A direct variation function contains the points $(2,14)$ and $(4,28)$. Which equation represents the function?
A. $y=\frac{x}{14}$
B. $y=\frac{x}{7}$
C. $y=7x$
D. $y=14x$
Answer (2)
Substitute the given points into the direct variation equation y = k x .
Solve for the constant of variation k using one of the points: k = 2 14 = 7 .
Verify the value of k using the other point: k = 4 28 = 7 .
Write the equation of the direct variation function: y = 7 x . The final answer is y = 7 x .
Explanation
Understanding the Problem We are given that a direct variation function contains the points ( 2 , 14 ) and ( 4 , 28 ) . We need to find the equation that represents this function. A direct variation function has the form y = k x , where k is the constant of variation.
Finding the Constant of Variation To find the constant of variation k , we can use either of the given points. Let's use the point ( 2 , 14 ) . Substituting these values into the equation y = k x , we get: 14 = k e q 2
Solving for k Now, we solve for k by dividing both sides of the equation by 2: k = 2 14 = 7
Verifying k We can verify this value of k using the other point ( 4 , 28 ) . Substituting these values into the equation y = k x , we get: 28 = k e q 4
Solving for k (again) Solving for k by dividing both sides of the equation by 4: k = 4 28 = 7
Writing the Equation Since the value of k is the same for both points, we can write the equation of the direct variation function as: y = 7 x
Examples
Direct variation is a fundamental concept in many real-world scenarios. For example, the distance you travel at a constant speed varies directly with the time you spend traveling. If you travel at 60 miles per hour, the equation representing this relationship is d = 60 t , where d is the distance and t is the time. Similarly, the amount you earn at a fixed hourly rate varies directly with the number of hours you work. If you earn 15 p er h o u r , t h ee q u a t i o ni s e = 15h , w h ere e i syo u re a r nin g s an d h$ is the number of hours worked. Understanding direct variation helps in predicting outcomes and managing resources effectively.
The equation representing the direct variation function that passes through the points ( 2 , 14 ) and ( 4 , 28 ) is y = 7 x . Therefore, the correct answer is C. y = 7 x .
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