You buy a car in 2005 for $7860 and it depreciates linearly. You sell the car for $5040 in 2011....

what does the domain represent in the function?
what does the range represent in this function?
what is the slope of the linear representation?
what is the function that models the linear representation?
using the model, make a prediction for what you could sell the car for today?

To answer these questions, we need to analyze the given information and understand the concept of linear depreciation.

1. What does the domain represent in the function?
The domain represents the input values or the independent variable of the function. In this case, the domain represents the years. We are given the years 2005 and 2011, so the domain would be [2005, 2011].

2. What does the range represent in this function?
The range represents the output values or the dependent variable of the function. In this case, the range represents the value of the car at a specific year. We are given the values $7860 and $5040, so the range would be [$7860, $5040].

3. What is the slope of the linear representation?
To find the slope, we use the formula:
slope = (change in y) / (change in x)
The change in y is $5040 - $7860 = -$2820 (since the value decreased)
The change in x is 2011 - 2005 = 6
Therefore, the slope = (-$2820) / 6 = -$470 or -$470/year.

4. What is the function that models the linear representation?
The linear function that models this situation would be:
y = mx + b,
where y represents the value of the car in dollars at a given year, x represents the year, m represents the slope, and b represents the y-intercept (initial value).
Using the given information, we have:
y = -470x + b
We can now find the y-intercept by substituting the values (x, y) = (2005, $7860):
$7860 = -470 * 2005 + b
Solving for b, we get:
b = $9500
So, the function that models the linear representation of the car's value is:
y = -470x + $9500.

5. Using the model, make a prediction for what you could sell the car for today?
To make a prediction for the current year, substitute the current year into the function and solve for y.
Let's assume the current year is 2021.
y = -470(2021) + $9500
y = -$949,670 + $9500
y ≈ $-940,170

Based on the linear model, the prediction for what you could sell the car for today would be approximately -$940,170. However, keep in mind that this value is negative, indicating that its value has significantly decreased.