Question 50·Medium·Two-Variable Data: Models and Scatterplots
A car dealership recorded the resale value of a certain model at several points after the car was first purchased.
| Months since purchase | Resale value ($) |
|---|---|
| 5 | 19,500 |
| 10 | 18,000 |
| 15 | 16,500 |
| 20 | 15,000 |
| 25 | 13,500 |
Assuming a linear model best fits the data, what is the predicted resale value, in dollars, of this model 40 months after purchase?
(Express the answer as an integer)
For linear model questions based on a table, first check that the change in the dependent variable (like value) is constant for equal changes in the independent variable (like time); this confirms a linear relationship. Then compute the rate of change (slope) using any two convenient points. To predict a future value, either (1) extend the pattern from the closest known point by multiplying the slope by the additional time and adding/subtracting that change, or (2) write the linear equation and substitute the desired x‑value. Choose the method that uses the smallest, simplest numbers to save time and reduce errors.
Hints
Look for a pattern in the table
Compare the resale values as the months increase by 5. How much does the value change each time the months go up by 5?
Convert the 5‑month change to a 1‑month change
Once you know how much the value changes in 5 months, divide that number by 5 to find the change per month.
Extend the pattern to 40 months
The table stops at 25 months. How many months is it from 25 to 40? Use the monthly rate of change to find how much the value will change in that time, and then adjust the 25‑month value accordingly.
Desmos Guide
Enter the data as a table
In Desmos, click the "+" icon and choose "Table." In the first column (x1), enter 5, 10, 15, 20, 25. In the second column (y1), enter 19500, 18000, 16500, 15000, 13500.
Fit a linear model to the data
In a new expression line, type y1 ~ m x1 + b. Desmos will perform a linear regression and show values for m (the slope) and b (the intercept).
Use the regression equation to predict the value at 40 months
In another expression line, define f(x) = m x + b. Then either type f(40) or add a new table with x = 40 in the first column and f(x) in the second column. The corresponding output is the predicted resale value at 40 months.
Step-by-step Explanation
Find the rate of change (slope) from the table
Look at how the resale value changes every time the months increase by 5.
From 5 to 10 months:
- Value goes from 19,500 to 18,000, a change of .
From 10 to 15 months:
- Value goes from 18,000 to 16,500, again a change of .
The same change happens every 5 months, so the rate of change is constant.
Per month, the value changes by
So the car loses 300 dollars in value each month.
Determine how many more months to go beyond the table
The table gives the value up to 25 months, where the value is 13,500 dollars.
We need the value at 40 months.
Number of months from 25 to 40 is
So we must extend the linear pattern forward 15 more months.
Find the total change in value from 25 to 40 months
We already found that the value decreases by 300 dollars per month.
Over 15 months, the total decrease is
So from 25 months to 40 months, the car will lose 4,500 dollars in value.
Apply the change to the known value at 25 months
At 25 months, the value is 13,500 dollars.
We subtract the total decrease from this amount:
So, the predicted resale value of the car 40 months after purchase is 9000 dollars.