To determine which graph accurately models the function \( f(x) = 3(1.25)^x \) representing the population growth of a swarm of fruit flies, we need to analyze the properties of the function and compare them with the options available.
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Exponential Growth: The function is an exponential growth function starting at \( f(0) = 3(1.25)^0 = 3 \). This means that on Day 0 (x=0), the population is 3.
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Growth Rate: The base of the exponent (1.25) indicates the population is increasing by 25% each day. Therefore, the population will grow gradually.
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Behavior of the Graph:
- At \( x = 1 \): \( f(1) = 3(1.25)^1 = 3.75 \)
- At \( x = 2 \): \( f(2) = 3(1.25)^2 = 4.6875 \)
- At \( x = 3 \): \( f(3) = 3(1.25)^3 \approx 5.859375 \)
- At \( x = 4 \): \( f(4) = 3(1.25)^4 \approx 7.322 \)
- At \( x = 5 \): \( f(5) = 3(1.25)^5 \approx 9.153 \)
- At \( x = 6 \): \( f(6) = 3(1.25)^6 \approx 11.44 \)
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Expected Values: By Day 6, we can expect the population to be around 11.44, which is below 25, but shows consistent growth increasing from 3 and gradually approaching larger values.
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Identify the Correct Graph: Look for the graph that:
- Starts at (0, 3)
- Shows gradual increases in population over time
- Does not exceed the y-values significantly past 25, remaining within the range of 0 to 25
Without the actual images, I can't specify which option is correct. However, you should apply this analysis on the provided graphs. Option that starts at (0, 3) and gradually increases while maintaining the trend of exponential growth should be the correct graph representing the population of fruit flies as per the given function \( f(x) = 3(1.25)^x \).
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