To determine the x-intercepts of both the original function \( y = \frac{1}{2}x - 4 \) and the vertically dilated function with a factor of \( k = 6 \), we can start by finding the x-intercept of the original function.
Original Function:
To find the x-intercept, we set \( y = 0 \):
\[ 0 = \frac{1}{2}x - 4 \]
Now, we solve for \( x \):
\[ \frac{1}{2}x = 4 \]
Multiplying both sides by 2:
\[ x = 8 \]
Thus, the x-intercept of the original function \( y = \frac{1}{2}x - 4 \) is \( (8, 0) \).
Vertically Dilated Function:
The vertically dilated function can be represented as:
\[ y = k \left( \frac{1}{2}x - 4 \right) = 6 \left( \frac{1}{2}x - 4 \right) \]
Distributing the \( k = 6 \):
\[ y = 3x - 24 \]
Now, we find the x-intercept of this dilated function by setting \( y = 0 \):
\[ 0 = 3x - 24 \]
Now, we solve for \( x \):
\[ 3x = 24 \]
Dividing both sides by 3:
\[ x = 8 \]
Thus, the x-intercept of the dilated function \( y = 3x - 24 \) is also \( (8, 0) \).
Summary of X-Intercepts:
- Original function \( y = \frac{1}{2}x - 4 \): x-intercept \( (8, 0) \)
- Vertically dilated function \( y = 3x - 24 \): x-intercept \( (8, 0) \)
Conclusion:
Both functions have the same x-intercept, which is \( (8, 0) \). The correct answer from the options provided is:
- The original function has an x-intercept of (8,0) and the dilated function has an x-intercept of (8,0).