Which linear function represents the line given by the point-slope equation $y-8=\frac{1}{2}(x-4)$?
A. $f(x)=\frac{1}{2} x+4$
B. $f(x)=\frac{1}{2} x+6$
C. $f(x)=\frac{1}{2} x-10$
D. $f(x)=\frac{1}{2} x-12$
Answer (2)
Distribute the slope: y − 8 = 2 1 ( x − 4 ) becomes y − 8 = 2 1 x − 2 .
Isolate y by adding 8 to both sides: y = 2 1 x − 2 + 8 .
Simplify to slope-intercept form: y = 2 1 x + 6 .
Express as a linear function: f ( x ) = 2 1 x + 6 , so the answer is f ( x ) = 2 1 x + 6 .
Explanation
Understanding the Problem The problem gives us a line in point-slope form and asks us to find the equivalent linear function in slope-intercept form. The point-slope form is given by y − y 1 = m ( x − x 1 ) , where ( x 1 , y 1 ) is a point on the line and m is the slope. The slope-intercept form is given by y = m x + b , where m is the slope and b is the y-intercept. Our goal is to rewrite the given equation in slope-intercept form and then express it as a linear function.
Distributing the Slope We start with the given equation: y − 8 = 2 1 ( x − 4 ) . To convert this to slope-intercept form, we need to isolate y on one side of the equation. First, distribute the 2 1 on the right side:
Simplifying the Equation y − 8 = 2 1 x − 2 1 ( 4 ) which simplifies to y − 8 = 2 1 x − 2 .
Isolating y Next, add 8 to both sides of the equation to isolate y : y = 2 1 x − 2 + 8 .
Slope-Intercept Form Simplify the equation: y = 2 1 x + 6 .
Linear Function Finally, replace y with f ( x ) to express the equation as a linear function: f ( x ) = 2 1 x + 6 .
Examples
Linear functions are incredibly useful in everyday life. For example, imagine you're saving money. If you start with $100 and save 20 e a c h w ee k , t h e t o t a l am o u n t yo u ha v ec anb ere p rese n t e d b y a l in e a r f u n c t i o n f(x) = 20x + 100 , w h ere x$ is the number of weeks. This helps you predict how much money you'll have in the future. Similarly, understanding linear functions helps in calculating distances, costs, and many other real-world scenarios.
The linear function represented by the point-slope equation y − 8 = 2 1 ( x − 4 ) is f ( x ) = 2 1 x + 6 . Therefore, the correct answer is option B. f ( x ) = 2 1 x + 6 .
;
