To determine which graph correctly displays the population growth of beetles according to the model \( f(x) = 350(1.22)^x \), we need to calculate the values of \( f(x) \) for the given months (0 to 5).
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For \( x = 0 \): \[ f(0) = 350(1.22)^0 = 350 \]
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For \( x = 1 \): \[ f(1) = 350(1.22)^1 = 350 \times 1.22 \approx 427 \]
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For \( x = 2 \): \[ f(2) = 350(1.22)^2 = 350 \times 1.4884 \approx 520 \]
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For \( x = 3 \): \[ f(3) = 350(1.22)^3 = 350 \times 1.8049 \approx 631 \]
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For \( x = 4 \): \[ f(4) = 350(1.22)^4 = 350 \times 2.2087 \approx 772 \]
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For \( x = 5 \): \[ f(5) = 350(1.22)^5 = 350 \times 2.698 \approx 944 \]
Now, the population of beetles is increasing over time. The values of \( f(x) \) at \( x = 0, 1, 2, 3, 4, 5 \) are respectively:
- 350, 427, 520, 631, 772, 944.
This shows a clear increasing trend.
Given the options:
- A and C describe decreasing curves, which cannot be correct.
- D involves an inappropriate scale and represents a flat range for beetle population counts which is incorrect.
Thus, the correct response is B, as it depicts an increasing curve with an appropriate scale for the population growth of beetles.
1 answer