Question

which linear function represents the line given by the point - slope equation $y - 8=\frac{1}{2}(x - 4)$?
$\bigcirc$ $f(x)=\frac{1}{2}x + 4$
$\bigcirc$ $f(x)=\frac{1}{2}x + 6$
$\bigcirc$ $f(x)=\frac{1}{2}x - 10$
$\bigcirc$ $f(x)=\frac{1}{2}x - 12$

Explanation:

Step1: Expand the point - slope form

We start with the point - slope equation \(y - 8=\frac{1}{2}(x - 4)\). Using the distributive property \(a(b - c)=ab - ac\), where \(a=\frac{1}{2}\), \(b = x\) and \(c = 4\), we get \(y-8=\frac{1}{2}x-\frac{1}{2}\times4\).
Simplifying \(\frac{1}{2}\times4 = 2\), so the equation becomes \(y - 8=\frac{1}{2}x-2\).

Step2: Solve for y (or f(x))

To isolate \(y\), we add 8 to both sides of the equation.
\(y=\frac{1}{2}x-2 + 8\).
Simplifying \(-2 + 8=6\), we get \(y=\frac{1}{2}x + 6\). Since \(y = f(x)\), then \(f(x)=\frac{1}{2}x+6\).

Answer:

B. \(f(x)=\frac{1}{2}x + 6\)