To identify the linear function that represents the line given by the point-slope equation [tex]\( y - 8 = \frac{1}{2}(x - 4) \)[/tex], follow these steps:
1. Start with the given point-slope equation:
[tex]\[
y - 8 = \frac{1}{2}(x - 4)
\][/tex]
2. Distribute the slope (which is [tex]\(\frac{1}{2}\)[/tex]) on the right-hand side:
[tex]\[
y - 8 = \frac{1}{2}x - \frac{1}{2} \cdot 4
\][/tex]
3. Calculate [tex]\(\frac{1}{2} \cdot 4\)[/tex]:
[tex]\[
y - 8 = \frac{1}{2}x - 2
\][/tex]
4. Isolate [tex]\(y\)[/tex] to convert the equation into slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[
y = \frac{1}{2}x - 2 + 8
\][/tex]
5. Combine the constants on the right-hand side:
[tex]\[
y = \frac{1}{2}x + 6
\][/tex]
Thus, the equation of the line in slope-intercept form is [tex]\( y = \frac{1}{2}x + 6 \)[/tex].
So, the correct linear function representing the given point-slope equation is:
[tex]\[
f(x) = \frac{1}{2} x + 6
\][/tex]
