Mr. Shaw graphs the function [tex]f(x)=-5 x+2[/tex] for his class. The line contains the point [tex](-2,12)[/tex]. What is the point-slope form of the equation of the line that he graphed?
A. [tex]y-12=-5(x+2)[/tex]
B. [tex]y-12=2(x+2)[/tex]
C. [tex]y+12=2(x-2)[/tex]
D. [tex]y+12=-5(x-2)[/tex]
Answer (2)
The point-slope form of a line is y − y 1 = m ( x − x 1 ) .
Identify the slope m = − 5 from the given function f ( x ) = − 5 x + 2 .
Identify the point ( x 1 , y 1 ) = ( − 2 , 12 ) on the line.
Substitute the slope and the point into the point-slope form to get y − 12 = − 5 ( x + 2 ) .
The point-slope form of the equation of the line is y − 12 = − 5 ( x + 2 ) .
Explanation
Understanding the Problem We are given the function f ( x ) = − 5 x + 2 and the point ( − 2 , 12 ) on the line. We want to find the point-slope form of the equation of the line.
Recalling Point-Slope Form The point-slope form of a line 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 of the line.
Identifying the Slope From the given function f ( x ) = − 5 x + 2 , we can identify the slope m as the coefficient of x , which is − 5 .
Identifying a Point on the Line We are given the point ( − 2 , 12 ) on the line, so we can use x 1 = − 2 and y 1 = 12 .
Substituting and Simplifying Substituting the slope m = − 5 and the point ( x 1 , y 1 ) = ( − 2 , 12 ) into the point-slope form, we get:
y − 12 = − 5 ( x − ( − 2 ))
Simplifying the equation, we have:
y − 12 = − 5 ( x + 2 )
Final Answer Therefore, the point-slope form of the equation of the line is y − 12 = − 5 ( x + 2 ) .
Examples
The point-slope form is useful in many real-world applications. For example, if you know the rate at which a savings account is growing (the slope) and the amount in the account at a particular time (a point), you can use the point-slope form to write an equation that models the amount in the account at any time. This allows you to predict future savings or determine how long it will take to reach a specific savings goal. Another example is determining the equation of a road given its slope and a point on the road on a map.
The point-slope form of the equation of the line represented by the function f ( x ) = − 5 x + 2 and passing through the point ( − 2 , 12 ) is y − 12 = − 5 ( x + 2 ) . Thus, the correct option is A.
;
