To find the linear function that represents the line given by the point-slope equation [tex]\( y - 8 = \frac{1}{2}(x - 4) \)[/tex], we need to transform it into the slope-intercept form [tex]\( y = mx + b \)[/tex].
Here are the steps:
1. Start with the point-slope form equation:
[tex]\[
y - 8 = \frac{1}{2}(x - 4)
\][/tex]
2. Distribute the [tex]\(\frac{1}{2}\)[/tex] on the right-hand side:
[tex]\[
y - 8 = \frac{1}{2} x - \frac{1}{2} \cdot 4
\][/tex]
[tex]\[
y - 8 = \frac{1}{2} x - 2
\][/tex]
3. Isolate [tex]\(y\)[/tex] by adding 8 to both sides of the equation:
[tex]\[
y - 8 + 8 = \frac{1}{2} x - 2 + 8
\][/tex]
[tex]\[
y = \frac{1}{2} x + 6
\][/tex]
Thus, the linear function that represents the line is:
[tex]\[
r(x) = \frac{1}{2} x + 6
\][/tex]
Therefore, the correct option is:
[tex]\[
r(x) = \frac{1}{2} x + 6
\][/tex]
