Teacher Guide: Exponential Decay

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

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10 questions on Exponential Functions. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives

Write exponential decay models in the form $y = a(1 - r)^t$ from a percent decrease
Distinguish the decay rate from the decay factor and convert between them
Identify growth or decay from the base of an exponential function
Calculate values and the percent remaining after $n$ periods in depreciation, medicine and cooling contexts
Compare linear and exponential decrease and explain why their graphs differ

Prerequisites

• Introduction to Exponential Functions: the form $f(x) = a \cdot b^x$
• Exponential Growth: growth factor $1 + r$ and percent increase
• Converting percentages to decimals
• Evaluating powers with a calculator

Discussion Starters

1. A shop says a phone 'loses 20% every year'. Will it be worth nothing after 5 years? Why or why not?
2. Why does a new car lose more money in its first year than in its fifth year?
3. Would you rather a debt shrink by 100 dollars a month or by 10% a month? Does the answer depend on the size of the debt?
4. Why might a doctor tell you to take a medicine every 8 hours instead of once a day?
5. Can you think of something that decays exponentially but never quite reaches zero?

Common Misconceptions

A 15% decrease means the base is 0.15

Remediation: Ask 'what fraction is left?' after one period. With 100 units and a 15% loss, 85 remain, so the multiplier is 0.85. Have students say 'lose 15, keep 85' before writing any formula.

A percent decrease takes away the same amount every period

Remediation: Build a table side by side: subtract 200 each year versus multiply by 0.8 each year from 1000. Students see the percent losses shrink (200, 160, 128...) while the linear losses stay the same.

Exponential decay ends at zero after enough periods

Remediation: Use the sequence visualizer with ratio 0.8 and many terms. Each bar is 80% of the previous one, so it can never be zero. Connect this to the horizontal asymptote $y = 0$ .

Differentiation Ideas

For Struggling Students:

• Start with factors that give clean numbers (0.5, 0.8, 0.9) and short tables before using the formula
• Use the phrase 'lose r, keep 1 - r' and a two-column table (lost, kept) for each problem
• Allow a calculator with prepared expressions like 1000 × 0.8^3

For On-Level Students:

• Model depreciation, medicine and cooling problems with $y = a(1 - r)^t$
• Compute the percent remaining after $n$ periods
• Compare a linear and an exponential decrease and find when one overtakes the other

For Advanced Students:

• Find the decay rate from two data points using roots, for example $b^2 = 0.81$
• Investigate the time it takes to lose half of the value (preview of half-life)
• Research how the simple cooling model relates to Newton's law of cooling

Standards Alignment

HSF-LE.A.1c (CCSS.MATH.CONTENT.HSF.LE.A.1.C)
Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another
HSF-IF.C.8b (CCSS.MATH.CONTENT.HSF.IF.C.8.B)
Use the properties of exponents to interpret expressions for exponential functions and classify them as representing exponential growth or decay
HSF-LE.B.5 (CCSS.MATH.CONTENT.HSF.LE.B.5)
Interpret the parameters in a linear or exponential function in terms of a context

Lesson Resources

visualGeometric Decay Bars
Sequence visualizer with first term 100 and ratio 0.8: students watch each bar shrink by 20% and read the percent remaining
activityCar Lot Depreciation
Groups receive car prices and yearly loss rates, build decay tables and decide which car keeps its value best
activityLinear or Exponential?
Card sort of tables and descriptions into constant-amount versus constant-percent decrease
worksheetDecay in Daily Life
Depreciation, medicine and cooling problems with stated rounding

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

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.

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.

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

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.

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

Look at the base

b

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.

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.

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.

Teacher Guide: Exponential Decay

Use this lesson with your class

For Teachers

For Struggling Students:

For On-Level Students:

For Advanced Students:

Lesson Content

Definition

Worked Examples

Common Mistakes

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

The Natural Base e