To find the exponential function that models the value of the car over time, we need to determine the decay factor based on the values given in the table.
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Initial Value (when x=0):
- The car is worth $11,000 at the start (x=0).
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Value after 1 Year (y=1):
- At x=1, the value is $9,350.
- To find the decay factor, we can set up the following equation based on the general model \( v(x) = v(0) \cdot r^x \): \[ 9,350 = 11,000 \cdot r^1 \]
- Solving for \( r \): \[ r = \frac{9,350}{11,000} = 0.850 \]
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Value after 2 Years (y=2):
- At x=2, the value is $7,948.
- We can verify our decay factor: \[ v(2) = 11,000 \cdot (0.85)^2 \] \[ v(2) = 11,000 \cdot 0.7225 = 7,947.5 \quad \text{(which is approximately 7,948)} \]
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Value after 3 Years (y=3):
- At x=3, the value is $6,755.
- Verify again: \[ v(3) = 11,000 \cdot (0.85)^3 \] \[ v(3) = 11,000 \cdot 0.614125 = 6,755 \quad \text{(exact)} \]
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Exponential Function:
- We can conclude that the values are modeled by the function: \[ v(x) = 11,000 \cdot (0.85)^x \]
- This matches option b: \( y = 11000 \cdot (0.85)^x \).
Thus, the correct answer is b. \( y = 11000 \cdot (0.85)^x \).