To model the insect population growth, we can observe the values you've provided and look for a pattern. The population appears to show exponential growth, specifically it seems to be increasing by a consistent multiplication factor.
First, let's fill in the missing population values for each month:
- For \(x = 0\), \(y = 12\) (initial population)
- For \(x = 1\), \(y = 18\)
- For \(x = 2\), \(y = 27\)
- For \(x = 3\), \(y = 40.5\)
Finding the Multiplication Factor
To determine the growth factor, we can divide each population value by the population from the previous month:
-
From \(x = 0\) to \(x = 1\): \[ \frac{18}{12} = 1.5 \]
-
From \(x = 1\) to \(x = 2\): \[ \frac{27}{18} = 1.5 \]
-
From \(x = 2\) to \(x = 3\): \[ \frac{40.5}{27} = 1.5 \]
The multiplication factor (growth rate) is consistently 1.5.
Writing the Exponential Function
The general form of an exponential growth function is given by: \[ y = y_0 \cdot r^x \] where:
- \(y\) is the population at time \(x\),
- \(y_0\) is the initial population,
- \(r\) is the growth factor,
- \(x\) is the number of months.
In our case:
- \(y_0 = 12\)
- \(r = 1.5\)
Thus, the exponential function that describes this scenario is: \[ y = 12 \cdot (1.5)^x \]
Final Table of Values
Here is the completed table with the population values for months 0 to 3:
| months, \(x\) | population, \(y\) | |----------------|--------------------| | 0 | 12 | | 1 | 18 | | 2 | 27 | | 3 | 40.5 |
Summary
The exponential function that models the insect population is: \[ y = 12 \cdot (1.5)^x \]