To determine which graph accurately models the function \( f(x) = 3(1.25)^x \), we need to understand how this exponential growth function behaves over the given time range.
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Identify Important Features of the Function:
- The initial value (when \( x = 0 \)): \[ f(0) = 3(1.25)^0 = 3 \]
- The growth factor is \( 1.25 \), meaning for each day, the population increases by 25%.
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Calculate Values at Key Points:
- \( f(1) = 3(1.25)^1 = 3 \cdot 1.25 = 3.75 \)
- \( f(2) = 3(1.25)^2 = 3 \cdot 1.5625 = 4.6875 \)
- \( f(3) = 3(1.25)^3 = 3 \cdot 1.953125 \approx 5.859375 \)
- \( f(4) = 3(1.25)^4 \approx 7.3 \)
- \( f(5) = 3(1.25)^5 \approx 9.13 \)
- \( f(6) = 3(1.25)^6 \approx 11.41 \)
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Characteristics of the Graph:
- The graph starts at \( (0, 3) \).
- The population will increase gradually more steeply due to the exponential growth.
- The values will remain under 25 until about \( x = 10 \) (beyond the given range).
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Comparing with Each Option:
- Look for a graph that starts at approximately 3 on the y-axis when \( x = 0 \) and shows a gradual increase that reflects the calculations provided above.
Without seeing the specific options, based on this information, select the option that:
- Starts at around \( (0, 3) \).
- Reflects an upward curve starting slow but increasing more steeply after the initial point.
If you have access to the options, choose the one that best matches these criteria. Otherwise, you may need to evaluate based on general features of the graphs presented.