A car was purchased for \( \$ 16,500 \). Each year since, the resale value has decreased by \( 25 \% \). Let \( t \) be the number of years since the purchase. Let \( y \) be the resale value of the car, in dollars. Write an exponential function showing the relationship between \( y \) and \( t \).

The resale value of the car can be modeled by an exponential decay function. Since the car's value decreases by \( 25\% \) each year, it retains \( 75\% \) of its value at the end of each year. Therefore, the function can be expressed as: \[ y = 16500 \times (0.75)^t \] This equation shows that after \( t \) years, the resale value \( y \) is equal to the initial value of \( 16,500 \) multiplied by \( 0.75 \) raised to the power of the number of years \( t \). Realizing the power of exponential decay is fascinating! The way a car depreciates can dramatically affect its market value over time. In just a few years, that \( \$ 16,500 \) car might only be worth a fraction of its original cost. Knowing this makes it easier for buyers and sellers to make informed decisions about transactions, as understanding depreciation allows for better budgeting and planning ahead! When calculating depreciation, one common mistake is confusing the percentage decrease with the remaining value. Remember, if a car depreciates by \( 25\% \), it retains \( 75\% \) of its value, not just dropping straight down \( 25\% \) from the original price each year. Keep a close eye on the numbers, and use this exponential function to anticipate the car's worth after several years!