = 12x - 4 \] Adding 4 to both sides: \[ 4 = 12x \] Dividing both sides by 12: \[ x = \frac{4}{12} = \frac{1}{3} \]
Thus, the x-intercept of the original function is: \[ \left(\frac{1}{3}, 0\right) \]
2. Finding the x-intercept of the vertically dilated function
The vertically dilated function with a factor of \( k = 6 \) is: \[ y = 6(12x - 4) = 72x - 24 \] To find the x-intercept for this function: \[ 0 = 72x - 24 \] Adding 24 to both sides: \[ 24 = 72x \] Dividing both sides by 72: \[ x = \frac{24}{72} = \frac{1}{3} \]
Thus, the x-intercept of the dilated function is: \[ \left(\frac{1}{3}, 0\right) \]
Conclusion
After calculating, we find that:
- The original function \( y = 12x - 4 \) has an x-intercept of \(\left(\frac{1}{3}, 0\right)\)
- The vertically dilated function \( y = 72x - 24 \) also has an x-intercept of \(\left(\frac{1}{3}, 0\right)\)
None of the responses provided match the correct x-intercept calculations.