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 2 1 on the right side of the equation: 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 the equation: y = 2 1 x + 6 .
Express as a linear function: f ( x ) = 2 1 x + 6 . The final answer is f ( x ) = 2 1 x + 6 .
Explanation
Understanding the problem The equation of the line is given in point-slope form: y − 8 = 2 1 ( x − 4 ) . We need to convert this equation to the slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept. The linear function can be written as f ( x ) = y = m x + b .
Distribute First, distribute the 2 1 on the right side of the equation:
y − 8 = 2 1 x − 2 1 ( 4 )
Simplify Simplify the right side of the equation:
y − 8 = 2 1 x − 2
Isolate y Isolate y by adding 8 to both sides of the equation:
y = 2 1 x − 2 + 8
Simplify Simplify to get the equation in slope-intercept form:
y = 2 1 x + 6
Final Answer Express the equation as a linear function:
f ( x ) = 2 1 x + 6
So, the linear function that represents the line given by the point-slope equation y − 8 = 2 1 ( x − 4 ) is f ( x ) = 2 1 x + 6 .
Examples
Understanding linear functions is crucial in many real-world applications. For instance, imagine you're tracking the distance a car travels over time at a constant speed. The equation f ( x ) = 2 1 x + 6 could represent this, where f ( x ) is the distance in kilometers, x is the time in minutes, 2 1 is the speed in km/min, and 6 is the initial distance from a starting point. By understanding this function, you can easily calculate the car's position at any given time, which is essential for navigation and logistics.
The linear function for the point-slope equation y − 8 = 2 1 ( x − 4 ) is f ( x ) = 2 1 x + 6 . Therefore, the correct answer is option B. This represents the relationship in slope-intercept form with a slope of 2 1 and a y-intercept of 6.
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