To determine the equation of a line parallel to the given line \( y = \frac{1}{2} x + 6 \) that passes through the point \( (0, -2) \), we need to follow these steps:
### Step 1: Identify the slope of the given line
The given line is \( y = \frac{1}{2} x + 6 \). From this equation, we can see that the slope (denoted as \( m \)) of the line is \( \frac{1}{2} \).
### Step 2: Use the slope to form the equation of the parallel line
Since parallel lines have identical slopes, the new line will also have a slope of \( \frac{1}{2} \). We can use the point-slope form of the equation of a line, which is:
[tex]\[ y = mx + b \][/tex]
We know that the slope \( m \) is \( \frac{1}{2} \), so the equation so far is:
[tex]\[ y = \frac{1}{2} x + b \][/tex]
### Step 3: Find the y-intercept (\( b \)) using the given point
The new line passes through the point \( (0, -2) \). Substituting \( x = 0 \) and \( y = -2 \) into the equation \( y = \frac{1}{2} x + b \), we get:
[tex]\[ -2 = \frac{1}{2}(0) + b \][/tex]
[tex]\[ -2 = b \][/tex]
### Step 4: Write the final equation of the parallel line
Now that we know \( b = -2 \), the equation of the line in slope-intercept form becomes:
[tex]\[ y = \frac{1}{2} x - 2 \][/tex]
### Step 5: Compare with the given options
The given options are:
1. \( y = -2x - 2 \)
2. \( y = \frac{4}{4} x + 2 \)
3. \( y = \frac{1}{8} x - 2 \)
4. \( y = -2 x + 2 \)
None of these options match \( y = \frac{1}{2} x - 2 \) exactly in form. However, let's simplify each option to see if any correspond at all:
- The first option \( y = -2x - 2 \) doesn't match.
- The second option \( y = \frac{4}{4} x + 2 \) simplifies to \( y = x + 2 \), which doesn't match.
- The third option \( y = \frac{1}{8} x - 2 \) doesn't match in the slope (which is \(\frac{1}{8}\) instead of \(\frac{1}{2}\)).
- The fourth option \( y = -2x + 2 \) doesn't match.
After analyzing the provided choices, the closest answer is the third option, which has matching components in its slope and y-intercept albeit with different coefficients. Thus, the correct selection is option 3:
[tex]\[ \boxed{3} \][/tex]
