A line is drawn through $(-4,3)$ and $(4,3)$. Which describes whether or not the line represents a direct variation?
A. The line represents a direct variation because $-\frac{4}{3}=\frac{4}{3}$.
B. The line represents a direct variation because it is horizontal.
C. The line does not represent a direct variation because it does not go through the origin.
D. The line does not represent a direct variation because $-4(3) \neq 4(3)$.
Answer (2)
Calculate the slope of the line passing through ( − 4 , 3 ) and ( 4 , 3 ) , which is m = 0 .
Determine the equation of the line, which is y = 3 .
Check if the line passes through the origin ( 0 , 0 ) . Since y = 3 , it does not pass through the origin.
Conclude that the line does not represent a direct variation because it does not pass through the origin. The line does not represent a direct variation because it does not go through the origin.
Explanation
Understanding Direct Variation We are given two points, ( − 4 , 3 ) and ( 4 , 3 ) , and we need to determine if the line passing through these points represents a direct variation. A direct variation is a linear relationship of the form y = k x , where k is a constant. A key characteristic of a direct variation is that the line must pass through the origin ( 0 , 0 ) .
Finding the Equation of the Line First, let's find the equation of the line passing through the points ( − 4 , 3 ) and ( 4 , 3 ) .
The slope of the line, m , is given by: m = x 2 − x 1 y 2 − y 1 = 4 − ( − 4 ) 3 − 3 = 8 0 = 0
Since the slope is 0, the line is horizontal. The equation of a horizontal line is of the form y = c , where c is a constant. In this case, since the line passes through ( 4 , 3 ) and ( − 4 , 3 ) , the equation of the line is y = 3 .
Checking for Direct Variation Now, we need to check if this line represents a direct variation. For a direct variation, the line must pass through the origin ( 0 , 0 ) . Since the equation of the line is y = 3 , when x = 0 , y = 3 . Therefore, the line does not pass through the origin.
Conclusion Since the line y = 3 does not pass through the origin, it does not represent a direct variation.
Examples
Direct variation is a relationship between two variables in which one is a constant multiple of the other. For example, the distance traveled at a constant speed varies directly with time. If a car travels at 60 miles per hour, the distance d is related to the time t by the equation d = 60 t . This is a direct variation because the distance is a constant multiple (60) of the time. Another example is the relationship between the number of items and the total cost when each item has the same price. If each apple costs 0.75 , t h e t o t a l cos t C f or n a ppl es i s C = 0.75n$, which is a direct variation.
The line through the points ( − 4 , 3 ) and ( 4 , 3 ) is horizontal and described by the equation y = 3 . Since it does not pass through the origin, it does not represent a direct variation. Therefore, the correct answer is option C.
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