The SAT tests exponential vs linear in three predictable ways: (1) given a table, identify the model (check differences, then ratios); (2) given a word problem with percentages, write the formula (growth rate → multiplier 1 + r; decay rate → 1 − r); (3) compare two functions and identify when one overtakes the other. The #1 trap is students writing y = a(r)^x when the correct form is y = a(1 + r)^x — SAT answer choices always include both versions.
Another classic: percent growth sounds slow but compounds. If a question says 'the population grows by 5% per year', many students mistakenly use 0.05 as the multiplier (which would shrink the population to nothing). The multiplier is 1.05.
Strategy: when the problem mentions doubling, halving, percent increase/decrease over time, or compound interest, you're in exponential territory. When it mentions a constant rate like 'per minute' or 'each year she adds', you're in linear territory.
The Intuition
Linear growth adds the same amount each step; exponential growth multiplies by the same factor each step. Two savings accounts: Account A adds
0 each week (linear); Account B starts at
and doubles each week (exponential). After 10 weeks Account A has
00, but Account B has
,024. The SAT's favorite trap is a table where you have to spot whether the pattern has constant DIFFERENCES (linear) or constant RATIOS (exponential).
20 SAT-styled questions. Pick an answer to see the explanation immediately.
1.A population of bacteria doubles every hour. If there are 300 bacteria now, how many will there be in 4 hours?
2.A savings account earns 5% annual interest, compounded yearly. Which equation gives the balance B after t years starting from
,000?
3.Which table represents an exponential function?
4.A car loses 20% of its value each year. If it was bought for $30,000, which expression gives its value after 5 years?
5.Functions f(x) = 100 + 50x and g(x) = 100(1.10)^x. At approximately what x does g first exceed f?
6.A sample of a radioactive substance has a half-life of 10 days. If it starts at 800 grams, how many grams remain after 30 days?
7.A town's population is modeled by P(t) = 5000(1.03)^t, where t is years since 2020. What is the annual percent growth rate?
8.Which equation models exponential DECAY?
9.A bacteria culture triples every 4 hours. Starting with 50 bacteria, how many are present after 12 hours?
10.The value y is modeled by y = 50 · 2^(t/5), where t is in years. How often does y double?
11.Linear function A adds 20 each year. Exponential function B grows by 10% each year. Both start at 100. Which is larger at year 5?
12.In the equation y = 800(0.85)^t, what does 0.85 represent?
13.A bank account balance is given by B = 1000(1.02)^(4t), where t is years. How often does the interest compound?
14.Which is the better long-term investment: Account A earning $5 per month, or Account B earning 0.5% per month compounded? Both start at
,000.
15.A chart shows: year 0: 200, year 1: 220, year 2: 242, year 3: 266.2. Which model fits?
16.A store's revenue is $50,000 in year 0 and grows by 6% per year. Which expression gives revenue in year 8?
17.Which function grows fastest as x becomes very large?
18.A car is worth
4,000 new. Its value drops to
8,000 after 1 year. If the depreciation is exponential, what is the annual decay rate?
19.A tree doubles in height every 3 years. It is currently 2 meters tall. What equation gives height H after t years?
20.Two cities grow their populations: City X adds 2,000 people each year, and City Y grows by 3% per year. Both start at 50,000. Which statement is true?
Frequently Asked Questions
How do I tell linear from exponential from a table?
Compute the differences between consecutive y-values. If they are constant, it's linear. If not, compute the ratios. If those are constant, it's exponential.
Does exponential always mean fast growth?
No — if the growth factor is between 0 and 1, it's exponential decay and approaches zero. Still exponential.
Why does exponential eventually beat linear?
Exponential growth compounds: the bigger it gets, the faster it grows. Linear growth rate is constant.