Markus bought a car for $40,000

Question

$
40
,
000
and it decreases in value by 15%
15
%
every year.

Part A: Write an exponential function that represents the value of the car at the end of every year. Use the variable x
𝑥
in your equation. Do not include a comma in your equation. Simplify the expression in the parenthesis.

f(x)=
𝑓
(
𝑥
)
=

Answer 1

To express the value of the car at the end of every year using an exponential decay function, we start with the initial value of the car and the rate at which it decreases.

The initial value of the car is $40,000. The car decreases in value by 15% each year, which means that at the end of each year, it retains 85% (100% - 15%) of its value.

We can represent this mathematically as follows:

1. The initial value is \( V_0 = 40000 \).
2. The rate of value retention is \( 0.85 \) (since it retains 85% of its value).

The exponential function representing the value of the car after \( x \) years can be described as:

\[ f(x) = V_0 \cdot (retention \ rate)^x \]

Substituting the initial value and the retention rate, we get:

\[ f(x) = 40000 \cdot (0.85)^x \]

Thus, the exponential function that represents the value of the car at the end of every year is:

\[ \boxed{f(x) = 40000(0.85)^x} \]