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

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

   Hint (a):

   If you start with $100\%$ and decrease by $20\%$, what percentage remains? This is the multiplier.

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.

   Answer (b):

   $y = 23500\left(0.8\right)^{x}$

3. How much will your car be worth in four years?

   Hint (c):

   Let $x = 4$ in the equation from part (b).

4. What is the average rate of change in the value of your car during the first four years? Explain what the average rate of change means in this context.

   Hint (d):

   What was the total change in value during the first four years? Divide this number by $4$.

5. In how many years will the trade-in value of your car be $$6000$?

   Step (e):

   Solve: $6000 = 23500\left(0.8\right)^{x}$

   More Help (e):

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

6. If your car is really $2.7$ years old now, what was its value when it was new?

   Hint (f):

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