Exponential Growth and Decay

Introduction

Hello students 👋 In this lesson, you will learn how quantities can grow or shrink by the same percentage over time. This is called exponential growth and exponential decay. Unlike linear change, where a quantity changes by the same amount each step, exponential change multiplies by the same factor each step. That makes it very important in real life, from population growth to medicine in the body, from bank interest to radioactive decay.

By the end of this lesson, you should be able to:

explain the key ideas and terms in exponential growth and decay,
use exponential models to solve problems,
connect this topic to sequences, algebra, and financial modelling,
interpret results using technology such as graphing tools or spreadsheets 📈,
understand how exponential models appear in IB Mathematics: Applications and Interpretation SL.

A good way to think about it is this: if a quantity keeps changing by the same percentage, then its pattern is usually exponential.

What Exponential Growth and Decay Mean

Exponential growth happens when a quantity increases by the same percentage over equal time intervals. Exponential decay happens when a quantity decreases by the same percentage over equal time intervals. For example, if a town’s population grows by $3\%$ each year, then each year the population is $1.03$ times the previous year’s population. If a laptop battery loses $12\%$ of its charge each hour, then each hour it keeps only $0.88$ of what it had before.

The general exponential model is

$$y = a(b)^t$$

where:

$a$ is the initial value,
$b$ is the growth or decay factor,
$t$ is the number of time periods,
$y$ is the value after $t$ periods.

If $b>1$, the model represents growth. If $0<b<1$, the model represents decay. For growth by a rate $r$, the factor is $b = 1+r$. For decay by a rate $r$, the factor is $b = 1-r$.

Example: if a plant is $20$ cm tall and grows by $5\%$ each week, then its height after $t$ weeks is

$$h = 20(1.05)^t$$

This formula shows multiplication, not repeated addition. That is the main difference from linear growth. In linear growth, a quantity might increase by $2$ every day, giving a pattern like $2,4,6,8. In exponential growth, a quantity might increase by $20\%$ every day, giving a pattern like $100,120,144,172.8$.

Exponential Sequences and Algebraic Representation

Exponential models are closely related to sequences. A sequence is an ordered list of numbers. In this topic, exponential patterns usually form a geometric sequence, where each term is found by multiplying the previous term by a constant ratio.

If the first term is $u_1$ and the common ratio is $r$, then the $n$th term is

$$u_n = u_1r^{n-1}$$

This is the same structure as an exponential function, except the variable may be the term number $n$ instead of time $t$. For example, if a culture of bacteria starts with $500$ bacteria and doubles every hour, then the number after $n$ hours is

$$u_n = 500(2)^{n}$$

if $n=0$ represents the starting time. If $n=1$ represents the first hour, then the sequence form is

$$u_n = 500(2)^{n-1}$$

Both forms are correct if the meaning of the variable is clear.

students, this is why algebra matters here: you must be careful with the index, the starting value, and what the variable stands for. In IB Mathematics: Applications and Interpretation SL, clear notation is very important because it helps you translate real situations into mathematical models.

A useful skill is rewriting percentages as factors. For growth by $p\%$, the factor is

$$1+\frac{p}{100}$$

For decay by $p\%$, the factor is

$$1-\frac{p}{100}$$

So a $7\%$ increase becomes $1.07$, and a $15\%$ decrease becomes $0.85$.

Solving Problems with Exponential Models

Many exam questions ask you to find an unknown value, such as time, rate, or starting amount. Exponential equations often require algebraic manipulation. If the variable is in the exponent, logarithms are usually needed.

For example, suppose a phone worth $1200$ dollars loses $18\%$ of its value each year. Its value after $t$ years is

$$V = 1200(0.82)^t$$

If you want to know when the phone’s value falls below $500$, solve

$$1200(0.82)^t < 500$$

First divide by $1200$:

$$(0.82)^t < \frac{500}{1200}$$

Then use logarithms to find $t$.

Taking natural logs gives

$$t\ln(0.82) < \ln\left(\frac{500}{1200}\right)$$

Because $\ln(0.82)$ is negative, the inequality reverses when dividing. This is an important detail ⚠️.

The idea of solving for time also appears in finance. If money is invested with compound interest, the model is exponential growth. For example, if $P$ dollars are invested at annual rate $r$, compounded once per year, then after $t$ years:

$$A = P(1+r)^t$$

This is a financial model that fits the number and algebra topic because it combines algebraic expressions, sequences, and real-world applications.

Interpreting Graphs and Using Technology

Graphs are one of the best ways to understand exponential change. The graph of $y=a(b)^x$ has a curved shape, not a straight line. For growth, it rises slowly at first and then more quickly. For decay, it falls quickly at first and then levels off toward $0$.

A key feature is the horizontal asymptote. For exponential decay of the form $y=a(b)^x$ with $0<b<1$, the graph approaches the line

$$y=0$$

but does not usually cross it. That means the quantity gets very small, but in the model it does not become negative.

Technology helps you explore this. On a graphing calculator or spreadsheet, students, you can:

enter an exponential formula,
make a table of values,
compare it with linear models,
estimate when a value reaches a certain level,
check whether the model fits real data.

For example, if a scientist measures the amount of a medicine in the bloodstream, the data may show exponential decay. A spreadsheet can help determine whether the decay factor is close to constant from one time period to the next. This is a strong example of technology-supported interpretation, which is an important part of IB Mathematics: Applications and Interpretation SL.

Real data is not always perfectly exponential. Sometimes the pattern changes because of outside effects, measurement error, or limits in the situation. For that reason, models are simplifications of reality, not exact copies of it.

Real-World Applications and Common Mistakes

Exponential growth and decay appear in many situations:

population growth,
bacteria reproduction 🦠,
radioactive decay,
depreciation of cars or electronics,
compound interest,
cooling processes in some simplified models,
drug concentration in the body.

One common mistake is confusing additive and multiplicative change. If something increases by $50$ each year, that is not exponential. If it increases by $5\%$ each year, that is exponential.

Another mistake is using the wrong factor. For a $20\%$ increase, the factor is $1.2$, not $0.2$. For a $20\%$ decrease, the factor is $0.8$, not $1.2$.

A third mistake is forgetting the starting value. If a population starts at $800$ and grows by $4\%$ per year, the model is

$$P = 800(1.04)^t$$

not just $P = (1.04)^t$.

Let’s look at a comparison. Suppose two savings plans both start at $1000$.

Plan A adds $50$ each year.
Plan B grows by $5\%$ each year.

After one year, Plan A has $1050$ and Plan B has $1050$. After two years, Plan A has $1100$, while Plan B has

$$1000(1.05)^2 = 1102.5$$

The exponential plan grows more slowly at first, but it can overtake the linear plan later. This shows why exponential change can become very powerful over time.

Conclusion

Exponential growth and decay describe situations where change happens by repeated multiplication. The key model is

$$y = a(b)^t$$

with $b>1$ for growth and $0<b<1$ for decay. This topic connects directly to geometric sequences, algebraic manipulation, financial models, and technology-based interpretation. In IB Mathematics: Applications and Interpretation SL, you are expected to understand the meaning of the model, use it correctly in context, and interpret the results clearly.

students, whenever you see a real situation involving constant percentage change, think exponential. Ask: What is the initial value? What is the factor? What does the variable represent? Those questions will help you build strong mathematical reasoning and solve problems accurately.

Study Notes

Exponential change means multiplication by a constant factor each time period.
Growth factor: $b = 1+r$.
Decay factor: $b = 1-r$.
General model: $y = a(b)^t$.
Geometric sequence term formula: $u_n = u_1r^{n-1}$.
Growth occurs when $b>1$.
Decay occurs when $0<b<1$.
Exponential graphs are curved, not straight lines.
Exponential decay often approaches the horizontal asymptote $y=0$.
Logarithms are useful when solving for a variable in the exponent.
Real-world examples include population, interest, depreciation, and radioactive decay.
Technology can be used to graph, table, and interpret exponential models.
Always check whether the situation is linear or exponential before modelling.