Which linear function represents the line given by the point-slope equation [tex]\( y - 8 = \frac{1}{2}(x - 4) \)[/tex]?

A. [tex]\( f(x) = \frac{1}{2} x + 4 \)[/tex]

B. [tex]\( f(x) = \frac{1}{2} x + 6 \)[/tex]

C. [tex]\( f(x) = \frac{1}{2} x - 10 \)[/tex]

D. [tex]\( f(x) = \frac{1}{2} x - 12 \)[/tex]



Answer :

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]