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7-138.

A rule-of-thumb used by car dealers is that the trade-in value of a car decreases by 20% of its value each year.

1. Explain how the phrase “decreases by 20% of its value each year” tells you that the trade-in value varies exponentially with time (i.e., can be represented by an exponential function).

“Decreases by $20\%$” means that you multiply by $0.8$ each year.A sequence with a multiplier is geometric and has an exponential rule.

2. Suppose the initial value of your car is $23,500. Write an equation expressing the trade-in value of your car as a function of the number of years from now. $y=23500(0.8)^x$ 3. How much will your car be worth in four years? Use the equation that you found in part (b). 4. In how many years will the trade-in value of your car be$6000?

$6000=23500(0.8)^x$

Divide by $23,500$ first then take the log of both sides.

5. If your car is really 2.7 years old now, what was its trade-in value when it was new?

This was $2.7$ years ago, so let $x=−2.7$ in the equation.