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 points ( − 8 , − 6 ) and ( 12 , 9 ) to solve for the constant of variation k .
Calculate k using the first point: k = − 8 − 6 = 4 3 .
Substitute k into the direct variation equation: y = 4 3 x . Therefore, the equation representing the function is y = 4 3 x .
Explanation
Understanding the Problem We are given that a direct variation function contains the points ( − 8 , − 6 ) and ( 12 , 9 ) . 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. We will use the given points to find the constant of variation k .
Finding the Constant of Variation Using the point ( − 8 , − 6 ) , we can write the equation − 6 = k ( − 8 ) . Solving for k , we get k = − 8 − 6 = 4 3 Using the point ( 12 , 9 ) , we can write the equation 9 = k ( 12 ) . Solving for k , we get k = 12 9 = 4 3 Since the value of k is the same for both points, we can be confident that this is a direct variation.
Writing the Equation Now, we substitute the value of k into the equation y = k x to obtain the equation of the direct variation function: 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're 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, in cooking, the amount of ingredients you need often varies directly with the number of servings you want to make. If a recipe for 4 people requires 2 cups of flour, then for 8 people, you'll need 4 cups, demonstrating a direct variation.
The equation representing the direct variation function containing the points ( − 8 , − 6 ) and ( 12 , 9 ) is y = 4 3 x . This was determined by finding the constant of variation k from the given points. Therefore, the correct choice is Option C: y = 4 3 x .
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