Come up with and describe two scenarios: one that models a direct variation situation and one that models an inverse variation situation. Do not state explicitly which scenario is which, but provide at least four data pairs for each situation. Your classmates will have to determine which of the scenarios is a direct variation and which is an inverse variation, and the value of k for each.
Scenario 1: Direct Variation Situation
In this scenario, we have two variables (let's call them x and y) that vary directly with each other. This means that as x increases, so does y, and as x decreases, y decreases proportionally. The relationship between x and y can be represented by the equation y = kx, where k is the constant of variation.
Data Pairs:
1. (2, 4)
2. (4, 8)
3. (6, 12)
4. (8, 16)
In this scenario, we observe that as x doubles, y also doubles, indicating a direct variation relationship. To find the value of k, we can choose any of the data pairs and plug them into the equation y = kx. For example, using the first data pair (2, 4), we have 4 = 2k. Solving for k, we find k = 2.
Scenario 2: Inverse Variation Situation
In this scenario, we have two variables (let's call them x and y) that vary inversely with each other. This means that as x increases, y decreases, and as x decreases, y increases in a way that their product remains constant. The relationship between x and y can be represented by the equation xy = k, where k is the constant of variation.
Data Pairs:
1. (1, 4)
2. (2, 2)
3. (4, 1)
4. (8, 0.5)
In this scenario, we observe that as x doubles, y is halved, indicating an inverse variation relationship. To find the value of k, we can choose any of the data pairs and multiply the values of x and y. For example, using the first data pair (1, 4), we have 1 * 4 = k. Therefore, k = 4.
In summary, scenario 1 represents direct variation, where y = kx, with a value of k = 2. Scenario 2 represents inverse variation, where xy = k, with a value of k = 4.