Which table shows exponential decay?

A 2-column table has 4 rows. The first column is labeled x with entries 1, 2, 3, 4. The second column is labeled y with entries 16, 8, 4, 2.
A 2-column table has 4 rows. The first column is labeled x with entries 1, 2, 3, 4. The second column is labeled y with entries 16, 128, 8, 4.
A 2-column table has 4 rows. The first column is labeled x with entries 1, 2, 3, 4. The second column is labeled y with entries 16, 12, 9, 7.
A 2-column table has 4 rows. The first column is labeled x with entries 1, 2, 3, 4. The second column is labeled y with entries 16, 8, 3, 1.

1 answer

To determine which table shows exponential decay, we need to identify the table where the values of \( y \) decrease by a consistent multiplicative factor (typically between 0 and 1) as \( x \) increases.

Let's analyze each table:

- **Table 1:**
- \( x: 1, 2, 3, 4 \)
- \( y: 16, 8, 4, 2 \)
- Here, the pattern is \( y \) is halved each time \( x \) increases by 1. This indicates an exponential decay with a factor of \(\frac{1}{2}\).

- **Table 2:**
- \( x: 1, 2, 3, 4 \)
- \( y: 16, 128, 8, 4 \)
- Here, there is no consistent multiplicative factor. The values do not follow an exponential decay or growth pattern.

- **Table 3:**
- \( x: 1, 2, 3, 4 \)
- \( y: 16, 12, 9, 7 \)
- Here, the values decrease but not by a consistent multiplicative factor. This is more linear or polynomial rather than exponential.

- **Table 4:**
- \( x: 1, 2, 3, 4 \)
- \( y: 16, 8, 3, 1 \)
- Here, the values decrease but not by a consistent multiplicative factor. This table does not show a clear exponential pattern.

Based on the analysis, **Table 1** shows exponential decay:
- The \( y \)-values are decreasing by a consistent factor of \(\frac{1}{2}\) as \( x \) increases.