A direct variation function contains the points (-8, -6) and (12, 9). Which equation represents the function?
A direct variation function is represented by the equation \(y = kx\), where \(k\) is a constant of variation. This function passes through the origin \((0,0)\) and represents a linear relationship. To determine the equation, we need to find the constant \(k\) using the given coordinates \((-8, -6)\) and \((12, 9)\).
We substitute the coordinates of one of the given points into the equation \(y = kx\) to solve for \(k\). Using the point \((12, 9)\):
\[9 = k \times 12\]
Dividing both sides by \(12\), we find:
\[k = \frac{9}{12} = \frac{3}{4}\]
Now, we verify this value with the other point \((-8, -6)\):
\[-6 = k \times (-8) \implies k = \frac{-6}{-8} = \frac{3}{4}\]
Since both points yield the same constant \(k = \frac{3}{4}\), the equation of the function is \(y = \frac{3}{4}x\).
The calculated equation \(y = \frac{3}{4}x\) matches one of the provided options. The correct equation represents the relationship where \(y\) is directly proportional to \(x\) with a slope of \(\frac{3}{4}\). Thus, the final answer is \(y = \frac{3}{4}x\).
In real-life scenarios, direct variation describes relationships where two quantities change at the same rate. For example, if you are buying apples at a fixed price per pound, the total cost \(y\) is directly proportional to the weight \(x\) of the apples (\(y = kx\)). By knowing the price for one specific purchase, you can determine the constant of variation \(k\) (the price per pound) and predict the cost for any other weight.