To graph the original function \( y = \frac{1}{2}x - 4 \) and the vertically dilated function with a factor of \( k = 6 \), we first need to write the equation for the vertically dilated function.
Step 1: Write the vertically dilated function
A vertical dilation by a factor of \( k \) means that we multiply the entire function by \( k \). Therefore, the vertically dilated function is:
\[ y = 6 \left( \frac{1}{2}x - 4 \right) = 3x - 24 \]
Step 2: Find the x-intercepts
The x-intercept occurs when \( y = 0 \).
For the original function \( y = \frac{1}{2}x - 4 \):
Set \( y = 0 \):
\[ 0 = \frac{1}{2}x - 4 \]
Add 4 to both sides:
\[ 4 = \frac{1}{2}x \]
Multiply both sides by 2:
\[ x = 8 \]
So, the x-intercept of the original function is \( (8, 0) \).
For the vertically dilated function \( y = 3x - 24 \):
Set \( y = 0 \):
\[ 0 = 3x - 24 \]
Add 24 to both sides:
\[ 24 = 3x \]
Divide both sides by 3:
\[ x = 8 \]
So, the x-intercept of the vertically dilated function is also \( (8, 0) \).
Summary:
- The x-intercept of the original function \( y = \frac{1}{2}x - 4 \) is \( (8, 0) \).
- The x-intercept of the vertically dilated function \( y = 3x - 24 \) is also \( (8, 0) \).
Both functions have the same x-intercept at \( x = 8 \).
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