Autumn is thinking about buying a car for $38,000. 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: 32,000, 26,000, 20,000 car 2 (value in dollars) 32,300, 27,455, 23,336.75 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 5 years. will there be any significant difference in the value of either car after 5 years? explain your answer, and show the value of each car after 5 years. (4 points)

Part A. In order to know the function, the first step is to plot the points for both cars. In the picture, this is represented by the blue and orange lines for Car 1 and Car 2, respectively. When you connect the dots, it forms a smooth line. Therefore, both functions are linear.

Part B. In this case, you have to find the m and b for the standard form of a linear equation, y=mx + b, wherein m is the slope and b is the y-intercept. But, you can use MS Excel to make a trendline and know the linear equation from it through linear regression. The equation for car 1 is y=-6000x+38000 and that for car 2 is y=-4481.6x+36661.

Part C. In the formulated equations in Part B, we use this to know the value of y by replacing x = 5 years.

Car 1:
y = -6000(5)+38000 = $8,000
Car 2:
y=-4481.6(5)+36661 = $14,253

There is a significant difference, obviously. For Car 1, the value dropped from $20,000 to $8000 in just a year. For Car 2, the value dropped from $23,336.75 to $14,253 in just a year.

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