Lesson 6.4: Exponential Growth and Decay

Introduction

In this lesson, we will explore the concept of exponential growth and decay, which are fundamental ideas in mathematics that appear in various real-world situations. By the end of this lesson, you, students, will be able to:

Understand the shape of the graph of the function $y = a^x$ for exponential growth and decay.
Read and interpret an exponential growth or decay graph.
Recognize everyday examples of exponential growth, such as population growth, and decay, such as depreciation of objects.
Interpret values taken from an exponential graph in context.

Hook

Imagine a population of rabbits in a field. If conditions are right, this population can double every month. This rapid increase is an example of exponential growth. On the other hand, consider a car that depreciates in value over time. Initially, it might lose a large part of its value, but over the years, this loss happens at a decreasing rate, which is an example of exponential decay. Understanding these concepts not only helps us in mathematics but also provides insight into various fields such as biology, economics, and environmental studies.

Understanding Exponential Functions

An exponential function is a mathematical function of the form:

$ y = a^x$

where:

$y$ is the output of the function,
$a$ is a positive constant known as the base of the exponential,
$x$ is the exponent, which can be any real number.

Exponential Growth

When the base $a > 1$, the function exhibits exponential growth. Let's analyze the function deeper.

Shape of the Graph

The graph of an exponential growth function looks like this:

It approaches the x-axis as $x$ decreases, but never touches it (this is called the asymptote).
As $x$ increases, $y$ increases rapidly, creating a steep incline.

Example of Exponential Growth

Let’s take an example to illustrate this. Suppose we have a function defined as:

$ y = 2^x$

Now, we can compute the values for specific values of $x$:

For $x = 0$: $y = 2^0 = 1$
For $x = 1$: $y = 2^1 = 2$
For $x = 2$: $y = 2^2 = 4$
For $x = 3$: $y = 2^3 = 8$

Plotting these points:

For $(0, 1)$, $(1, 2)$, $(2, 4)$, and $(3, 8)$ shows a steep rising curve, characteristic of exponential growth.

Exponential Decay

When the base $0 < a < 1$, the function demonstrates exponential decay. Let’s investigate how this functions.

Shape of the Graph

The graph of an exponential decay function looks as follows:

It starts at a high value when $x$ is negative and decreases towards the x-axis, creating a gradual slope.
Just like the growth function, it approaches the x-axis but never actually touches it.

Example of Exponential Decay

Consider the function defined as:

$ y = \left(\frac{1}{2}$

$ight)^x$

Now, let’s compute the values for specific values of $x$:

For $x = 0$: $y = $\left($$\frac{1}{2}

ight)^0 = 1

For $x = 1$: $y = $\left($$\frac{1}{2}

ight)^1 = $\frac{1}{2}$

For $x = 2$: $y = $\left($$\frac{1}{2}

ight)^2 = $\frac{1}{4}$

For $x = 3$: $y = $\left($$\frac{1}{2}

ight)^3 = $\frac{1}{8}$

Here, plotting the points: $(0, 1)$, $(1, \frac{1}{2})$, $(2, \frac{1}{4})$, and $(3, \frac{1}{8})$ shows a smooth decline, characteristic of exponential decay.

Real-World Examples of Exponential Growth and Decay

Population Growth

Exponential growth can be observed in populations of organisms, like the aforementioned rabbits. If we assume a population of rabbits doubles every month, we can express this growth with the function:

$ P(t) = P_0 \times 2^t$

where:

$P(t)$ is the population at time $t$,
$P_0$ is the initial population,
$t$ is the time in months.

For example, if the initial population $P_0 = 10$ rabbits, after 3 months, the population will be:

P(3) = $10 \times 2^3$ = $10 \times 8$ = 80 \ \text{rabbits}.

Depreciation of Value

Exponential decay can often be seen in assets, such as cars, which lose value over time. Typically modeled as:

V(t) = V_$0 \times$ (1 - r)^t

where:

$V(t)$ is the value of the car at time $t$,
$V_0$ is the initial value,
$r$ is the decay rate (expressed as a decimal), and
$t$ is time in years.

For a car that costs \$20,000 and depreciates at a rate of 15% ($r = 0.15$), the value after 5 years would be:

V(5) = $20000 \times$ (1 - 0.15)^$5 \approx 20000$ $\times 0$.$4407 \approx 8814$ \ $\text{USD}$.

Interpreting Graphs of Exponential Functions

When presented with a graph of an exponential function, here are some key aspects to consider:

Identify the base: If it grows quickly, the base is greater than 1. If it decays, the base is between 0 and 1.
Asymptote: Always look for where the graph approaches the x-axis but never quite reaches.
Initial value: The value of $y$ when $x=0$ gives insight into the starting point of the graph.
Behavior: Note how quickly the graph rises (for growth) or falls (for decay).

Conclusion

In the lesson, we have explored the fundamentals of exponential growth and decay, along with their graphical representations. We have examined how these concepts apply in real-world contexts such as population dynamics and asset depreciation. Understanding these principles will pave the way for deeper exploration in both mathematical theory and practical applications.

Study Notes

Exponential growth occurs with the function $y = a^x$ where $a > 1$.
Exponential decay occurs with the function $y = a^x$ where $0 < a < 1$.
Real-world examples include rabbit populations illustrating growth and asset depreciation demonstrating decay.
Key features of exponential graphs include their asymptotes and steepness depending on growth or decay.
Interpreting points from exponential graphs provides insights into the real-world phenomena they represent.