To determine the x-intercepts of the original function \( y = \frac{1}{2}x - 4 \) and its vertically dilated function \( y = 6(\frac{1}{2}x - 4) \), we first find the x-intercept of the original function.
Step 1: Find the x-intercept of the original function
An x-intercept occurs when \( y = 0 \).
Setting the original function to zero: \[ 0 = \frac{1}{2}x - 4 \]
Now, solve for \( x \) by adding 4 to both sides: \[ 4 = \frac{1}{2}x \]
Next, multiply both sides by 2 to isolate \( x \): \[ x = 8 \]
Thus, the x-intercept of the original function is \( (8, 0) \).
Step 2: Find the x-intercept of the dilated function
The vertically dilated function with a factor \( k = 6 \) is: \[ y = 6\left(\frac{1}{2}x - 4\right) \]
Setting this new function to zero for the x-intercept: \[ 0 = 6\left(\frac{1}{2}x - 4\right) \]
Since \( 6 \) is not zero, we can simplify the equation: \[ 0 = \frac{1}{2}x - 4 \]
This is the same equation as earlier, which we already solved. Therefore, we get: \[ 4 = \frac{1}{2}x \] \[ x = 8 \]
Thus, the x-intercept of the vertically dilated function is also \( (8, 0) \).
Conclusion
- The original function has an x-intercept of \( (8, 0) \).
- The dilated function also has an x-intercept of \( (8, 0) \).
The correct response to the problem 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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