Question

Which description matches the function represented by the values in this table?

X | Y
1 | 972
2 | 324
3 | 108
4 | 36
5 | 12

linear decay
exponential decay
exponential growth
linear growth

Answer 1

To determine the type of function represented by the values in the table, we can analyze the behavior of \( Y \) as \( X \) increases.

The values from the table are:

\[
\begin{array}{|c|c|}
\hline
X & Y \\
\hline
1 & 972 \\
2 & 324 \\
3 & 108 \\
4 & 36 \\
5 & 12 \\
\hline
\end{array}
\]

Observing the values:

- As \( X \) increases from 1 to 2, \( Y \) decreases from 972 to 324.
- As \( X \) increases from 2 to 3, \( Y \) decreases from 324 to 108.
- As \( X \) increases from 3 to 4, \( Y \) decreases from 108 to 36.
- As \( X \) increases from 4 to 5, \( Y \) decreases from 36 to 12.

The pattern shows that \( Y \) is decreasing, and the **ratios of successive values** of \( Y \) can provide insight into the type of decay.

Calculating the ratios:

- \( \frac{324}{972} = \frac{1}{3} \)
- \( \frac{108}{324} = \frac{1}{3} \)
- \( \frac{36}{108} = \frac{1}{3} \)
- \( \frac{12}{36} = \frac{1}{3} \)

Since the ratio of successive \( Y \) values is constant at \( \frac{1}{3} \), this indicates that the function is **exponential decay**.

Thus, the correct description that matches the function represented by the values in this table is **exponential decay**.