Let's start by finding the x-intercepts of the original function and the vertically dilated function.
The original function is given as: \[ y = \frac{1}{2}x - 4 \]
To find the x-intercept, we set \( y = 0 \) and solve for \( x \): \[ 0 = \frac{1}{2}x - 4 \] \[ \frac{1}{2}x = 4 \] \[ x = 4 \cdot 2 \] \[ x = 8 \]
So the x-intercept of the original function is \( (8, 0) \).
Next, let's find the vertically dilated function. The vertically dilated function with a factor \( k = 6 \) is: \[ y = 6\left(\frac{1}{2}x - 4\right) \] Let's simplify this: \[ y = 6 \cdot \frac{1}{2}x - 6 \cdot 4 \] \[ y = 3x - 24 \]
Now we can find the x-intercept of the dilated function by setting \( y = 0 \) again: \[ 0 = 3x - 24 \] \[ 3x = 24 \] \[ x = \frac{24}{3} \] \[ x = 8 \]
So the x-intercept of the dilated function is also \( (8, 0) \).
In summary:
- The original function has an x-intercept of \( (8, 0) \).
- The dilated function has an x-intercept of \( (8, 0) \).
The correct response is: The original function has an x-intercept of (8, 0), and the dilated function has an x-intercept of (8, 0).
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