To complete the chart and find the correct graph for the beetle population growth modeled by the equation \( f(x) = 350(1.22)^x \), we need to calculate the population for \( x = 0, 1, 2, 3, 4, 5 \).
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For \( x = 0 \): \[ f(0) = 350(1.22)^0 = 350 \times 1 = 350 \]
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For \( x = 1 \): \[ f(1) = 350(1.22)^1 = 350 \times 1.22 = 427 \]
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For \( x = 2 \): \[ f(2) = 350(1.22)^2 \approx 350 \times 1.4884 \approx 520.94 \approx 521 \]
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For \( x = 3 \): \[ f(3) = 350(1.22)^3 \approx 350 \times 1.8047 \approx 631.645 \approx 632 \]
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For \( x = 4 \): \[ f(4) = 350(1.22)^4 \approx 350 \times 2.1336 \approx 746.76 \approx 747 \]
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For \( x = 5 \): \[ f(5) = 350(1.22)^5 \approx 350 \times 2.6014 \approx 910.49 \approx 910 \]
Now we can summarize the values in the chart:
| x (Months) | f(x) (Beetles) | |------------|----------------| | 0 | 350 | | 1 | 427 | | 2 | 521 | | 3 | 632 | | 4 | 747 | | 5 | 910 |
With these values, we can observe that the population of beetles is increasing exponentially.
In terms of the graphs:
- The correct graph should be an increasing curve (since the population is increasing) and not a decreasing curve.
- The y-axis should have values greater than 350, extending upwards to around 1000 or more to accommodate the values calculated above, particularly for \( f(5) \).
Thus, the right image would be the one that shows an increasing curve with an arrow extending upwards, corresponding with the growth in beetle population over time.