Question

In 2006, a car was purchased for $17,000. Each year since, the resale value has decreased by 21%.

Let t be the number of years since 2006. Let y be the value of the car, in dollars.
Write an exponential function showing the relationship between y and t

Answer 1

We start with a value of 17000 in 2006. For reasons that will soon be clear, let's write it as

`f(0)=17000=17000\cdot (0.79)^0`

One year later, the resale value has decreased by 21%, which implies that the resale value is 79% of the previous one: the resale value in 2007 is

`f(1)=f(0)\cdot 0.79 = 17000\cdot (0.79)^1`

Move again one year forward: the resale price in 2008 is 79% of the resale price in 2007:

`f(2)=f(1)\cdot 0.79 = (17000\cdot 0.79)\cdot 0.79 = 17000\cdot (0.79)^2`

So, as you can see, after t years from 2006, the resale value will be multiplied by 0.79 t times, leading to the function

`f(t) = 17000\cdot(0.79)^t`