The dots are what happened. The line is what the model says should have happened — and almost every SAT question here is about the gap between the two.
Which of the following is the best interpretation of 1,150 in this context?
A scatterplot relates x, the number of years since 2010, and y, the average price, in dollars, of a used sedan. The line of best fit is y = 21,500 − 1,150x.
1,150 is the slope, and a slope is a rate: it is how much y changes for a one-unit change in x. One unit of x is one year, so the price falls by about $1,150 per year, every year. A names the intercept instead — 21,500 is the price at x = 0, the year 2010. C spends the whole drop at once; the slope applies once per year, so over 10 years the model loses 10 × 1,150 = $11,500. D describes a residual, the gap between one actual data point and the line, which is a different quantity entirely — you will meet it in step 5.
The line of best fit is the single straight line that passes as close as possible to every point at once. It does not have to touch a single dot, and usually it touches none.
Ten months of revenue, ten dots. The trend is clearly upward and roughly straight, which is exactly the condition that makes a straight line of best fit worth drawing.
“According to the line of best fit, what is the predicted revenue in month 7?” is a question about the line only. The dot at month 7 is irrelevant to it.
go up from x = 7 until you hit the line
then straight across to the y-axis
y = 3(7) + 14 = 21 + 14 = 35
or use the equation, if the item gives you one
predicted revenue ≈ $35,000
the axis is in thousands — put the units back
“The predicted value”, “the value given by the line of best fit” and “the value the model estimates” all mean the line. “The actual value”, “the data point” and “the recorded revenue” all mean the dot. The month-7 dot sits at 33, so the two readings differ by 2 — and that difference is usually the answer to the next question.
Pick two places where the line crosses a grid corner cleanly. Never use two data points — they are not on the line.
the line passes through (2, 20) and (10, 44)
two clean grid readings, far apart
slope = 44 − 2010 − 2 = 248 = 3
rise over run, in the axis units
at x = 0 the line is at y = 14
where it meets the vertical axis
y = 3x + 14
check month 10: 3(10) + 14 = 44 ✓
| Number | What it is | What it means here |
|---|---|---|
| 14 | the y-intercept | the store took in about $14,000 in the month it opened |
| 3 | the slope | revenue rose by about $3,000 for each additional month |
Say the slope as a rate with both units — “thousands of dollars per month”. Choices that drop the “per month” are describing an amount, not a rate, and are wrong for that reason alone.
Interpolating means predicting at an x the data already covers. Extrapolating means running the line out past the data, where nothing was ever measured to support it.
3(7) + 14 = 35
months 1 to 10 were all observed — trustworthy
3(40) + 14 = 134
$134,000 a month, on ten months of evidence
data range: x = 1 to x = 10
the first and last dots set the honest range
x = 7 is inside it
interpolation, and the usual SAT ask
x = 40 is four times past the last dot
nothing says the trend keeps going that long
If an item asks what the line of best fit predicts at month 40, the answer is 134 — do the arithmetic. Extrapolation is not a reason to refuse. It matters when the question changes shape: “which statement is best supported by the data”, where a claim about month 40 is not supported, or “over which interval is the model appropriate”, where the answer is the interval the dots cover.
Same x, two y-values: the one you observed and the one the line claims. Subtract them in that order — actual minus predicted.
x = 1: line gives 5(1) + 10 = 15, dot is at 22
the hollow circle is the prediction
residual = 22 − 15 = +7
dot above the line → the model under-predicted
x = 3: line gives 5(3) + 10 = 25, dot is at 18
same subtraction, other direction
residual = 18 − 25 = −7
dot below the line → the model over-predicted
It is the opposite, and the wording flips every year. If the real value beat the line, the line was too small there: it under-predicted. Say it out loud with the numbers — “the store actually made 22, the model only said 15” — rather than trusting the words “above” and “over” to agree. They do not.
A best-fit line always leaves some dots above and some below; the positives and negatives roughly cancel. An item asking “for how many points does the line predict a value greater than the actual value” is simply asking you to count the dots sitting below the line.
Some items hand you a table instead of a picture and ask for the line of best fit. Typing beats eyeballing.
make a table in Desmos: x1, y1
one row per data point
type y1 ~ mx1 + b
the tilde, not an equals sign — that is the regression
read m and b from the panel
m is the slope, b the intercept, already rounded for you
Once the line is plotted, a prediction is one click: type the x-value into a point and Desmos reports the y on the line. And if you already have the equation, plotting it over the table shows at a glance which dots sit above it and which sit below — the count that residual items ask for.
Which of the following best describes the relationship between the line and the point?
For a scatterplot relating x, the number of days a seedling has been growing, and y, its height in millimeters, the line of best fit is y = 4.2x + 31. One of the data points has coordinates (12, 76).
Predicted: 4.2(12) + 31 = 50.4 + 31 = 81.4. Actual: 76. The prediction is 81.4 − 76 = 5.4 higher than what really happened, so the model over-predicted and the residual is −5.4. B subtracts the right pair the wrong way round and turns an over-prediction into an under-prediction — the single most common slip on these. C forgets the + 31, using 50.4 as the prediction and reporting 76 − 50.4 = 25.6. D strips the intercept off the data point instead, 76 − 31 = 45, which is not a comparison of any two values at x = 12.
Hint: write the two numbers down before you compare them — predicted, then actual. The order of the subtraction is the whole question.
