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. $y=\frac{x}{5}$
D. $y=3 x$
Answer (2)
Find the constant of variation k using one of the given points in the direct variation equation y = k x .
Substitute x = − 9 and y = − 3 into y = k x to get − 3 = k ( − 9 ) , which gives k = 3 1 .
Write the equation of the direct variation function as y = 3 1 x .
The equation that represents the direct variation function is y = 3 x .
Explanation
Understanding the Problem 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.
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 ( − 9 , − 3 ) . Substituting x = − 9 and y = − 3 into the equation y = k x , we get: − 3 = k ( − 9 ) To solve for k , we divide both sides of the equation by − 9 :
k = − 9 − 3 = 3 1 So, k = 3 1 .
Writing the Equation Now that we have found the constant of variation k = 3 1 , we can write the equation of the direct variation function as: y = 3 1 x This can also be written as: y = 3 x .
Final Answer Comparing the equation y = 3 x with the given options, we see that it matches the option y = 3 x . Therefore, the equation that represents the direct variation function is 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 drive at 60 miles per hour, the distance you cover is d = 60 t , where t is the time in hours. 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 to make the same recipe for 8 people, you would need 4 cups of flour, demonstrating a direct variation.
The direct variation equation based on the points ( − 9 , − 3 ) and ( − 12 , − 4 ) is y = 3 x after determining the constant of variation k = 3 1 . However, this specific equation is not listed among the available options provided. This indicates a potential oversight in the question's options.
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