Problem 9 - Algebra 2

Question

Is the relationship between the values in each table a direct variation, an inverse variation, or neither? Write equations to model the direct and inverse variations. $$ \begin{array}{|c|c|c|c|c|}\hline x & {0.5} & {2.1} & {3.5} & {11} \\ \hline y & {1} & {4.2} & {7} & {22} \\ \hline\end{array} $$

Step-by-Step Solution

Answer

The relationship between the values in each table is a direct variation, not an inverse variation. The equation modeling the direct variation is \(y = 2x\). There is no equation modeling an inverse variation since the relationship does not display this type of variation.

Step 1: Identifying Direct Variation

Check if the values follow a direct variation by dividing each y value by its corresponding x value (i.e., y/x): For (0.5, 1), 1/0.5 = 2. For (2.1, 4.2), 4.2/2.1 = 2. For (3.5, 7), 7/3.5 = 2. For (11, 22), 22/11 = 2. So, the pairs in the table do follow a direct variation, as the value of y/x is constant, i.e., 2.

Step 2: Writing Equation for Direct Variation

The equation representing the direct variation can be found using the formula y = kx. Here, k, the constant of variation, is 2 as determined above. So, for our case, our direct variation equation becomes, y = 2x.

Step 3: Identifying Inverse Variation

Even though we have already identified this as a direct variation, for completeness, we'll test for inverse variation. For inverse variation, xy should remain constant: For (0.5, 1), 0.5*1 = 0.5. For (2.1, 4.2), 2.1*4.2 = 8.82. The value of xy is not constant, so it's not an inverse variation.

Step 4: No Inverse Variation

Since the values of xy are not constant across the pairs, we can confidently say that there's no inverse variation between x and y in the given data set. Hence, no equation for inverse variation will be derived.

Key Concepts

Inverse Variation

In mathematics, inverse variation describes a relationship between two variables where one variable increases as the other decreases. This means the product of the variables remains constant. We can express this relationship with the formula \(x \cdot y = k\), where \(k\) is the constant of variation.

Consider two variables, \(x\) and \(y\), in a situation where they are inversely proportional. If you multiply each pair of these variables, the product should yield the same constant every time. However, when we tested the data from the exercise:

• For \((0.5, 1)\), the product \(0.5 \cdot 1 = 0.5\)
• For \((2.1, 4.2)\), the product \(2.1 \cdot 4.2 = 8.82\)

Although these calculations show different products, indicating that the values do not follow an inverse variation, it’s important to test for such relationships to confirm true mathematical models. In the exercise, since the product was not consistent, we concluded there was no inverse variation.

Constant of Variation

A constant of variation is a number that defines the proportional relationship between two variables. In direct variation, it is also known as the "constant of proportionality." This constant can be found from the equation \(y = kx\), where \(k\) represents the constant of variation.

In the exercise problem, by dividing \(y\) by \(x\) for each pair:

• \(1/0.5 = 2\)
• \(4.2/2.1 = 2\)

We see that the division of each pair gives us the same constant, 2. This indicates that \(y\) is directly proportional to \(x\), and 2 is the constant of variation for this relationship.

Hence, the equation that models the direct variation is \(y = 2x\). Understanding the role of the constant is crucial, as it allows us to establish clear models and relationships in mathematical problems.

Mathematical Modeling

Mathematical modeling is an important tool that helps us represent real-world situations using mathematical concepts and equations. It allows us to simplify complex scenarios and make predictions or understand patterns within a dataset.

In the context of the exercise, mathematical modeling helped us categorize the relationship between \(x\) and \(y\) values as a direct variation, which is a simple form of modeling. By testing for constant ratios (direct) and constant products (inverse), models can guide us in understanding the underlying structures of data.

Scientific fields and industries leverage mathematical models to solve problems, optimize systems, and make informed decisions. By grasping this concept through simple exercises, students build a foundation for more advanced applications. Recognizing whether a relationship is direct, inverse, or neither is a critical first step in creating an effective model.