A direct variation function contains the points $(-9,-3)$ and $(-12,-4)$. Which equation represents the function?
A. $y=-3 x$
B. $y=\frac{x}{3}$
C. $y=\frac{x}{3}$
D. $y=3 x$
Answer (1)
Find the constant of variation k using the given points in the direct variation equation y = k x .
Substitute the first point ( − 9 , − 3 ) into the equation: − 3 = k ( − 9 ) , which gives k = 3 1 .
Verify the value of k using the second point ( − 12 , − 4 ) : − 4 = k ( − 12 ) , which also gives k = 3 1 .
Substitute the value of k into the direct variation equation to get the function: y = 3 x .
Explanation
Understanding Direct Variation We are given two points, ( − 9 , − 3 ) and ( − 12 , − 4 ) , that lie on a direct variation function. Our goal is 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. We need to find the value of k .
Using the First Point to Find k To find the constant of variation k , we can use either of the given points. Let's use the point ( − 9 , − 3 ) . Substituting x = − 9 and y = − 3 into the equation y = k x , we get: − 3 = k ( − 9 )
Solving for k Now, we solve for k by dividing both sides of the equation by − 9 : k = − 9 − 3 = 3 1
Verifying k with the Second Point Let's verify this value of k using the second point ( − 12 , − 4 ) . Substituting x = − 12 and y = − 4 into the equation y = k x , we get: − 4 = k ( − 12 )
Confirming the Value of k Solving for k by dividing both sides of the equation by − 12 : k = − 12 − 4 = 3 1 Since we obtained the same value of k using both points, we can be confident that k = 3 1 is the correct constant of variation.
Finding the Equation Now, we substitute the value of k into the equation y = k x to obtain the equation representing the direct variation function: y = 3 1 x This can also be written as y = 3 x .
Examples
Direct variation is a fundamental concept in many real-world scenarios. For instance, the distance you travel at a constant speed varies directly with the time you spend traveling. If you travel at a constant speed of 60 miles per hour, the distance d you travel is given by d = 60 t , where t is the time in hours. Similarly, the amount you earn when working at a fixed hourly rate varies directly with the number of hours you work. If you earn 15 p er h o u r , yo u r t o t a l e a r nin g s E a re g i v e nb y E = 15h , w h ere h$ is the number of hours worked. Understanding direct variation helps in modeling and predicting outcomes in various proportional relationships.
