Question

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 mc018-2.jpg








direct variation; y =


direct variation; y = 12x


inverse variation; xy = 12


inverse variation; = 12

Answer 1

To determine whether the data represents a direct variation or an inverse variation, we need to analyze how \(y\) changes with respect to \(x\).

1. **Direct Variation**: This occurs if \(y = kx\) for some constant \(k\).
2. **Inverse Variation**: This occurs if \(xy = k\) for some constant \(k\).

Looking at the provided \(x\) and \(y\) values:

- For \(x = 2\), \(y = 6\)
- For \(x = 4\), \(y = 3\)

Now, let's calculate \(xy\) for the given pairs:

- When \(x = 2\), \(xy = 2 \times 6 = 12\)
- When \(x = 4\), \(xy = 4 \times 3 = 12\)

Since both products equal 12, this confirms that the data shows an inverse variation.

Therefore, we can write the equation for inverse variation as:

\[
xy = 12
\]

You can also express this relationship explicitly for \(y\):

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

Thus, the data represents an **inverse variation**, and the equation modeling the data is:

\[
xy = 12
\]

Or

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