A residual compares an observed value with the value predicted by a line at the same input. To find the greatest absolute residual, use Desmos to evaluate the given line at each input, then compare the sizes of the gaps between predictions and observations. Don't fit a new line or pick the largest observed output; neither tells you which point is farthest from the given line.
Hints
- Hint 1
A residual is observed output minus predicted output at the same input. Keep each week's push-up count paired with the prediction for that week. What does the given line predict for each week in the table?
- Hint 2
The absolute value of a residual is its size without its sign. A point below the line can have a bigger gap than one above it. After subtracting each prediction from its observation, which gap has the greatest size?
Step-by-step
Compare the gaps from the given line
Step 1Translate absolute residual into a calculation
A residual is the observed push-up count minus the line's prediction . So the absolute residual is . Absolute value measures the size of the gap, even when the observation is below the line.
- Step 2
Store the given prediction rule
Type in Desmos. This names the given line , so will give its predicted push-up count after weeks. There's no need to fit another line.
- Step 3
Keep the observations paired
Enter the weeks in a Desmos table's column and the observed push-up counts in its column. Each same-row pair is one observation, so the count stays paired with week .
- Step 4
Find the line's predictions
Type to evaluate the line at every week in the table. Desmos prints . These are predictions, in week order, not the push-up counts the coach recorded.
- Step 5
Compare the absolute residuals
Type . Desmos prints in the same week order. The largest gap, , is the third entry: at weeks, the coach recorded push-ups while the line predicted . So has the greatest absolute residual. Choice C.