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

A. [tex]r(x) = \frac{1}{2}x + 4[/tex]
B. [tex]r(x) = \frac{1}{2}x + 6[/tex]
C. [tex]f(x) = \frac{1}{2}x - 10[/tex]
D. [tex]y(x) = \frac{1}{2}x - 12[/tex]

Sagot :

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]