To complete the chart based on the function \( f(x) = 350(1.22)^x \), we will calculate the population of beetles for each month (x) from 0 to 5.
-
For \( x = 0 \): \[ f(0) = 350(1.22)^0 = 350 \times 1 = 350 \]
-
For \( x = 1 \): \[ f(1) = 350(1.22)^1 = 350 \times 1.22 = 427 \]
-
For \( x = 2 \): \[ f(2) = 350(1.22)^2 = 350 \times 1.4884 \approx 521 \]
-
For \( x = 3 \): \[ f(3) = 350(1.22)^3 = 350 \times 1.810584 \approx 633 \]
-
For \( x = 4 \): \[ f(4) = 350(1.22)^4 = 350 \times 2.204487 \approx 771 \]
-
For \( x = 5 \): \[ f(5) = 350(1.22)^5 = 350 \times 2.684962 \approx 939 \]
Now, we can fill in the chart:
| Months (x) | Beetles (f(x)) | |------------|----------------| | 0 | 350 | | 1 | 427 | | 2 | 521 | | 3 | 633 | | 4 | 771 | | 5 | 939 |
Plotting the Points
The points from the completed table are:
- (0, 350)
- (1, 427)
- (2, 521)
- (3, 633)
- (4, 771)
- (5, 939)
Determining the Correct Graph
The population of beetles is increasing over time due to the growth rate of 22% per month. This suggests the graph should display an exponential growth curve.
Among the given responses:
- The first option describes an increasing curve with an arrow on the top over a wide range.
- The second option describes a decreasing curve, which is incorrect.
- The third option describes an increasing curve but with a smaller range of the x-axis, from 0 to 10, which should fit our results.
- The fourth option once again describes a decreasing curve, which is also incorrect.
The most suitable graph representing this scenario is Option 3: "An increasing curve with an arrow on the top is plotted on a coordinate plane. The x-axis for months ranges from 0 to 10 in increments of 1. The y-axis for Beetles ranges from 0 to 2000 in increments of 250."
This graph correctly reflects the exponential growth of the population of which beetles are part, given the calculated values.