Which linear function represents the line given by the point-slope equation $y+1=-3(x-5)$ ?

A. $f(x)=-3 x-6$
B. $f(x)=-3 x-4$
C. $f(x)=-3 x+16$
D. $f(x)=-3 x+14$

Distribute the -3: y + 1 = − 3 x + 15 .
Isolate y: y = − 3 x + 14 .
Express as a linear function: f ( x ) = − 3 x + 14 .
The linear function representing the line is f ( x ) = − 3 x + 14 ​ .

Explanation

Understanding the Problem We are given the point-slope form of a linear equation: y + 1 = − 3 ( x − 5 ) . Our goal is to rewrite this equation in the slope-intercept form, which is f ( x ) = m x + b , where m is the slope and b is the y-intercept. This will allow us to identify the correct linear function from the given options.

Distributing the -3 First, we distribute the − 3 on the right side of the equation:


y + 1 = − 3 x + 15

Isolating y Next, we isolate y by subtracting 1 from both sides of the equation:

y = − 3 x + 15 − 1
y = − 3 x + 14

Expressing as a Linear Function Now, we can express the equation as a linear function f ( x ) :

f ( x ) = − 3 x + 14
Comparing this with the given options, we find that the correct linear function is f ( x ) = − 3 x + 14 .

Final Answer Therefore, the linear function that represents the given line is f ( x ) = − 3 x + 14 .

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 $3 and the per-mile rate is 2 , t h e t o t a l cos t y f or a r i d eo f x mi l esc anb ere p rese n t e d b y t h e l in e a r f u n c t i o n y = 2x + 3$. Understanding linear functions helps you predict and calculate costs in various scenarios.

Answered by GinnyAnswer | 2025-07-03

The linear function corresponding to the point-slope equation y + 1 = − 3 ( x − 5 ) is f ( x ) = − 3 x + 14 . Therefore, the correct answer is option D. This represents the line in slope-intercept form, where the slope is -3 and the y-intercept is 14.
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Answered by Anonymous | 2025-07-04