Question

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

• $f(x)=\frac{1}{2}x + 4$
• $f(x)=\frac{1}{2}x + 6$
• $f(x)=\frac{1}{2}x - 10$
• $f(x)=\frac{1}{2}x - 12$

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 \(y-8=\frac{1}{2}x-\frac{1}{2}\times4\).
Simplify \(\frac{1}{2}\times4 = 2\), so the equation becomes \(y-8=\frac{1}{2}x - 2\).

Step2: Solve for y

To get the equation in slope - intercept form (\(y=mx + b\) or \(f(x)=mx + b\)), we add 8 to both sides of the equation.
\(y=\frac{1}{2}x-2 + 8\).
Simplify \(-2 + 8=6\), so \(y=\frac{1}{2}x + 6\), which is \(f(x)=\frac{1}{2}x+6\).

Answer:

\(f(x)=\frac{1}{2}x + 6\) (the option with this function, e.g., if the options are labeled as above, it would be the second option: \(f(x)=\frac{1}{2}x + 6\))