Exponential models are where the SAT quietly separates students who only memorized y = mx + b from students who understand rates. The good news: the whole topic reduces to one form and one habit. The form is y = a·bˣ. The habit is read the base b.
The one form: y = a·bˣ
Every SAT exponential model is a version of y = a·bx:
- a is the initial amount — the value when x = 0 (the starting population, the price today, the principal).
- b is the growth factor per time step. If b > 1 the quantity grows; if 0 < b < 1 it decays.
- x is the number of time periods elapsed.
Almost every exponential question is really: "read a and b out of the story, then know which way b pushes."
The core skill: percent change → base b
This one conversion is behind most SAT exponential word problems. Write the percent as a decimal r, then:
- Growth: b = 1 + r. A 3% increase → b = 1.03; growing 20% → b = 1.20.
- Decay: b = 1 − r. A 3% decrease → b = 0.97; losing 5% → b = 0.95.
Worked example — percent change
A town of 8,000 people grows 2% per year. Model the population after t years.
a = 8000, and 2% growth → b = 1.02. So P = 8000 · (1.02)t. After 10 years: 8000 · (1.02)10 ≈ 8000 · 1.219 ≈ 9,752 people (approx.).
Half-life and doubling
Half-life means the quantity multiplies by 1/2 every fixed period. The model is y = a·(1/2)t/h, where h is the half-life and t is elapsed time. Doubling works the same way with a base of 2: y = a·2t/d, where d is the doubling time.
| Half-lives elapsed | Fraction remaining | Percent left |
|---|---|---|
| 1 | 1/2 | 50% |
| 2 | 1/4 | 25% |
| 3 | 1/8 | 12.5% |
| n | (1/2)ⁿ | 100 · (1/2)ⁿ % |
Compound interest
Yes, the SAT uses it. The formula is A = P(1 + r/n)nt, where P is the principal, r the annual rate as a decimal, n the number of compoundings per year, and t the number of years. Watch the units carefully:
- Compounded annually (n = 1) simplifies to A = P(1 + r)t — the plain exponential form.
- Compounded monthly means n = 12, so the rate per period is r/12 and the exponent is 12t.
- Compounded quarterly means n = 4 (r/4 and exponent 4t).
Worked example — compound interest
$1,000 is invested at 6% annual interest, compounded monthly. Value after 2 years?
A = 1000 · (1 + 0.06/12)12·2 = 1000 · (1.005)24 ≈ $1,127.16 (approx.).
Linear vs. exponential — the constant test
A frequent SAT question type: is this situation linear or exponential? The distinction is add vs. multiply:
| Type | Each step | Wording signal |
|---|---|---|
| Linear | Adds a fixed amount (constant difference) | "increases by 50 per year" |
| Exponential | Multiplies by a fixed factor (constant ratio) | "increases by 8% per year," "doubles" |
Given a table, check successive values: constant difference is linear; constant ratio (percentage) is exponential.
For Chinese-American families · 华人家庭视角
The recurring SAT trap is confusing a percent with a base: reading "grows 5%" as multiplying by 5 or 0.05 instead of 1.05. Once a student reliably converts a percent into b = 1 ± r, exponential questions stop being a distinct category and become plug-in problems.
SAT 反复设的陷阱是把"百分比"和"底数"搞混：把"增长 5%"读成乘 5 或乘 0.05，而不是乘 1.05。一旦学生能稳定地把百分比转成 b = 1 ± r，指数题就不再是一类独立题型，而只是代入计算。
Quick reference
- Model: y = a·bˣ. a = initial amount (at x = 0); b = growth factor; b > 1 grows, 0 < b < 1 decays.
- Percent → base: growth b = 1 + r, decay b = 1 − r (r as a decimal).
- Half-life: y = a·(1/2)^(t/h). Doubling: y = a·2^(t/d).
- Compound interest: A = P(1 + r/n)^(nt); monthly → n = 12, exponent 12t.
- Linear adds a constant amount; exponential multiplies by a constant factor (percent).
Frequently asked questions
What is the exponential model on the SAT?
It is y = a·bˣ, where a is the initial amount (the value when x = 0) and b is the growth factor per time step. If b > 1 the quantity grows; if 0 < b < 1 it decays. The exponent x is the number of time periods. Almost every SAT exponential question is a matter of correctly reading a and b out of a story and knowing which way b pushes.
How do I turn a percent change into the base b?
Add or subtract the percent from 1. A 3% increase per year means b = 1.03; a 3% decrease per year means b = 0.97; growing 20% means b = 1.20; losing 5% means b = 0.95. The rule is b = 1 + r for growth and b = 1 − r for decay, where r is the percent written as a decimal. This single conversion is behind most SAT exponential word problems.
How does half-life work in an exponential model?
Half-life means the quantity multiplies by 1/2 every fixed period, so the model is y = a·(1/2)^(t/h), where h is the half-life and t is elapsed time. After one half-life you have 50% left, after two half-lives 25%, after three 12.5%, and so on. Doubling time works the same way with a base of 2: y = a·2^(t/d).
What is the compound interest formula and does the SAT use it?
Yes. The formula is A = P(1 + r/n)^(nt), where P is the principal, r is the annual rate as a decimal, n is the number of compoundings per year, and t is years. If interest compounds once a year (n = 1) it simplifies to A = P(1 + r)^t, the plain exponential form. Watch the units: 'compounded monthly' means n = 12 and the exponent is 12t.
How do I tell linear from exponential growth on the SAT?
Linear growth adds a fixed amount each step (constant difference — +5 every year). Exponential growth multiplies by a fixed factor each step (constant ratio — ×1.05 every year). In a table, check whether successive values differ by a constant amount (linear) or a constant percentage/ratio (exponential). Wording like 'increases by 50 units per year' is linear; 'increases by 8% per year' or 'doubles' is exponential.
How Lighthouse Math teaches this
Lighthouse Math (灯塔数学) is a diagnostic-driven SAT Math program for Chinese-American families in the Boston / Arlington MA area. We don't assign more volume — we map the two or three specific structural mechanisms silently capping a student's score, then close them. Every concept above is taught bilingually so the underlying math is understood, not just translated.
我们用中英双语讲清机制本身，而不是靠刷题量。先做诊断，找到隐形封顶你孩子分数的 2–3 个结构性卡点，再逐一打破 —— 让孩子真正理解数学，而不只是背题型。