https://file.aiquickdraw.com/mathbot/file/1726246340325uvtauzju.jpg

The provided image contains data representing the value of a car over time, with the goal of finding an exponential regression equation that models the car's depreciation. Let's extract the data from the table and perform the exponential regression.

### Data from the Table:
- Years (x): 0, 1, 2, 3, 4, 5, 6
- Value in Dollars (y): 15800, 13136, 11018, 9337, 8034, 6115, 4876

We need to calculate an exponential regression model of the form:
\[
y = a \cdot b^x
\]
where:
- $y$ is the value of the car,
- $x$ is the number of years,
- $a$ is the initial value of the car (at $x = 0$),
- $b$ is the growth/decay factor we need to determine through regression.

Let me calculate this equation and determine the value of the car after 12 years.The exponential regression equation for the car's value is:

\[
y = 15879.301 \cdot 0.832^x
\]

Using this equation, the value of the car after 12 years is approximately **\$1747.07**, rounded to the nearest cent.

Let me know if you would like a detailed breakdown of the process!

### Additional Questions:
1. How does the value of $b$ affect the car's depreciation rate?
2. What is the significance of the coefficient $a$ in an exponential model?
3. How would the model change if the car appreciated in value instead of depreciating?
4. Can this model be used to predict the value of the car beyond 12 years? Why or why not?
5. How does the exponential model compare to a linear model in predicting depreciation?

### Tip:
Exponential regression is ideal for modeling quantities that decrease by a constant percentage over time, such as depreciation.

The accompanying table shows the value of a car over time that was purchased for 15800 dollars, where x is years and y is the value of the car in dollars. Write an exponential regression equation for this set of data, rounding all coefficients to the nearest thousandth. Using this equation, determine the value of the car, to the nearest cent, after 12 years.

Learn how to calculate the exponential regression for a car's depreciation over time. This example involves using the exponential decay formula and finding the car's value after 12 years, based on a starting value of $15,800. Step-by-step guide with rounded coefficients and regression details included.

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