Table of Contents
Understanding Exponential Decay
In the parent chapter on Exponential Functions, you saw the general idea of functions that change by a fixed percentage (or factor) per unit of time or per step. In this chapter we focus specifically on the case where quantities decrease over time: exponential decay.
Exponential decay models situations where a quantity shrinks by the same proportion in each equal time interval. The key feature is that the rate of decrease is proportional to the current amount.
Basic Form of Exponential Decay
A typical discrete-time exponential decay model looks like:
$$
y = a \cdot b^x
$$
where
- $a$ is the initial value (often when $x = 0$),
- $b$ is the base, with $0 < b < 1$ for decay,
- $x$ is the independent variable (often time).
Because $0 < b < 1$, as $x$ increases, the value $b^x$ gets smaller and smaller, so $y$ decreases.
A common way to write $b$ is in terms of a decay rate $r$:
$$
b = 1 - r,
$$
where $r$ is the decay rate per unit of $x$ (often per unit of time), written as a decimal. Then
$$
y = a(1 - r)^x.
$$
For example, if something loses $8\%$ of its value each year, then $r = 0.08$ and
$$
y = a(1 - 0.08)^x = a(0.92)^x.
$$
Identifying Decay from a Formula
Given an exponential function, you can decide whether it represents growth or decay by examining the base:
- If $b > 1$: exponential growth (covered in the growth chapter).
- If $0 < b < 1$: exponential decay.
So, in expressions like:
- $y = 500(0.7)^x$: decay (base $0.7$).
- $y = 200\left(\dfrac{4}{5}\right)^x$: decay (base is $0.8$).
- $y = 50(1.03)^x$: growth (base $1.03$).
When the formula is written as $y = a(1 - r)^x$:
- If $0 < r < 1$, then $1 - r$ is between $0$ and $1$, so the function describes decay.
- The percentage decay rate is $100r\%$.
Graph Features of Exponential Decay
For a basic decay function $y = a(1 - r)^x$ with $a > 0$ and $0 < r < 1$:
- The graph passes through $(0, a)$ since $y(0) = a(1 - r)^0 = a$.
- As $x$ increases, $y$ decreases and gets closer and closer to $0$ but does not cross it. The $x$-axis ($y = 0$) is a horizontal asymptote.
- The function is decreasing for all real $x$.
- For positive $a$, all function values are positive: $y > 0$ for all $x$.
If $a < 0$, the graph is reflected across the $x$-axis, but the “decay” behavior in magnitude is similar.
Percentage Decay and the Base
Often the problem describes a percentage decrease, such as “decreases by $15\%$ per year.” You convert that description into the base:
- If a quantity decreases by $r$ (as a decimal) each step, the base is $1 - r$.
- If the problem gives a percentage $p\%$, then $r = \dfrac{p}{100}$.
Examples:
- Decreases by $25\%$ per month:
- $r = 0.25$,
- $b = 1 - 0.25 = 0.75$,
- model: $y = a(0.75)^x$.
- Decreases by $3\%$ per year:
- $r = 0.03$,
- base is $0.97$,
- model: $y = a(0.97)^x$.
Writing Decay Models from Verbal Descriptions
In decay problems, you typically know:
- an initial value,
- a decay percentage (or a multiplier),
- a time variable and sometimes a time unit.
The steps to create a model are:
- Identify the initial value $a$.
- Find the decay rate $r$ (as a decimal) or the base $b$.
- Determine how $x$ is measured (years, hours, etc.).
- Write $y = a(1 - r)^x$ or $y = ab^x$.
Example:
“A car is bought for \$24{,}000 and loses $12\%$ of its value each year. Let $x$ be the number of years since purchase. Model the value of the car.”
- Initial value: $a = 24000$.
- Decay rate: $r = 0.12$.
- Base: $1 - 0.12 = 0.88$.
- Model:
$$
V = 24000(0.88)^x,
$$
where $V$ is the value in dollars after $x$ years.
Half-Life and Related Concepts
Many decay processes in science are described by a half-life. The half-life of a substance or quantity is the time required for it to reduce to half of its initial value.
If a quantity has a half-life of $h$ (units of time), and $x$ is time measured in the same units, a common model is:
$$
y = a\left(\dfrac12\right)^{x/h}.
$$
Interpretation:
- When $x = 0$, $y = a$ (initial amount).
- When $x = h$, $y = a\left(\dfrac12\right)^{1} = \dfrac{a}{2}$.
- When $x = 2h$, $y = a\left(\dfrac12\right)^{2} = \dfrac{a}{4}$, and so on.
This fits within the general exponential decay structure; here the base is $\left(\dfrac12\right)^{1/h}$ for each unit of time.
Half-life problems often ask:
- to find the amount left after a certain time; or
- to determine how long it takes for the quantity to fall to some fraction of the original.
Solving these typically uses techniques from the “Solving equations” chapters (for example, logarithms in Algebra II).
Comparing Linear and Exponential Decay
Although detailed comparison is treated in other chapters, it is useful here to distinguish exponential decay from simple linear decrease:
- Linear decay: loses the same amount in each time step (for example, “loses \$500 in value each year”).
- Exponential decay: loses the same percentage (the same proportion) in each time step (for example, “loses $10\%$ of its value each year”).
With exponential decay, the amount lost each year gets smaller over time, because it is always a fixed percentage of a shrinking quantity.
Typical Applications of Exponential Decay
Exponential decay models appear in many contexts. Some common examples:
- Depreciation of cars or electronics (value decreases by a fixed percentage per year).
- Radioactive decay (amount of a substance decreases over time following a half-life).
- Cooling processes under certain conditions (approaching the environment’s temperature).
- Drug concentration in the body (concentration dropping proportionally to the current amount).
- Decrease of a population when the death rate dominates and is proportional to the current population.
In each case, the key idea is: the rate of decrease at any moment is proportional to how much is currently present.
Reading and Interpreting Decay Graphs
From a graph of exponential decay, you should be able to read:
- The initial value: the $y$-intercept at $x = 0$.
- Whether the function is decay: it is decreasing and approaches a horizontal asymptote.
- The long-term behavior: it approaches but does not cross the asymptote (often the $x$-axis, $y = 0$).
When a vertical shift or horizontal shift is involved (topics of function transformations), the horizontal asymptote may change from $y = 0$ to another value such as $y = k$. In such cases, the quantity decays toward the level $y = k$ instead of toward $0$.
Discrete vs. Continuous Decay (Overview)
Exponential decay can be modeled in both discrete and continuous ways:
- Discrete models: $y = a(1 - r)^n$, where $n$ counts whole periods (like years or months).
- Continuous models: often use the base $e$ (Euler’s number) and have the form
$$
y = ae^{kt},
$$
where $k < 0$ for decay.
The detailed study of continuous decay and the number $e$ belongs in more advanced sections (exponential functions and later calculus), but you should recognize that a negative exponent coefficient ($k < 0$) indicates decay in the continuous model.
Summary
In exponential decay:
- The base satisfies $0 < b < 1$.
- The quantity changes by a fixed percentage (decay rate) per step.
- The function often has the form $y = a(1 - r)^x$.
- The graph is decreasing, positive (for $a > 0$), and approaches a horizontal asymptote.
- Half-life problems are an important special case where the quantity halves after a fixed time interval.