To determine which graph accurately models the function \( f(x) = 3(1.25)^x \), we first analyze the function.
This function represents exponential growth, starting at an initial population of 3 when \( x = 0 \). As \( x \) (Time in Days) increases, the function grows by a factor of 1.25 each day.
Now we can calculate some specific values:
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For \( x = 0 \): \[ f(0) = 3(1.25)^0 = 3(1) = 3 \]
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For \( x = 1 \): \[ f(1) = 3(1.25)^1 = 3(1.25) = 3.75 \]
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For \( x = 2 \): \[ f(2) = 3(1.25)^2 = 3(1.5625) = 4.6875 \]
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For \( x = 3 \): \[ f(3) = 3(1.25)^3 = 3(1.953125) \approx 5.859375 \]
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For \( x = 4 \): \[ f(4) = 3(1.25)^4 = 3(2.44140625) \approx 7.32421875 \]
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For \( x = 5 \): \[ f(5) = 3(1.25)^5 = 3(3.0517578125) \approx 9.1552734375 \]
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For \( x = 6 \): \[ f(6) = 3(1.25)^6 = 3(3.814697265625) \approx 11.444091796875 \]
When visually inspecting the graphs labeled Option #1, Option #2, Option #3, and Option #4, we will look for a graph that starts at approximately 3 when \( x = 0 \) and rises slightly more than previous points as \( x \) increases (showing exponential growth).
If you provide me with the descriptions or properties of the curves present in the four options (like their starting points, slopes, shapes, etc.), I can help you choose the correct option among them.
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