Understand SAT exponential functions y = a·b^x: interpret growth and decay, convert percent rates to factors, read a and b, and separate exponential from linear models.
$a$ = initial value, $b$ = factor per step.
The output when $x = 0$, since $b^0 = 1$.
A rate of $r$ (as a decimal) increasing; e.g. $6\%$ growth → $b = 1.06$.
Decreasing; e.g. $6\%$ decay → $b = 0.94$.
An exponential function has the form , where the variable sits in the exponent. That single structural fact — the variable is an exponent, not a base — separates exponential behavior from linear behavior and drives every SAT question in this family. In a linear model you add the same amount each step; in an exponential model you multiply by the same factor each step. Master that contrast and most of these questions solve themselves.
In there are exactly two numbers to interpret:
The size of sets the direction:
| Value of | Behavior | Meaning |
|---|---|---|
| Growth | Quantity increases each step | |
| Constant | No change | |
| Decay | Quantity shrinks each step |
The most tested micro-skill here is converting a percent rate into a multiplier. If a quantity grows by each year, you keep the original and add , so the factor is . If it decays by each year, you keep only what remains, so . Read it backward and you fail the item: means a decrease (because remains), not a decrease.
A colony of bacteria starts at cells and increases by every hour. The starting amount is , and a increase means each hour the colony becomes of its previous size, so . The model is
After hours,
about cells. Notice the count did not climb by a fixed amount each hour — it rose by , then , then about . That accelerating jump is the fingerprint of exponential growth.
The SAT repeatedly asks you to decide whether a situation is linear or exponential. Ask one question: does the quantity change by a constant amount or by a constant percent/factor?
A table settles it too. If successive outputs share a common difference (each is the previous plus a fixed number), the model is linear. If they share a common ratio (each is the previous times a fixed number), it is exponential. For each term doubles the last — a common ratio of — so .
A favorite question shows a model such as and asks what a number means. Here is the initial amount, and says the quantity keeps of its value each period — equivalently, it drops by per period. Being able to translate into "a decay" in plain words is exactly what these questions reward.
Sometimes you are given two data points and must find . If a quantity is at and at , then , so and — a growth rate per year. Whenever a factor is trapped under a power, isolate it and take the matching root.
A population is modeled by . What is the initial population, and what is the yearly percent change?
A bacteria colony starts at cells and grows by per hour. Write a model and find the count after hours.
A sample of mg of a substance decays by each hour. How much remains after hours, to the nearest tenth of a milligram?
A sequence of outputs at is . Write a model, then predict the output at .
A car’s value follows (in hundreds of dollars, in years). By what percent does the value change each year, and what is after years?
An exponential function passes through and . Find and then .
Using the bare rate as the factor: writing for a increase.
A increase keeps and adds , so . Using would erase almost everything each step, which is a decay.
Treating a "constant percent change" as a linear model, e.g. .
A constant percent means multiplying each step, which is exponential: . Only a constant amount per step is linear.
Reading decay backward: interpreting as a decrease.
means remains, so the quantity decreases by each period. Subtract the factor from to get the percent lost.
Placing the initial value at instead of .
The coefficient is the value at , because . At the quantity has already been multiplied by once.
A savings account contains $800 and its balance grows by 3% each year. Which function models the balance after years?
A car is worth $24{,}000 and loses 15% of its value each year. To the nearest dollar, what is the car worth after 2 years?
The function models a quantity. What is the value of the quantity when ?
A town’s population starts at and increases by 12% every year. Which function correctly models the population after years?
In the model , what does the number represent?
A bacteria culture starts at 300 cells and doubles every hour. After how many hours will the culture first reach 2{,}400 cells?
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