Two different relationships are shown below.

Relationship 1: xy = 3

Relationship 2: A graph is shown that has the following points plotted: one comma three, two comma six, three comma nine, four comma twelve.

a) Identify each relationship as direct or inverse variation.
b) Calculate the constant of each relationship and explain how you arrived at your answer.
c) Using complete sentences, explain how the two relationships are different, and how they are similar.

a) To identify whether a relationship is direct or inverse variation, we need to examine the equation or the pattern of the relationship.

For Relationship 1 (xy = 3), it is an inverse variation. In an inverse variation, the product of the variables is constant, which means that as one variable increases, the other variable decreases. In this case, as x increases, y decreases.

For Relationship 2 (the graph), we need to observe the pattern of the plotted points. In this case, as the x-values increase, so do the y-values. This indicates a direct variation. In a direct variation, as one variable increases, the other variable also increases.

b) To calculate the constant of each relationship, we need to rewrite the equations in a form that highlights the constant of variation.

For Relationship 1 (xy = 3), we can rewrite it as y = 3/x. The constant of variation is 3 since it remains the same when x and y change. In this case, y varies inversely with x, and the constant of variation is 3.

For Relationship 2 (graph), we can calculate the constant of variation by selecting any two points and computing the ratio of their y-values to their x-values.

Let's take the points (1, 3) and (2, 6):
The ratio for these two points is 3/1 = 6/2 = 3. Therefore, the constant of variation is 3. In this case, y varies directly with x, and the constant of variation is also 3.

c) The two relationships are different in their behavior. Relationship 1 shows an inverse variation, where as x increases, y decreases. This is represented by the equation y = 3/x. Relationship 2 shows a direct variation, where as x increases, y also increases. This is shown by the plotted points on the graph.

However, both relationships have a constant of variation of 3. This means that the product of the variables in Relationship 1 and the ratio of the variables in Relationship 2 is always equal to 3. This common constant shows that although they have different patterns, they still exhibit a consistent relationship between the variables.