Question

the manufacturers suggested retail price (msrp) for a particular car is $25,430, and it is expected to be worth $19,970 in 2 years
(a) find a linear depreciation function for this car
(b) estimate the value of the car 3 years from now
(c) at what rate is the car depreciating?

(a) what is the linear depreciation function for this car?
f(x) =
(simplify your answer. do not include the $ symbol in your answer.)

Explanation:

Step1: Define linear depreciation function

A linear depreciation function has the form \( f(x) = mx + b \), where \( x \) is the number of years, \( m \) is the slope (depreciation rate per year), and \( b \) is the initial value (MSRP). The initial value \( b = 25430 \) (when \( x = 0 \)). After \( x = 2 \) years, the value is \( 19970 \).

Step2: Calculate the slope \( m \)

The slope \( m \) is calculated as the change in value over the change in time:
\( m=\frac{19970 - 25430}{2 - 0}=\frac{-5460}{2}=-2730 \)

Step3: Write the linear function

Substitute \( m = -2730 \) and \( b = 25430 \) into \( f(x)=mx + b \):
\( f(x)=-2730x + 25430 \)

Answer:

\( f(x)=-2730x + 25430 \)