A direct variation function contains the points $(-8,-6)$ and $(12,9)$. Which equation represents the function?
A. $y=-\frac{4}{3} x$
B. $y=\frac{3}{4} x$
C. $y=\frac{3}{4} x$
D. $y=\frac{4}{3} x$
Answer (2)
Recognize that a direct variation function has the form y = k x .
Use the given point ( − 8 , − 6 ) to find the constant of variation: k = − 8 − 6 = 4 3 .
Verify the constant of variation using the point ( 12 , 9 ) : k = 12 9 = 4 3 .
Write the equation of the direct variation function: y = 4 3 x . The final answer is y = 4 3 x .
Explanation
Understanding the Problem We are given that the function is a direct variation and passes through the points ( − 8 , − 6 ) and ( 12 , 9 ) . 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 .
Finding the Constant of Variation We can use either of the given points to find the constant of variation k . Let's use the point ( − 8 , − 6 ) . Substituting these values into the equation y = k x , we get: − 6 = k ( − 8 ) To solve for k , we divide both sides of the equation by − 8 :
k = − 8 − 6 = 4 3 So, k = 4 3 .
Verifying the Constant of Variation Now let's verify this value of k using the other point ( 12 , 9 ) . Substituting these values into the equation y = k x , we get: 9 = k ( 12 ) To solve for k , we divide both sides of the equation by 12 :
k = 12 9 = 4 3 So, k = 4 3 . Since we obtained the same value of k using both points, we can be confident that this is the correct constant of variation.
Writing the Equation Now that we have found the constant of variation k = 4 3 , we can write the equation of the direct variation function as: y = 4 3 x Thus, the equation that represents the direct variation function is y = 4 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 are driving at a steady 60 miles per hour, the equation d = 60 t represents this relationship, 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 , yo u re a r nin g s e c anb ere p rese n t e d a s e = 15h , w h ere h$ is the number of hours worked. Understanding direct variation helps in predicting outcomes and managing resources in various everyday situations.
The equation that represents the direct variation function passing through the given points is y = 4 3 x . The correct answer is option B. This was confirmed by finding and verifying the constant of variation using both provided points.
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