To determine if your table is correct and to find the exponential function rule, let's analyze the values given and look for a pattern.
From the table:
- At \( t = 0 \) hours, \( y = 2 \)
- At \( t = 1 \) hour, \( y = 6 \)
- At \( t = 2 \) hours, \( y = 18 \)
The relationship suggests that the number of good deeds is increasing exponentially. We can assume the general form of the exponential function is:
\[ y = a \cdot b^t \]
where:
- \( a \) is the initial amount of good deeds,
- \( b \) is the growth factor,
- \( t \) is the time in hours.
From the data provided:
- When \( t = 0 \): \[ y = a \cdot b^0 = a \quad \text{so } \quad a = 2 \]
- When \( t = 1 \): \[ y = 2 \cdot b^1 = 6 \quad \Rightarrow \quad b = \frac{6}{2} = 3 \]
- When \( t = 2 \): \[ y = 2 \cdot b^2 = 18 \quad \Rightarrow \quad 2 \cdot b^2 = 18 \quad \Rightarrow \quad b^2 = 9 \quad \Rightarrow \quad b = 3 \]
So both calculations give us the same growth factor \( b = 3 \).
Now substituting \( a \) and \( b \) back into the general equation, we have:
\[ y = 2 \cdot 3^t \]
Thus, the equation for the exponential function that describes the number of good deeds over time is:
\[ y = 2 \cdot 3^t \]
Summary
Your table appears to be correct:
-
Table:
- Time (Hours): 0, 1, 2
- Number of Good Deeds: 2, 6, 18
-
Exponential Function Rule: \[ y = 2 \cdot 3^t \]