To create an exponential equation from the points \((0, 10)\), \((1, 10)\), and \((2, 0)\), we start by using the general form of an exponential equation:
\[ y = a \cdot b^x \]
where \(a\) is the initial value when \(x = 0\) and \(b\) is the base of the exponential function.
Given the points:
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From \((0,10)\): \[ y = a \cdot b^0 \implies 10 = a \cdot 1 \implies a = 10 \] Thus, we can revise the equation to: \[ y = 10 \cdot b^x \]
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From \((1,10)\): \[ y = 10 \cdot b^1 \implies 10 = 10 \cdot b \implies b = 1 \]
With \(b = 1\), the equation becomes: \[ y = 10 \cdot 1^x \implies y = 10 \]
This equation represents a constant function and does not fit all three points since it does not go through (2,0). This suggests we need to consider if a different model fits better, such as a modified exponential function, which might include a decay.
Let's explore a different form like \(y = 10 \cdot e^{-kx}\).
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From \((0, 10)\): \[ 10 = 10 \cdot e^{-k \cdot 0} \implies 10 = 10 \cdot 1 \implies \text{(True)} \]
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From \((1, 10)\): \[ 10 = 10 \cdot e^{-k \cdot 1} \implies 1 = e^{-k} \] This implies \(k = 0\) which leads us back to \(y = 10\).
Next, let's consider something different, possibly a piecewise or more complex function, but if we wish to maintain exponential behaviors or shifts, we might look at a decaying function:
Using the points provided, we can reinterpret with a horizontal translation or attempted fitting to let:
If we consider points of intersection via logarithmic considerations to find arbitrary fit, we have: \[ y = 10 e^{-kx} = 0 \text{ requires guessing of k} \] Given more points, we could optimize further or explore transformations, but without further context or constraints (e.g., trailing linear conditions), we wouldn’t fit the typical exponential model cleanly with current data.
In conclusion, due to inconsistency in the points provided within an exponential format, we'd explore other possible models or adjust exponential characteristics toward polynomial fitting or other decay formats to best represent a curve or linear trends if necessary. An exponential model pragmatically from the data provided might remain constrained here. Would you need to consider a different model classification or confirmation for a specific fitting?