SAT Math Review · Problem-Solving and Data Analysis
Linear and exponential models of data
Real measurements never land exactly on a curve. Your job is to say which curve they are trying to land on — and then to say only what that model actually supports.
Which of the following equations best models the data?
A biologist counts the lily pads on a pond once a week. At weeks 0, 1, 2, 3, 4 the counts are 18, 29, 46, 74, 118. Which equation best models the number of pads, y, in terms of the number of weeks, x?
Divide down the column: 2918 ≈ 1.61, 4629 ≈ 1.59, 7446 ≈ 1.61, 11874 ≈ 1.59. The ratio is about 1.6 every week, so the count is exponential with starting value 18 and growth factor 1.6: y = 18(1.6)x. A is the linear reading — the average weekly increase is 118 − 184 = 25, but the actual increases are 11, 17, 28 and 44, nowhere near constant, and A would predict only 68 pads in week 2. C uses 0.6, the part that is added rather than the whole multiplier, so it shrinks the pond to 2 pads by week 4. D swaps the two roles: 1.6 as the starting count and 18 as the growth factor gives 1.6 × 18⁴ ≈ 168,000 pads.
1
Straight or curved: the scatter itself tells you which model to fit.
If the cloud of dots runs along a straight path, fit a line. If it bends — steepening or flattening as it goes — fit an exponential. The direction, up or down, does not decide it; the bend does.
even band → linesteep then flat → exponentialnever forced through the first dot
Both data sets are perfectly ordinary measurements, and neither sits exactly on its fit. The first climbs by 36 hundred tickets across nine days at a steady pace; the second loses most of its light in the first two feet and then keeps losing a quarter of whatever is left.
2
From a table: subtract first, and if that fails, divide.
Here are the first five depth readings from the lake, in lumens. Equal steps in x, so both tests are fair.
Depth x (ft)
Light L (lumens)
Difference
Ratio
0
500
—
—
1
372
−128
0.744
2
283
−89
0.761
3
211
−72
0.746
4
160
−51
0.758
differences: −128, −89, −72, −51
shrinking fast — not a constant difference
ratios: 372500 = 0.744, 283372 = 0.761
divide each reading by the one above it
211283 = 0.746, 160211 = 0.758
all four sit within 0.75 ± 0.02
constant ratio ≈ 0.75 → exponential
round the ratios to one sensible value
Measured ratios are never identical — that is the point
In a made-up table the ratios come out to exactly 1.5 four times. In a data table they come out 0.744, 0.761, 0.746, 0.758, and you are meant to call that constant. Compare the wobble to the trend: the ratios move by 2%, while the differences fall from 128 to 51, a collapse of 60%. Whichever column is steadier names the model.
What forcing a line would do here
Take the first difference, −128 lumens per foot, and run it out: at 6 feet the line predicts 500 − 128(6) = −268 lumens. Negative light. A quantity that keeps falling but can never reach zero is exponential decay, and a line will always drive it through the axis eventually.
3
Build the equation: a is the value at x = 0, b is the ratio.
Every exponential model on the SAT has the shape y = a(b)x. The data hands you both letters.
a = 500
the reading at the surface, x = 0
b ≈ 0.75
the steady ratio from the table
L = 500(0.75)x
check x = 2: 500 × 0.5625 = 281 vs 283 measured ✓
b > 1 is growth, 0 < b < 1 is decay. Turn b into a percent by comparing it to 1: 0.75 is 0.25 below 1, so the light decreases by about 25% for every additional foot of depth. A base of 1.6 would be 0.6 above 1: an increase of about 60% per step.
Fitted model
b
What the data is doing
18(1.6)x
1.6
growing about 60% per week
240(1.04)x
1.04
growing about 4% per year
500(0.75)x
0.75
losing about 25% per foot
96(0.5)x
0.5
halving each step
0.75 is not “a 75% decrease”
0.75 is the fraction that survives each step. The decrease is 1 − 0.75 = 0.25, or 25%. Choices are routinely built from the base itself — “decreases by 75%”, “decreases by 0.75 lumen per foot” — and both are the right number wearing the wrong meaning.
4
“The model predicts” means substitute — never look for the dot.
When a stem says “based on the model”, “the model estimates” or “according to the line of best fit”, it is asking for the value of the equation, even if a measurement is sitting right there on the graph.
L(6) = 500(0.75)6
the question is about the curve, not the dot
0.756 = 0.177978…
one keystroke in Desmos, no rounding yet
L(6) = 89.0 lumens
the reading at 6 feet was 91 — close, not equal
“the model predicts” → the equation“the measurement was” → the dot“closest to” → round at the end
Round only once, at the last line. Rounding 0.756 to 0.18 first gives 90 lumens, which is close enough here but drifts badly once the exponent is large — and “closest to” choices are often only a few units apart.
5
A model is only evidence inside the range of the data.
The lake readings run from 0 to 8 feet. The equation happily accepts x = 60, but nothing was ever measured there, so the number it returns is arithmetic rather than evidence.
inside: 6 feet
500(0.75)6 ≈ 89
between measured depths — supported
outside: 20 feet
500(0.75)20 ≈ 1.6
far past the last dot — arithmetic only
the exponential never reaches 0
it keeps a quarter of a quarter of a quarter…
the ticket line never stops climbing
4x + 6 at x = 200 is 806 hundred tickets
but the venue holds 12,000 seats
the situation has a ceiling the model does not
Answer the arithmetic; doubt the claim
“What does the model predict at 20 feet?” has the answer 1.6 lumens, and you should give it. “Which statement is best supported by the data?” is a different question, and a statement about 20 feet is not supported — the choice you want is the one that stays inside 0 to 8 feet. Read which of the two is being asked before you compute anything.
6
Desmos will fit either model and let you compare them.
Put the table in once; test both shapes against it in under a minute.
table columns x1, y1
the dots appear as you type
y1 ~ mx1 + b
the straight fit
y1 ~ a·cx₁
the exponential fit — a and c come out in the panel
look at which one hugs the dots
and at R², which is closer to 1 for the better fit
The faster move on a multiple-choice item
You rarely need the regression at all. Plot the data, then type the four candidate equations and see which curve runs through the cloud — a wrong base or a swapped pair of constants misses by an order of magnitude and is visible instantly. Testing choices against a plot is usually quicker than deriving the model from scratch.
✓
Check: four decisions, one data set.
which modelratios ≈ 0.75, differences wildexponential ✓
the equationL = 500(0.75)xa from x = 0, b from the ratio ✓
in words1 − 0.75 = 0.25down about 25% per foot ✓
the predictionL(6) ≈ 89 lumensinside 0–8 ft, so it is supported ✓
Remember the flow
Subtract, then divide · a at x = 0, b from the ratio · compare b to 1 · stay inside the data
Steadier column wins. The model predicts; the data measured.
Now you try
Practice problem
Which of the following is closest to the number the model predicts?
Weekly downloads of an app, y, in thousands, x months after release were recorded for 0 ≤ x ≤ 6 and are well modeled by y = 250(1.3)x. According to the model, which is closest to the weekly downloads, in thousands, 9 months after release?
Substitute x = 9: 1.39 ≈ 10.604, so y ≈ 250 × 10.604 ≈ 2,651 thousand. A treats the growth as a fixed 30% of the starting value, 250 + 9(75) = 925, which is the linear reading of an exponential model. B stops at the edge of the data, computing 250(1.3)6 ≈ 1,207 — the last month actually measured, not month 9. C uses the exponent 8, the classic off-by-one from counting the months between release and month 9 instead of the months after release. Note that month 9 is three months past the data, so the arithmetic is sound but a claim about real downloads there would not be supported by these observations.
Hint: the exponent counts periods since x = 0, and nothing about “outside the data range” changes what the equation returns. Compute first, then decide whether the question wanted a value or a claim.
