Question

3.
Does the data in the table represent a direct variation or an inverse variation? Write an equation to model the data in the table.

x 2 4 8 12
y 6 3 2/3 1

Answer 1

To determine whether the data represents a direct variation or an inverse variation, we need to analyze the relationship between \(x\) and \(y\).

1. **Direct Variation**: This occurs when \(y = kx\), where \(k\) is a constant. In this case, as \(x\) increases, \(y\) also increases (or decreases) proportionally.

2. **Inverse Variation**: This occurs when \(y = \frac{k}{x}\), where \(k\) is a constant. In this case, as \(x\) increases, \(y\) decreases proportionally.

Now, let's review the data given:

\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
2 & 6 \\
4 & 3 \\
8 & \frac{2}{3} \\
12 & 1 \\
\hline
\end{array}
\]

Next, we can check if the relationship is an inverse variation by calculating \(xy\) for each pair of values:

- For \(x = 2\) and \(y = 6\): \(xy = 2 \times 6 = 12\)
- For \(x = 4\) and \(y = 3\): \(xy = 4 \times 3 = 12\)
- For \(x = 8\) and \(y = \frac{2}{3}\): \(xy = 8 \times \frac{2}{3} = \frac{16}{3} \approx 5.33\) (which is not constant)
- For \(x = 12\) and \(y = 1\): \(xy = 12 \times 1 = 12\)

Since \(xy\) is not constant across all pairs (it varies for \(x = 8\)), we need to investigate the pattern more closely to determine if it indicates a more specific type of function.

In examining the relationship between \(x\) and \(y\), we find that:

- When \(x\) doubles from:
- \(2\) to \(4\): \(y\) goes from \(6\) to \(3\) (halved)
- \(4\) to \(8\): \(y\) goes from \(3\) to \(\frac{2}{3}\) (approximately a decrease of a third)
- \(8\) to \(12\): \(y\) goes from \(\frac{2}{3}\) to \(1\) (also a decrease, but not proportional)

Since the values of \(xy\) are not consistent for all pairs of data, it may suggest that the function is not purely inverse variation either, but it approximates inverse behavior for some portions.

### Conclusion:
The data does not represent a strict inverse variation since not all products \(xy\) equal a constant, nor does it represent direct variation because the values do not increase together proportionately.

However, an approximate inverse relationship can be suggested due to decreasing trends, particularly noted that as \(x\) increases, \(y\) decreases. The general model can be settled as:

\[
y = \frac{12}{x}
\]

This displays an inverse behavior for the groups \(2, 4, 12\), while the value for \(x = 8\) does not maintain consistent relations in this case.