Which linear function represents the line given by the point-slope equation $y-2=4(x-3)$?
A. $f(x)=6 x-1$
B. $f(x)=8 x-6$
C. $f(x)=4 x-14$
D. $f(x)=4 x-10
Answer (1)
Distribute the 4 in the point-slope equation: y − 2 = 4 x − 12 .
Add 2 to both sides to isolate y : y = 4 x − 10 .
Express the equation as a linear function: f ( x ) = 4 x − 10 .
The linear function that represents the line is f ( x ) = 4 x − 10 .
Explanation
Understanding the Problem We are given the point-slope equation of a line: y − 2 = 4 ( x − 3 ) . Our goal is to find the linear function f ( x ) that represents this line. The options are in the form f ( x ) = m x + b , where m is the slope and b is the y-intercept. We need to convert the given point-slope equation into slope-intercept form.
Distributing the 4 To convert the point-slope equation to slope-intercept form, we first distribute the 4 on the right side of the equation:
y − 2 = 4 ( x − 3 )
y − 2 = 4 x − 12
Isolating y Next, we add 2 to both sides of the equation to isolate y :
y − 2 + 2 = 4 x − 12 + 2
y = 4 x − 10
Expressing as a Linear Function Now we have the equation in slope-intercept form: y = 4 x − 10 . We can express this as a linear function: f ( x ) = 4 x − 10 .
Final Answer Comparing our result with the given options, we find that the correct linear function is f ( x ) = 4 x − 10 .
Examples
Linear functions are used in many real-world applications. For example, a taxi fare can be modeled as a linear function of the distance traveled. If the initial fare is $3 and the cost per mile is 2 , t h e t o t a l f a re f(x) f or x mi l esc anb ere p rese n t e d a s f(x) = 2x + 3$. Similarly, the relationship between temperature in Celsius and Fahrenheit is a linear function. Understanding linear functions helps in predicting and analyzing various phenomena in everyday life.
