Question

Autumn is thinking about buying a car. The table below shows the projected value of two different cars for three years.


Number of years 1 2 3
Car 1 (value in dollars) 38,000 32,000 26,000
Car 2 (value in dollars) 38,000 32,300 27,455


Part A: What type of function, linear or exponential, can be used to describe the value of each of the cars after a fixed number of years? Explain your answer. (2 points)

Part B: Write one function for each car to describe the value of the car f(x), in dollars, after x years. (4 points)

Part C: Autumn wants to purchase a car that would have the greatest value in 6 years. Will there be any significant difference in the value of either car after 6 years? Explain your answer, and show the value of each car after 6 years. (4 points)

Answer 1

PART A

The value of car A decreases by 6000 every year. Since the decrease is the same every year, the function is linear

The value of car B decreases by the ratio of $\frac{17}{20}$ every year. Since the decrease is by the same ratio every year, the function is exponential

PART B

Car 1: the function is $y=-6000x+44000$, where $y$ is the value after $x$ years. Negative 6000 shows the decrease every year and 44000 is the value of the car in Year 0

Car 2: the function is $y=(38000) ( \frac{17}{20}) ^{x-1}$, where $y$ is the value after $x$ years. 38000 is the value of the car after Year 1 and $\frac{17}{20}$ is the ratio of depreciation

PART C

Value of car 1 after 6 years is $-6000(6)+44000=8000$
Value of car 2 after 6 years is $(38000) ( \frac{17}{20}) ^{6-1} =16860.8$

There is a significant difference in the values of the cars after 6 years