To accurately model the population growth of the swarm of fruit flies using the given function \( f(x) = 3(1.25)^x \), we need to analyze the characteristics of this exponential growth function.
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Understanding the Function:
- The base of the exponent (1.25) is greater than 1, indicating that the function represents growth rather than decay.
- The initial population when \( x = 0 \) is \( f(0) = 3(1.25)^0 = 3 \).
- As \( x \) (representing time in days) increases, the value of \( f(x) \) will continue to increase.
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Evaluating at Specific Points:
- \( f(1) = 3(1.25)^1 = 3.75 \)
- \( f(2) = 3(1.25)^2 \approx 4.69 \)
- \( f(3) = 3(1.25)^3 \approx 5.86 \)
- \( f(4) = 3(1.25)^4 \approx 7.33 \)
- \( f(5) = 3(1.25)^5 \approx 9.16 \)
- \( f(6) = 3(1.25)^6 \approx 11.45 \)
From this evaluation, we see that:
- At day 4 (x=4), \( f(4) \approx 7.33 \), which is indeed between 5 and 10.
- At day 6 (x=6), \( f(6) \approx 11.45 \), which is above 10.
- Characteristics of the Graph:
- The graph of \( f(x) = 3(1.25)^x \) will start at 3 when \( x = 0 \) and will increase gradually.
- It will not decline or go to zero and should not exhibit a sharp increase resembling a vertical asymptote but should rather rise steeply or gradually based on the specific constant multiplier.
Analyzing the Options:
- Option 1: Declines and reaches zero by day 3 — Not accurate.
- Option 2: Increases gradually and at day 4 is between 5 and 10, and day 6 is above 10 — Accurate.
- Option 4: Goes almost straight up — This may be somewhat accurate, but it doesn't align with the gradual increase defined by the equation.
Conclusion:
The graph that models this population growth accurately is Option 2.
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