Final answer:
To find the equation that represents the direct variation function, we use the points given to solve for the constant of variation, k. After testing all options, we find that the correct equation is Option C: y = 2/3x, as it works for both points provided.
Step-by-step explanation:
To determine which equation represents the direct variation function containing the points (-9, -3) and (-12, 4), we first need to understand that a direct variation is a linear equation of the form y = kx, where k is the constant of variation. To find k, we can use the two points provided by solving for k in either point. For instance, with the point (-9, -3), we can set up the equation:
-3 = k(-9)
From this, we solve for k by dividing both sides by -9:
k = -3 / -9 = 1/3
However, we need to check if this value of k works for the other point (-12, 4) as well:
4 = (1/3)(-12) = -4
Since this does not hold true, the value of k = 1/3 cannot represent the direct variation for both points. Let's test the options provided one by one:
- A) y = -3x does not work for (-12, 4) since -3(-12) = 36, not 4.
- B) y = 2 is a constant function and does not even involve x; hence it cannot be the correct equation.
- C) y = 2/3x works for both points. For (-9, -3): -3 = (2/3)(-9), and for (-12, 4): 4 = (2/3)(-12).
- D) y = 3/4x does not satisfy either point.
Therefore, the correct equation that represents the direct variation function is Option C: y = 2/3x.