To model the insect population using an exponential function, we can start from the information given:
- x: 0, 1, 2, 3
- y: 12, 18, 27, 41
We can observe that the growth appears to be accelerating, which suggests an exponential growth pattern.
The general form of the exponential function is:
\[ y = ab^x \]
where:
- \( y \) is the population at time \( x \),
- \( a \) is the initial population (when \( x = 0 \)),
- \( b \) is the growth factor (how much the population multiplies each time period).
From the data:
- When \( x = 0 \), \( y = 12 \). Therefore, \( a = 12 \).
So, we can start with:
\[ y = 12b^x \]
Next, we can find \( b \) using the data points we have. Let's use the point when \( x = 1 \):
\[ y = 18 \] \[ 18 = 12b^1 \] \[ b = \frac{18}{12} = \frac{3}{2} = 1.5 \]
Now, let's check if this value of \( b \) works with the other points.
- For \( x = 2 \):
\[ y = 12(1.5^2) \] \[ y = 12 \cdot 2.25 = 27 \] (This matches the given data)
- For \( x = 3 \):
\[ y = 12(1.5^3) \] \[ y = 12 \cdot 3.375 = 40.5 \] (This is approximately 41, which is within rounding error)
The model closely fits the data, so we can conclude that:
The exponential function that describes the insect population over time is:
\[ y = 12(1.5^x) \]
And the completed table reflects the values already provided:
| x | y | |---|----| | 0 | 12 | | 1 | 18 | | 2 | 27 | | 3 | 41 |