English:Exponential Growth and Decay

Introduction

Exponential change appears whenever a quantity is repeatedly multiplied by the same factor over equal intervals. It can model situations such as compound interest, early stages of population growth, depreciation, cooling approximations, and radioactive decay. In this aiMOOC for Grades 9–10, you will learn how to recognize, represent, calculate, graph, interpret, and critique exponential growth and decay.

The image compares linear, power, and exponential growth. Exponential growth may begin quietly, yet repeated multiplication can eventually make it outgrow many other patterns. An exponential model is not simply a model for something that is “very fast”; its defining feature is a constant multiplicative factor or, equivalently in equal intervals, a constant percentage rate.

By the end of the course, you should be able to:

1. Recognize exponential functions: Decide whether a table, graph, equation, or situation shows exponential change.
2. Connect percentages and factors: Translate a percent increase or decrease into a growth or decay factor.
3. Build exponential models: Write and use equations of the form $y=a\cdot b^x$.
4. Interpret graphs: Identify the initial value, growth or decay behavior, and horizontal asymptote.
5. Apply models: Solve realistic problems involving finance, populations, depreciation, and half-life.
6. Evaluate models: Explain when an exponential model is useful and when real-world limits make it unrealistic.

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Understanding Exponential Change

Repeated Multiplication

Suppose a quantity starts at 100 and increases by 20% each period. A 20% increase means that you keep the original 100% and add 20%, so you multiply by 1.20 each time:

Period	Value	Calculation
0	100	Initial value
1	120	$100\cdot 1.20$
2	144	$120\cdot 1.20$
3	172.8	$144\cdot 1.20$

The differences are not constant, but the ratio between consecutive values is constant. That constant ratio is the key signal of exponential change.

For comparison, a linear pattern such as 100, 120, 140, 160 adds the same amount each period. An exponential pattern such as 100, 120, 144, 172.8 multiplies by the same factor each period.

The Basic Exponential Model

A common model is

$y=a\cdot b^x$

where:

1. a is the initial value, because $b^0=1$ and therefore $y=a$ when $x=0$.
2. b is the growth factor or decay factor.
3. x is the number of equal time intervals or another independent variable.
4. y is the modeled quantity.

If $a>0$:

1. When $b>1$, the model shows exponential growth.
2. When $0<b<1$, the model shows exponential decay.
3. When $b=1$, the quantity stays constant rather than growing or decaying.

Every basic positive exponential function $y=a\cdot b^x$ has a positive y-intercept at $(0,a)$. Without vertical shifts, its graph approaches the x-axis, $y=0$, as a horizontal asymptote.

From Percent Rate to Factor

For a percent increase of $r$, written as a decimal, use

$b=1+r$.

For a percent decrease of $r$, use

$b=1-r$.

Examples:

Change per interval	Factor	Model behavior
6% increase	1.06	Exponential growth
25% increase	1.25	Exponential growth
8% decrease	0.92	Exponential decay
35% decrease	0.65	Exponential decay

A common mistake is to confuse a 20% decrease with a factor of 0.20. A 20% decrease means 80% remains, so the correct factor is 0.80.

Exponential Growth

Recognizing Growth

In exponential growth, equal increases in the input produce equal multiplicative changes in the output. The graph rises from left to right when the initial value is positive, and its slope becomes steeper as the output becomes larger.

For example, if a digital archive starts with 250 files and grows by 12% each month, then after $t$ months a simple model is

$A(t)=250(1.12{)}^t$.

After 6 months,

$A(6)=250(1.12{)}^6\approx 493.46$.

If the context counts whole files, you would interpret this as about 493 files. The model gives a mathematical estimate; the context tells you how to round.

Doubling Time

The doubling time is the amount of time required for a growing quantity to become twice as large. If a population doubles every 3 hours and begins at 80, a useful model is

$P(t)=80\cdot 2^{t/3}$.

After 6 hours, two doubling intervals have passed, so

$P(6)=80\cdot 2^2=320$.

Doubling-time form is especially convenient when the problem directly states how long each doubling takes.

Compound Interest as Exponential Growth

With compound interest, interest is added to the current balance, so later interest is calculated on a balance that already includes earlier interest. For annual compounding at rate $r$,

$A=P(1+r{)}^t$,

where $P$ is the principal and $t$ is measured in years.

For $n$ compounding periods per year, a standard model is

$A=P{(1+\frac{r}{n})}^{nt}$.

The graph illustrates how repeated percentage growth produces a curved rather than linear pattern. In real financial decisions, fees, taxes, changing rates, and deposits or withdrawals can make the actual balance differ from a simple model.

Biological Growth and Its Limits

Bacteria are often used to illustrate exponential growth because cells can reproduce by division. During a period when resources are abundant and environmental conditions are favorable, a population may approximately follow exponential growth. Real populations, however, cannot grow exponentially forever because resources and space are limited.

The bacterial growth curve shows that an exponential phase is only one stage. This is an important modeling lesson: a mathematically correct exponential rule can still become unrealistic outside the interval where its assumptions hold.

Exponential Decay

Repeated Percentage Decrease

Exponential decay happens when the same fraction of a quantity remains after each equal interval. If a device worth 900 currency units loses 18% of its value each year, then 82% remains each year. A simple model is

$V(t)=900(0.82{)}^t$.

After 4 years,

$V(4)=900(0.82{)}^4\approx 406.91$.

The model decreases quickly at first and then more slowly in absolute terms, because each percentage loss is taken from the current, smaller amount.

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Half-Life

The half-life of a decaying quantity is the time required for the quantity to fall to half its current value. If $N_0$ is the initial amount and the half-life is $h$, then

$N(t)=N_0{\left(\frac{1}{2}\right)}^{t/h}$.

If a sample begins at 640 units and has a half-life of 5 years, then after 15 years three half-lives have passed:

$N(15)=640{\left(\frac{1}{2}\right)}^3=80$.

Notice that the model approaches zero but does not reach zero at a finite time. In the basic model, the x-axis is a horizontal asymptote.

Building Models from Representations

From a Table

To test whether equally spaced data are exponential, calculate ratios of consecutive output values. Consider:

x	y	Ratio to previous value
0	50	—
1	60	1.2
2	72	1.2
3	86.4	1.2

Because the ratio is constantly 1.2, the model is

$y=50(1.2{)}^x$.

If the differences were constant instead of the ratios, a linear model would be more appropriate.

From Two Consecutive Values

If you know the initial value $a$ and the value after one interval, you can find the factor by division:

$b=\frac{\text{new value}}{\text{previous value}}$.

For example, if a quantity changes from 240 to 204 in one period,

$b=\frac{204}{240}=0.85$.

The factor 0.85 means 85% remains, which is a 15% decrease. The model is

$y=240(0.85{)}^x$.

From a Graph

When you read an exponential graph, look for:

1. the y-intercept, which gives the initial value in the basic model;
2. whether the graph rises or falls from left to right;
3. the horizontal asymptote;
4. the scale and units on both axes;
5. whether equal horizontal steps correspond to a roughly constant multiplicative change.

Do not classify a graph as exponential only because it is curved. Quadratic, cubic, logarithmic, and other functions are curved too.

Worked Examples

Example: Growth by a Percentage

A game community has 1,200 members and grows by 7% per month. A simple model is

$M(t)=1200(1.07{)}^t$.

After 10 months,

$M(10)\approx 1200(1.967151)\approx 2360.58$,

so the model predicts about 2,361 members.

Example: Decay by a Percentage

A machine component has an efficiency index of 100 and loses 4% of the index each year. The decay factor is 0.96, so

$E(t)=100(0.96{)}^t$.

After 8 years,

$E(8)\approx 72.14$.

The loss is not 32 percentage points by simple subtraction; each 4% decrease is applied to the current value.

Example: Comparing Linear and Exponential Change

Plan A starts at 500 and adds 75 each year:

$A(t)=500+75t$.

Plan B starts at 500 and grows by 12% each year:

$B(t)=500(1.12{)}^t$.

At first, the values can be fairly close. Over longer periods, the exponential model can pull away because its yearly increase itself grows as the current quantity grows. Comparing a linear and an exponential model is therefore a useful way to understand the effect of repeated percentage change.

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Technology, Data, and Model Checking

Using a Spreadsheet or Graphing Tool

A spreadsheet can make repeated calculations visible. Put time values in one column and use a formula such as $=250*(1.12{)}^t$ in the next. Fill the formula down, then create a scatter plot or line graph. A graphing calculator or graphing website can show the same model directly.

When using technology, you should still be able to explain:

1. what the initial value means;
2. what the factor means;
3. why the model represents growth or decay;
4. what the units are;
5. whether the model is reasonable over the chosen time interval.

Checking a Model

A useful model should match both the mathematics and the situation. Ask:

1. Are the time intervals equal?
2. Is the percentage change approximately constant?
3. Are the units consistent?
4. Does the model make sensible predictions in the time range you are using?
5. Are there real-world limits, such as limited resources, changing rates, or a minimum possible value?

Exponential models are powerful precisely because they are simple. Their simplicity also means that you must state their assumptions.

Common Misconceptions

1. Percent decrease versus factor: A 30% decrease uses a factor of 0.70, not 0.30.
2. Linear versus exponential: Constant difference suggests linear change; constant ratio suggests exponential change.
3. Fast versus exponential: “Exponential” describes a multiplicative pattern, not merely something that seems fast.
4. Initial value: In $y=a\cdot b^x$, the initial value is $a$, not $b$.
5. Approaching zero: A basic positive decay model approaches zero without reaching it at a finite input.
6. Model limits: A model can fit data well for a while and still fail when conditions change.

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Interactive Tasks

Quiz: Test Your Knowledge

What feature most clearly identifies exponential change in equally spaced data? (A constant ratio between consecutive values) (!A constant difference between consecutive values) (!A graph that is always a straight line) (!A value that changes only once)

A model uses a growth factor of 1.08. What does this mean per interval? (Eight percent growth) (!Eight percent decay) (!Eighty percent growth) (!One percent growth)

A quantity has a decay factor of 0.85. What percent is lost each interval? (Fifteen percent) (!Eighty five percent) (!Five percent) (!One hundred fifteen percent)

In the model y equals 120 times 1.3 to the power t, what is the initial value? (120) (!1.3) (!13) (!156)

A quantity starts at 200 and is multiplied by 1.5 for three periods. What is its value? (675) (!450) (!500) (!900)

A substance has a half-life of four years. What fraction remains after eight years? (One quarter) (!One half) (!Three quarters) (!One eighth)

What is the horizontal asymptote of the basic positive model y equals a times b to the power x? (The x axis) (!The y axis) (!The line x equals one) (!The line y equals a)

Why can compound interest produce exponential growth? (Interest is repeatedly applied to the current balance) (!The same amount is always added) (!The balance must double every year) (!The interest rate becomes zero)

Which sequence shows exponential growth? (5 then 10 then 20 then 40) (!5 then 10 then 15 then 20) (!4 then 7 then 10 then 13) (!20 then 18 then 16 then 14)

What type of change is modeled by a positive factor of 0.70? (Exponential decay) (!Exponential growth) (!Linear growth) (!Constant change)

Memory Game

Initial value	Value of the quantity when the input is zero
Growth factor	Multiplier greater than one used each equal interval
Decay factor	Positive multiplier smaller than one used each equal interval
Doubling time	Time required for a growing quantity to become twice as large
Half-life	Time required for a decaying quantity to become half as large
Horizontal asymptote	Line that the basic exponential graph approaches

Drag and Drop

Match the correct terms.	Topic
Repeated multiplication by one point one two	Twelve percent growth
Repeated multiplication by zero point seven six	Twenty-four percent decay
Value when time is zero	Initial value
Time needed to reduce to one half	Half-life
Constant addition each interval	Linear change

...

Crossword Puzzle

Growth	What word describes exponential change with a factor greater than one?
Decay	What word describes exponential change with a factor between zero and one?
Factor	What is the repeated multiplier in an exponential model called?
Doubling	What word completes the phrase for the time needed to become twice as large?
Halflife	What single term names the time needed for a decaying amount to become half as large?
Asymptote	What line does a basic positive exponential graph approach without reaching?

Cloze Text

Complete the text.

In the model y equals a times b to the power x, a is the . Exponential growth uses a factor . Exponential decay uses a positive factor . A constant percentage change produces over equal intervals. For a percentage increase written as a decimal, the factor is . During each half-life interval, a decaying quantity is . The x-axis is the of a basic positive exponential model. A real-world exponential model should only be trusted while its assumptions remain .

Open-Ended Tasks

Easy

1. Growth table: Create a table for a quantity that starts at 50 and grows by 20% for six periods; show the repeated factor and explain the pattern in two or three sentences.
2. Decay sketch: Draw or digitally create the graph of a quantity that starts at 300 and keeps 80% each period; label the initial value and horizontal asymptote.
3. Everyday exponential examples: Make a one-page poster with two plausible growth examples and two plausible decay examples, explaining why each example involves repeated multiplication rather than constant addition.
4. Video explanation: Record a 60–90 second video that teaches the difference between a growth factor and a decay factor using your own numerical example.

Standard

1. Spreadsheet model: Build a spreadsheet comparing a linear model that adds 50 per period with an exponential model that grows by 10% per period for 15 periods, then graph and interpret the difference.
2. Compound interest investigation: Compare annual and monthly compounding for the same principal and annual rate over several years, then explain why the balances differ.
3. Bacterial growth model: Research a classroom-safe example of bacterial population growth, construct a short exponential model, and explain at least two reasons why unlimited exponential growth eventually becomes unrealistic.
4. Half-life simulation: Use coins to simulate decay by tossing a group each round, removing one chosen result, recording the remaining count, and comparing your experimental graph with an ideal half-life model.

Advanced

1. Model from data: Find or collect a small data set that may show repeated percentage change, test consecutive ratios, build an exponential model if justified, and discuss the model's error.
2. Growth versus logistic growth: Create a diagram or short report showing why a population can begin with exponential growth but later slow as resources become limited.
3. Doubling time investigation: Compare several positive percentage growth rates, calculate how many equal intervals each model needs to approximately double, and describe the relationship you observe.
4. Public information critique: Find a news, science, business, or social-media claim that uses the word “exponential,” inspect any available data, and present an evidence-based judgment about whether the term is mathematically justified.

Learning Assessment

1. Model selection: Given a table with equal input intervals, decide whether a linear or exponential model is more appropriate and justify your choice using differences or ratios.
2. Equation construction: Build an exponential equation from an initial value and a stated percent increase or decrease, then explain the meaning of every parameter in context.
3. Representation transfer: Convert one exponential situation from words to a table, equation, and graph, and explain how the same initial value and factor appear in all four representations.
4. Scenario comparison: Compare a constant-addition plan with a constant-percentage plan over time, identify when one overtakes the other, and explain why their long-term behavior differs.
5. Half-life reasoning: Use a half-life model to predict a remaining amount after several intervals and work backward from a given remaining fraction to determine how many half-lives have passed.
6. Model critique: Evaluate an exponential prediction made far beyond the observed data range and identify at least two assumptions that could make the prediction unreliable.

Evidence of Learning

Evidence type	What successful learning can show
Knowledge	You can explain initial value, growth factor, decay factor, percent rate, doubling time, half-life, and horizontal asymptote.
Skills	You can calculate with exponential models, compare ratios, translate among words, tables, equations, and graphs, and use digital tools to test predictions.
Products	You can produce accurate tables, graphs, spreadsheet models, written explanations, posters, simulations, or short videos that communicate exponential reasoning.
Reasoning	You can justify whether a pattern is exponential, interpret parameters in context, select sensible units and rounding, and explain differences between linear and exponential change.
Transfer	You can apply exponential thinking to unfamiliar situations and evaluate whether a model remains reasonable when real-world conditions change.

OERs on the Topic

The following English Wikipedia pages provide open background reading on the two central ideas in this course.

Useful related open topics include Exponential function, Geometric progression, Compound interest, Half-life, Population growth, Logarithm, and Mathematical model.

Linked Learning Areas

Exponential Growth and Decay Exponential function Percent change Geometric sequence Graph of a function Compound interest Half-life Population growth Logarithm Mathematical model

Exponential growth and decay connects Algebra with Functions, Statistics, Financial mathematics, Biology, Physics, Chemistry, Computer science, and data literacy. These connections help you see that one mathematical structure can describe many different systems while still requiring careful attention to assumptions and context.