To complete the table and determine which graph correctly displays the situation modeled by the equation \( f(x) = 350(1.22)^x \), we can calculate the population of beetles for the first few months (0 to 5).
Let's calculate the values:
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When \( x = 0 \): \[ f(0) = 350(1.22)^0 = 350 \times 1 = 350 \]
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When \( x = 1 \): \[ f(1) = 350(1.22)^1 = 350 \times 1.22 = 427 \]
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When \( x = 2 \): \[ f(2) = 350(1.22)^2 = 350 \times 1.4884 \approx 520 \]
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When \( x = 3 \): \[ f(3) = 350(1.22)^3 = 350 \times 1.8156 \approx 636 \]
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When \( x = 4 \): \[ f(4) = 350(1.22)^4 = 350 \times 2.2135 \approx 775 \]
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When \( x = 5 \): \[ f(5) = 350(1.22)^5 = 350 \times 2.6860 \approx 940 \]
Now we can compile the values in a table:
| Month (x) | Beetles (f(x)) | |-----------|----------------| | 0 | 350 | | 1 | 427 | | 2 | 520 | | 3 | 636 | | 4 | 775 | | 5 | 940 |
From the calculations, we see that the population of beetles is increasing over time.
Now, looking at the graph options:
- Graph Option Descriptions:
- Graph 1: An increasing curve, but x-axis and y-axis ranges are not appropriate (too large).
- Graph 2: An increasing curve with appropriate x-axis (0 to 10) and a y-axis (0 to 2000) that accommodates the increasing population.
- Graph 3: A decreasing curve, which doesn't match our calculated increasing values.
- Graph 4: A repeating description of Graph 3, also a decreasing curve.
The correct graph that displays the increasing population of beetles, based on our calculations and the equation provided, is Graph 2.
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