Digital SAT · Math · Unit 3 · Problem Solving & Data

Lesson 3.5 · Scatterplots, Lines of Best Fit & Models

On the test: 1–3 questions a test — mostly “which equation best represents the line of best fit shown?”, read in seconds

Big idea. A scatterplot puts one dot for every pair (x, y). Its SHAPE tells a story: which way it goes, how tightly, straight or curved. A line of best fit summarizes a straight trend — its y-intercept and slope can be read straight off the graph, and it predicts. Most SAT questions are answered by three glances: where the line crosses the y-axis, which way it tilts, and how steeply.

🎛 Move it · the Scatter Lab

Move the two sliders until your line fits. Watch the gaps (residuals) and the misfit score; then compare with the best line. Try the curved data sets.

y-intercept b2slope m0.5

Your line: y = 2 + 0.5x · misfit (sum of squared residuals) = 59.59

Green gaps: the point is ABOVE the line (actual > predicted). Red: BELOW. The line of best fit balances them and keeps them small.

🧑‍🏫 Rule 1 · Every dot is one pair (x, y)step 1 of 6

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one dot = one data pair
$(x, y) = (3, 5.6)$
Go across to $x$ , up to $y$ . The plot shows how $y$ tends to change as $x$ changes.

🧑‍🏫 Rule 2 · What a line of best fit isstep 1 of 3

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residual
$residual = actual - predicted$
The vertical gap from a point to the line. Above the line: positive (green). Below: negative (red).

🧑‍🏫 Rule 3 · Its equation from the graph: intercept → sign → sizestep 1 of 5

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📋 The question · SAT · March 2024 · Math 800 Book Q5
$figure from the test$
The scatterplot shows the relationship between two variables, $x$ and $y$ . A line of best fit is also shown. Which of the following equations best represents the line of best fit shown?
1. A) $y = 3.5 + 1.8 x$
2. B) $y = - 2.4 + 1.55 x$
3. C) $y = 3.4 + 1.55 x$
4. D) $y = 2.4 - 1.5 x$
step 1 · the y-intercept
$x = 0 \Rightarrow y \approx 3.5$
Where the LINE crosses the $y$ -axis. Options with $- 2.4$ or $2.4$ are out — keep A ( $3.5$ ) and C ( $3.4$ ).

🧑‍🏫 Rule 4 · The slope from two points on the linestep 1 of 5

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📋 The question · SAT · October 2024 · Math 800 Book Q8
$figure from the test$
Which of the following is closest to the slope of this line of best fit?
1. A) $- 3.3$
2. B) $- 1.1$
3. C) $1.1$
4. D) $3.3$
pick two points ON THE LINE
$(0, 14.5) and (13.2, 0)$
October 2024: use where the line crosses the axes or grid corners — never the data dots (they are not on the line).

Fast lane · The 10-second method

For “which equation best represents the line”: (1) find the y-intercept — it usually knocks out two options; (2) rising or falling decides the sign; (3) only then estimate the size of the slope from two points on the line.

Desmos move · In Bluebook’s Desmos

Type each option (y = 3.4 + 1.55x …) and see which one lies on the drawn line, reading the axes of the picture. With the data points given in a table, type x₁, y₁ in a table and y₁ ~ m x₁ + b — Desmos returns the line of best fit itself.

Real question 1SAT · May US 2024 · Math 800 Book Q7

$figure from the test$

Which of the following equations best represents the line of best fit shown?

🧑‍🏫 Rule 5 · A prediction comes from the LINEstep 1 of 3

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📋 The question · SAT · October US 2024 · Math 800 Book Q16
$figure from the test$
The scatterplot shows the body length, in centimeters (cm), of a New Zealand fur seal from an age of 1 to 6 years old. A line of best fit is also shown. For a New Zealand fur seal at an age of 3 years old, what is the body length, in centimeters, predicted by the line of best fit, to the nearest 10 cm?
(type the answer)
up to the line, across
$x = 3 \Rightarrow y \approx 109 \approx 110$
October US 2024: from age 3 go up to the line (not to a dot), then across to the $y$ -axis: about 109 → 110 cm.

🧑‍🏫 Rule 6 · Slope and intercept in contextstep 1 of 3

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📋 The question · Gen Z Study · slope in context
The line of best fit $y = 12 + 3.5 x$ models the height $y$ , in centimeters, of a plant $x$ weeks after it was planted. What is the best interpretation of 3.5 in this context?
1. A)The plant’s height when it was planted
2. B)The predicted increase in height, in centimeters, per week
3. C)The number of weeks the plant grew
4. D)The plant’s height after 3.5 weeks
the slope
$y = 12 + 3.5 x$
3.5 is the predicted change in $y$ for each increase of 1 in $x$ : 3.5 cm more per week (B).

🧑‍🏫 Rule 7 · Shifted data, shifted linestep 1 of 4

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📋 The question · SAT · August US 2024 · Math 800 Book Q13
$figure from the test$
The scatterplot shows the relationship between two variables, $x$ and $y$ , for the data set P. A line of best fit for the data is also shown. Data set Q is created by adding 540 units to the value of $x$ for each data point from the data set P. Which of the following could be an equation of a line of best fit for the data set Q?
1. A) $y = - 3 + \frac{35}{1301} x$
2. B) $y = 9 + \frac{1}{131} x$
3. C) $y = 18 + \frac{1}{131} x$
4. D) $y = 26 + \frac{35}{1301} x$
the line for P
$y \approx 11.5 + 0.027 x$
August US 2024: from about 14.2 at $x = 100$ to 15.0 at $x = 130$ : slope $\frac{0.8}{30} \approx 0.027 \approx \frac{35}{1301}$ .

🧑‍🏫 Rule 8 · How far to trust itstep 1 of 2

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📋 The question · Gen Z Study · interpolation vs extrapolation
A line of best fit was made from data for children aged 4 to 12. For which age is a prediction from the line LEAST reliable?
1. A)5
2. B)8
3. C)11
4. D)30
interpolation vs extrapolation
$4 \leq x \leq 12 ✓, x = 30 \times$
Predicting inside the data’s range is reasonable; far outside it, the trend may not hold. D.

🧑‍🏫 Rule 9 · Read the shape, then the numbersstep 1 of 5

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📋 The question · SAT · March 2024 · Math 800 Book Q6
$figure from the test$
The scatterplot shows the relationship between two variables, $x$ and $y$ . Which of the equations is the most appropriate model for the data shown?
1. A) $y = x^{2} - 8$
2. B) $y = x^{2} + 8$
3. C) $y = (x - 8)^{2}$
4. D) $y = (x + 8)^{2}$
U-shape → quadratic
$y = x^{2} + 8$
March 2024: lowest point near (0, 8): no shift left or right, lifted 8. B. ( $(x - 8)^{2}$ would put the bottom at $x = 8$ .)

🧑‍🏫 Rule 10 · Read and comparestep 1 of 2

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📋 The question · SAT · August US 2023 · Math 800 Book Q3
$figure from the test$
The line graphs show the number of games won by two football teams from 2004 to 2010. For how many of the years shown was the number of games won by Team A greater than the number of games won by Team B?
1. A)1
2. B)2
3. C)6
4. D)7
compare two lines
$2004, 05, 07, 08, 09, 10$
August US 2023: Team A is higher in 6 of the 7 years (not in 2006). C.

🧑‍🏫 Rule 11 · The steepest changestep 1 of 1

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📋 The question · SAT · August 2024 · Math 800 Book Q12
$figure from the test$
During which of the following time periods did the greatest increase in recorded temperature take place?
1. A)From $x = 6$ to $x = 7$
2. B)From $x = 5$ to $x = 6$
3. C)From $x = 2$ to $x = 3$
4. D)From $x = 1$ to $x = 2$
biggest rise between two neighbours
$2 \to 3 : 60 \to 75 (+ 15)$
August 2024: compare the rises: 1→2 is a FALL, 5→6 a fall, 6→7 only +2. The steepest climb is from $x = 2$ to $x = 3$ : C.

🧑‍🏫 Rule 12 · Average rate of change and percentstep 1 of 3

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📋 The question · SAT · August US 2023 · Math 800 Book Q22
$figure from the test$
The same line graph (NSF grants to museums in Minnesota, 2000–2006). What is the approximate average rate of change of the amount, in millions of dollars per year, that the NSF granted to museums in Minnesota from 2000 to 2006?
1. A) $- 2.07$
2. B) $- 0.48$
3. C) $0.41$
4. D) $0.82$
only the ends matter
$\frac{f ( 2006 ) - f ( 2000 )}{2006 - 2000} = \frac{1 - 3.9}{6}$
August US 2023: the ups and downs between do not count.

Real question 2SAT · August US 2023 · Math 800 Book Q22

$figure from the test$

What is the approximate average rate of change of the amount, in millions of dollars per year, that the NSF granted to museums in Minnesota from 2000 to 2006?

Take with you

▸Read a scatterplot for direction (+ / − / none), form (straight / curved), strength (tight / loose) and outliers.
▸Line of best fit: through the middle of the cloud, residuals (actual − predicted) small and balanced.
▸“Which equation?” → y-intercept first, then the sign (rise/fall), then the size from two points ON THE LINE; mind the axis scales.
▸Predict by going up to the LINE; slope = change in y per 1 of x (with units), intercept = y at x = 0.
▸Adding a constant to every x keeps the slope and moves the intercept by −(slope × constant).
▸U-shape → quadratic, constant ratio / bending upward → exponential, straight → linear.
▸Line graphs: compare point by point; average rate of change uses only the two ends.
▸Do not trust a line far outside its data, and an association never proves a cause.

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