A residual is the actual output minus the output predicted by a line of best fit. When two points on the line are labeled, use them to build the line, then evaluate it at the data point’s input in Desmos. Keep the dot’s actual output separate from the line’s prediction. Reversing the subtraction gives the wrong sign.
Hints
- Hint 1
Find the line’s slope using the two labeled points on the line. Slope is the change in price divided by the change in mileage. Why wouldn’t you use the point labeled Car to find it?
- Hint 2
A line’s predicted price starts at a known price and changes by slope times the change in mileage. Start from the labeled point at 60,000 miles. What mileage difference takes you to Car?
- Hint 3
The actual price comes from Car’s dot; the predicted price comes from the line at the same mileage. A residual is actual minus predicted. Should the result be positive or negative when the dot is above the line?
Step-by-step
Build the line, then find the residual
Step 1Find the line’s rate of change
Use the two labeled points on the line, not Car’s dot. The slope is the change in price divided by the change in mileage. Type in Desmos. It shows , so the line predicts a price drop of $0.20 for each additional mile.
- Step 2
Write the line’s predicted price
At miles, the line predicts $12,500. At mileage , the change in mileage is , so multiply that change by the slope and add it to $12,500. Type the prediction function . Here means the price on the line at mileage , not Car’s actual price.
- Step 3
Predict the price at Car’s mileage
Car’s mileage is , so type . Desmos shows . That is the predicted price from the line; the dot’s $15,000 is the actual price.
- Step 4
Subtract predicted from actual
The given residual rule says actual minus predicted. Type ; Desmos shows . Car’s actual price is $1,500 above the line’s prediction, so its residual is positive. Choice A.