To determine which linear function represents the given point-slope equation [tex]\( y - 8 = \frac{1}{2}(x - 4) \)[/tex], we need to convert it into the slope-intercept form, [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
1. Start with the point-slope equation:
[tex]\[
y - 8 = \frac{1}{2}(x - 4)
\][/tex]
2. Distribute [tex]\(\frac{1}{2}\)[/tex] on the right side of the equation:
[tex]\[
y - 8 = \frac{1}{2} \cdot x - \frac{1}{2} \cdot 4
\][/tex]
Simplifying the multiplication on the right side:
[tex]\[
y - 8 = \frac{1}{2} x - 2
\][/tex]
3. Add 8 to both sides to solve for [tex]\( y \)[/tex]:
[tex]\[
y - 8 + 8 = \frac{1}{2} x - 2 + 8
\][/tex]
Simplifying this gives:
[tex]\[
y = \frac{1}{2} x + 6
\][/tex]
So, the function that represents the line is:
[tex]\[
f(x) = \frac{1}{2} x + 6
\][/tex]
Therefore, the correct linear function is:
[tex]\[
f(x)=\frac{1}{2} x+6
\][/tex]
