Is the relationship between the values in each table a...

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. $$ \begin{array}{|c|c|c|c|c|}\hline x & {0.5} & {2.1} & {3.5} & {11} \\ \hline y & {1} & {4.2} & {7} & {22} \\ \hline\end{array} $$

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.
$$
\begin{array}{|c|c|c|c|c|}\hline x & {0.5} & {2.1} & {3.5} & {11} \\ \hline y & {1} & {4.2} & {7} & {22} \\ \hline\end{array}
$$

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Key Concepts

Constant of Proportionality

The constant of proportionality, denoted as k, is the factor that relates the variables in both direct and inverse variation. In a direct variation (y = kx), k is found by dividing y by x, whereas in an inverse variation (y = k/x) it is found by multiplying x and y. Establishing the value of k is crucial for accurately modeling the relationship between variables.

Inverse Variation

Inverse variation refers to a relationship where the product of two variables is constant, which mathematically is represented as y = k/x. In this scenario, as one variable increases, the other decreases in such a way that their product remains equal to the constant k. This type of relationship is fundamentally different from direct variation and is identified by checking whether the product of corresponding pairs of values remains constant.

Direct Variation

Direct variation describes a relationship between two variables in which the ratio y/x remains constant for all values of x and y. In such cases, the relationship can be modeled by the equation y = kx, where k is the constant of proportionality. This means that as x increases or decreases, y changes at a constant rate determined by k.

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