Choose a linear function for the line represented by the point-slope equation $y-5=3(x-2)$.
A. $f(x)=3 x+1$
B. $f(x)=3 x-1$
C. $f(x)=8 x+10$
D. $f(x)=8 x-10$
Answer (2)
Distribute the 3 in the point-slope equation: y − 5 = 3 x − 6 .
Add 5 to both sides to isolate y: y = 3 x − 6 + 5 .
Simplify the equation: y = 3 x − 1 .
Express as a linear function: f ( x ) = 3 x − 1 . The final answer is f ( x ) = 3 x − 1 .
Explanation
Understanding the Problem The problem gives us the point-slope form of a linear equation and asks us to identify the equivalent linear function from a set of options. Our goal is to convert the point-slope form into the slope-intercept form, which is y = m x + b , and then express it as a function f ( x ) = m x + b .
Isolating y We start with the point-slope equation: y − 5 = 3 ( x − 2 ) . To convert this to slope-intercept form, we need to isolate y on one side of the equation.
Distributing the 3 First, distribute the 3 on the right side of the equation:
y − 5 = 3 x − 6
Adding 5 to both sides Next, add 5 to both sides of the equation to isolate y :
y = 3 x − 6 + 5
Simplifying the equation Simplify the equation:
y = 3 x − 1
Expressing as a linear function Now, express the equation as a linear function:
f ( x ) = 3 x − 1
Identifying the correct option Finally, we compare our result with the given options. The correct linear function is f ( x ) = 3 x − 1 .
Examples
Linear functions are used in many real-world applications, such as calculating the cost of a taxi ride. For example, a taxi company might charge a fixed fee plus a per-mile rate. If the fixed fee is $3 and the per-mile rate is 2 , t h e t o t a l cos t f(x) f or a r i d eo f x mi l esc anb ere p rese n t e d b y t h e l in e a r f u n c t i o n f(x) = 2x + 3$. This allows you to easily calculate the cost of any ride by simply plugging in the number of miles.
By converting the point-slope equation y − 5 = 3 ( x − 2 ) to slope-intercept form, we find that the linear function is f ( x ) = 3 x − 1 . The correct option from the provided choices is B. f ( x ) = 3 x − 1 .
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