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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)ty = 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 nn 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⋅bxf(x) = a \cdot b^x
  • • Exponential Growth: growth factor 1+r1 + 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=0y = 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)ty = a(1 - r)^t
  • • Compute the percent remaining after nn 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 b2=0.81b^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)ty = a(1 - r)^t
Where:
  • aa = initial value (the amount at t=0t = 0)
  • rr = decay rate, the percent lost per period, written as a decimal (15%→r=0.1515\% \to r = 0.15)
  • 1−r1 - r = decay factor bb, the part that is kept each period (1−0.15=0.851 - 0.15 = 0.85)
  • tt = number of time periods
In the general form y=a⋅bty = a \cdot b^t you can tell the type from the base:
  • b>1b > 1: exponential growth
  • 0<b<10 < b < 1: exponential decay
After tt periods, the fraction that remains is (1−r)t(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⋅0.92tf(t) = 300 \cdot 0.92^t and g(t)=50⋅1.03tg(t) = 50 \cdot 1.03^t.
1

Look at the base of ff

b=0.92b = 0.92 and 0<0.92<10 < 0.92 < 1 → ff shows decay
2

Find the decay rate of ff

r=1−b=1−0.92=0.08r = 1 - b = 1 - 0.92 = 0.08 → ff loses 8%8\% per period
3

Look at the base of gg

b=1.03>1b = 1.03 > 1, so r=1.03−1=0.03r = 1.03 - 1 = 0.03 → gg grows 3%3\% per period
4

Read the initial values

f(0)=300⋅0.920=300f(0) = 300 \cdot 0.92^0 = 300 and g(0)=50g(0) = 50 → Starting values 300300 and 5050

Common Mistakes

Using the decay rate as the base: writing 25000⋅0.15t25000 \cdot 0.15^t for a 15%15\% loss

Why it's wrong: 0.15t0.15^t keeps only 15%15\% each year, which means losing 85%85\% per year. The base must be the part that is kept.

Correct: Decay factor =1−r= 1 - r. A 15%15\% decrease gives 25000⋅0.85t25000 \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%20\% of the current value, not of the original value, so the losses shrink.

Correct: Multiply by 0.80.8 each year: 1000→800→640→512→409.6→327.681000 \to 800 \to 640 \to 512 \to 409.6 \to 327.68. After 5 years 327.68327.68 is left.

Computing a⋅1−rta \cdot 1 - r^t instead of a(1−r)ta(1 - r)^t

Why it's wrong: The exponent belongs to the whole decay factor. 1−0.152=0.97751 - 0.15^2 = 0.9775, but (1−0.15)2=0.7225(1 - 0.15)^2 = 0.7225.

Correct: Find 1−r1 - r first, then raise it to the power tt, then multiply by aa.

Believing a decaying quantity eventually reaches exactly zero

Why it's wrong: Multiplying a positive number by a factor like 0.80.8 always gives a positive number.

Correct: In the model y=a(1−r)ty = a(1 - r)^t the values get closer and closer to 00 (the horizontal asymptote y=0y = 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%30\% per year is worth 1000⋅0.72=4901000 \cdot 0.7^2 = 490 dollars after 2 years.

1Try It Yourself

A new phone costs 1200 dollars and loses 25%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⋅0.7521200 \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%15\% of a drug each hour, then after 4 hours 0.854≈0.5220.85^4 \approx 0.522, so about 52.2%52.2\% of the dose remains.

2Try It Yourself

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

How many milligrams are left after 4 hours?

Step 1: Write the mathematical expression

Calculate: 200⋅0.74200 \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°C90°C in a 20°C20°C room: if the difference of 7070 shrinks 10%10\% per minute, after 1 minute it is 6363, so the coffee is 83°C83°C.

3Try It Yourself

A cup of tea is 80°C80°C in a room at 20°C20°C. In a simple model the temperature difference shrinks by 20%20\% every minute.

What is the tea's temperature after 3 minutes?

Step 1: Write the mathematical expression

Calculate: 20+60⋅0.8320 + 60 \cdot 0.8^3

Key Takeaways

  • 1Exponential decay follows y=a(1−r)ty = a(1 - r)^t: the quantity loses the same percentage every period
  • 2The decay rate rr is the percent lost; the decay factor 1−r1 - r is the part kept (a 15%15\% decrease gives 0.850.85)
  • 3In y=a⋅bty = a \cdot b^t, a base 0<b<10 < b < 1 means decay and b>1b > 1 means growth
  • 4After tt periods the fraction remaining is (1−r)t(1 - r)^t; multiply by 100%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 rr is what you lose each period (for example 15%=0.1515\% = 0.15). The decay factor is what you keep: 1−r=0.851 - 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 bb in y=a⋅bty = a \cdot b^t. If b>1b > 1, the quantity grows (for example 1.041.04 means +4%+4\%). If 0<b<10 < b < 1, it decays (for example 0.960.96 means −4%-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=0y = 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)ty = a(1 - r)^t
Decay rate
The percent lost per period, written as a decimal rr (for example 0.150.15 for 15%15\%)
Decay factor
The part kept each period, b=1−rb = 1 - r, with 0<b<10 < b < 1
Initial value
The amount aa at time t=0t = 0
Depreciation
The loss of value of an object, such as a car or a phone, over time
Horizontal asymptote
The line y=0y = 0 that a decay graph approaches but never touches

Formula Card

Exponential Decay Formula

y=a(1−r)ty = 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−rb = 1 - r

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

Percent Remaining

(1−r)t×100%(1 - r)^t \times 100\%

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

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