How can I tell growth from decay just by looking at the formula?

Look at the base $b$ in $y = a \cdot b^t$. If $b > 1$, the quantity grows (for example $1.04$ means $+4\%$). If $0 < b < 1$, it decays (for example $0.96$ means $-4\%$).

Does a decaying quantity ever reach zero?

Not in the model. Each period keeps a fixed fraction of a positive number, so the result stays positive. The graph gets closer and closer to the line $y = 0$, its horizontal asymptote. In real life we stop counting once the amount is too small to matter.

The half-life is the time it takes for exactly half of the quantity to be left. It is a special case of exponential decay with its own formula, covered in the lesson Half-Life Problems.

Teacher Resources

Exponential Decay

Q: What is the difference between the decay rate and the decay factor?

The decay rate $r$ is what you lose each period (for example $15\% = 0.15$). The decay factor is what you keep: $1 - r = 0.85$. The decay factor is the number that goes into the base of the power.

Q: Does a decaying quantity ever reach zero?

Not in the model. Each period keeps a fixed fraction of a positive number, so the result stays positive. The graph gets closer and closer to the line $y = 0$, its horizontal asymptote. In real life we stop counting once the amount is too small to matter.

Q: What is half-life?

The half-life is the time it takes for exactly half of the quantity to be left. It is a special case of exponential decay with its own formula, covered in the lesson Half-Life Problems.

Learn how quantities shrink by a constant percentage, use the decay formula y = a(1 - r)^t and solve depreciation, medicine and cooling problems.

Advanced25 minLesson

Printable worksheet

Definition

Exponential decay happens when a quantity decreases by the same percentage in every equal time period. The amount lost gets smaller each time, because each new loss is a percentage of what is left.

y = a(1 - r)^t

Where:

$a$ = initial value (the amount at $t = 0$ )
$r$ = decay rate, the percent lost per period, written as a decimal ( $15\% \to r = 0.15$ )
$1 - r$ = decay factor $b$ , the part that is kept each period ( $1 - 0.15 = 0.85$ )
$t$ = number of time periods

In the general form

y = a \cdot b^t

you can tell the type from the base:

$b > 1$ : exponential growth
$0 < b < 1$ : exponential decay

After

t

periods, the fraction that remains is

(1 - r)^t

, no matter how big the starting amount is.

Try it now

Which function shows exponential decay?

Worked Examples

For each function, decide whether it shows growth or decay and find the percent change per period: $f(t) = 300 \cdot 0.92^t$ and $g(t) = 50 \cdot 1.03^t$ .

Look at the base of $f$

$b = 0.92$ and $0 < 0.92 < 1$ → $f$ shows decay

Find the decay rate of $f$

$r = 1 - b = 1 - 0.92 = 0.08$ → $f$ loses $8\%$ per period

Look at the base of $g$

$b = 1.03 > 1$ , so $r = 1.03 - 1 = 0.03$ → $g$ grows $3\%$ per period

Read the initial values

$f(0) = 300 \cdot 0.92^0 = 300$ and $g(0) = 50$ → Starting values $300$ and $50$

Answer: $f$ is exponential decay at $8\%$ per period (starting at $300$ ); $g$ is exponential growth at $3\%$ per period (starting at $50$ ).

Common Mistakes

Using the decay rate as the base: writing $25000 \cdot 0.15^t$ for a $15\%$ loss

Why it's wrong: $0.15^t$ keeps only $15\%$ each year, which means losing $85\%$ per year. The base must be the part that is kept.

Correct: Decay factor $= 1 - r$ . A $15\%$ decrease gives $25000 \cdot 0.85^t$ .

Subtracting the same amount every period: '20% of 1000 is 200, so after 5 years the value is 0'

Why it's wrong: That is linear thinking. Each year's loss is $20\%$ of the current value, not of the original value, so the losses shrink.

Correct: Multiply by $0.8$ each year: $1000 \to 800 \to 640 \to 512 \to 409.6 \to 327.68$ . After 5 years $327.68$ is left.

Computing $a \cdot 1 - r^t$ instead of $a(1 - r)^t$

Why it's wrong: The exponent belongs to the whole decay factor. $1 - 0.15^2 = 0.9775$ , but $(1 - 0.15)^2 = 0.7225$ .

Correct: Find $1 - r$ first, then raise it to the power $t$ , then multiply by $a$ .

Believing a decaying quantity eventually reaches exactly zero

Why it's wrong: Multiplying a positive number by a factor like $0.8$ always gives a positive number.

Correct: In the model $y = a(1 - r)^t$ the values get closer and closer to $0$ (the horizontal asymptote $y = 0$ ) but never reach it.

Interactive Visual

Function graph

a = 400r = 0.2

Tap or hover over the graph to read off values.

y = a(1 − r)^tAsymptote

Table of values

t	0	1	2	3	4	5
y	400	320	256	204.8	163.84	131.072

Sequence Explorer

First Term (a₁ = 100)

Common Ratio (r = 0.8)

100,

80,

64,

51.2,

40.96,

32.768,

26.214,

20.972

, ...

Formula

a_n = 100 × 0.8^(n-1)

a_1

100

a_5

40.96

a_10

13.422

a_20

1.441

Interactive Sandbox

Expression Calculator

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History

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Practice Problems

18 problems

Problem 1 of 18

Easy

Which function shows exponential decay?

Why It Matters

Exponential decay describes many things that shrink over time:

Depreciation: A new car or phone loses a percentage of its value every year, so it loses the most money at the start
Medicine: The body removes a fixed percentage of a drug each hour, which decides how often a dose is needed
Cooling: A hot drink cools quickly at first and then more and more slowly as it nears room temperature
Light and sound: Each meter of water or wall absorbs a percentage of the light or sound passing through

Knowing the decay formula lets you predict values, compare offers (a fixed amount off per year versus a percentage off) and understand why some things never quite reach zero.

Real World Applications

Depreciation of Phones and Laptops

Electronics lose a percentage of their value each year. Resale websites and insurers use decay models to estimate what a used device is worth.

Example:

A 1000-dollar laptop that loses $30\%$ per year is worth $1000 \cdot 0.7^2 = 490$ dollars after 2 years.

1Try It Yourself

A new phone costs 1200 dollars and loses $25\%$ of its value each year.

What is the phone worth after 2 years?

Step 1: Write the mathematical expression

Use the decay formula: $1200 \cdot 0.75^2$

Medicine Dosing

Doctors and pharmacists use exponential decay to know how much of a drug is still in the body and when the next dose should be taken.

Example:

If the body removes $15\%$ of a drug each hour, then after 4 hours $0.85^4 \approx 0.522$ , so about $52.2\%$ of the dose remains.

2Try It Yourself

A patient takes 200 mg of a medicine. The body removes $30\%$ of the remaining amount every hour.

How many milligrams are left after 4 hours?

Step 1: Write the mathematical expression

Calculate: $200 \cdot 0.7^4$

Cooling Drinks

Engineers and food scientists model cooling with decay: the temperature difference between an object and the room shrinks by a percentage each minute.

Example:

Coffee at $90°C$ in a $20°C$ room: if the difference of $70$ shrinks $10\%$ per minute, after 1 minute it is $63$ , so the coffee is $83°C$ .

3Try It Yourself

A cup of tea is $80°C$ in a room at $20°C$ . In a simple model the temperature difference shrinks by $20\%$ every minute.

What is the tea's temperature after 3 minutes?

Step 1: Write the mathematical expression

Calculate: $20 + 60 \cdot 0.8^3$

Key Takeaways

1Exponential decay follows $y = a(1 - r)^t$ : the quantity loses the same percentage every period
2The decay rate $r$ is the percent lost; the decay factor $1 - r$ is the part kept (a $15\%$ decrease gives $0.85$ )
3In $y = a \cdot b^t$ , a base $0 < b < 1$ means decay and $b > 1$ means growth
4After $t$ periods the fraction remaining is $(1 - r)^t$ ; multiply by $100\%$ for the percent remaining
5Linear decrease subtracts the same amount and can reach zero; exponential decrease loses less each period and only approaches zero
6Round only at the end and state the rounding (nearest cent, one decimal place)

Frequently Asked Questions

The decay rate

r

is what you lose each period (for example

15\% = 0.15

). The decay factor is what you keep:

1 - r = 0.85

. The decay factor is the number that goes into the base of the power.

Glossary

Exponential decay: A decrease by the same percentage over equal time periods, modeled by $y = a(1 - r)^t$
Decay rate: The percent lost per period, written as a decimal $r$ (for example $0.15$ for $15\%$ )
Decay factor: The part kept each period, $b = 1 - r$ , with $0 < b < 1$
Initial value: The amount $a$ at time $t = 0$
Depreciation: The loss of value of an object, such as a car or a phone, over time
Horizontal asymptote: The line $y = 0$ that a decay graph approaches but never touches

Formula Card

Exponential Decay Formula

y = a(1 - r)^t

$a$ = initial value, $r$ = decay rate as a decimal, $1 - r$ = decay factor, $t$ = number of periods.

Decay Factor from Percent

b = 1 - r

A decrease of $p\%$ per period gives $b = 1 - \frac{p}{100}$. Example: $15\% \to 0.85$.

Percent Remaining

(1 - r)^t \times 100\%

The percent of the starting amount left after $t$ periods, whatever the starting amount is.

Exponential Decay

Definition

Worked Examples

Common Mistakes

Interactive Visual

Function graph

Table of values

Sequence Explorer

Formula

Interactive Sandbox

Expression Calculator

History

Practice Problems

Why It Matters

Real World Applications

Depreciation of Phones and Laptops

Medicine Dosing

Cooling Drinks

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Exponential Decay Formula

Decay Factor from Percent

Percent Remaining

More in This Topic

Introduction to Exponential Functions

Exponential Growth

Graphing Exponential Functions

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