Exponential growth and decay

Exponential Growth and Decay

Exponential functions are important in many real-life situations. They can model growth or decay processes, such as population growth, radioactive decay, and interest calculations. The general form of an exponential function is:

f(t) = a × b^t

In this equation:

• f(t) is the value of the function at time t.
• a is the initial value (the value at time t = 0).
• b is the base of the exponential function, which determines the growth or decay rate.
• t is the time variable.

Exponential Growth

Exponential growth occurs when the value of a function increases over time. This happens when the base b is greater than 1. For example, if a population of bacteria doubles every hour, we can model this with an exponential function.

Let's consider an example where a population of bacteria starts with 100 individuals and doubles every hour. The initial value a is 100, and the base b is 2 (since it doubles).

The function can be written as:

f(t) = 100 × 2^t

To find the population after 5 hours, substitute t = 5 into the function:

f(5) = 100 × 2^5

Now calculate:

f(5) = 100 × 32

Thus:

f(5) = 3200

After 5 hours, the population will be 3,200 bacteria.

Exponential Decay

Exponential decay occurs when the value of a function decreases over time. This happens when the base b is between 0 and 1. A common example of exponential decay is radioactive decay.

Suppose a certain radioactive substance has a half-life of 3 years. This means that every 3 years, half of the substance will decay. If we start with 80 grams of the substance, we can model this decay process.

The initial value a is 80 grams, and the decay factor can be calculated from the half-life:

Since it halves every 3 years, the decay factor is:

b = 1/2

The function can be expressed as:

f(t) = 80 × (1/2)^(t/3)

Now, to find the amount remaining after 6 years, substitute t = 6 into the function:

f(6) = 80 × (1/2)^(6/3)

Calculate:

f(6) = 80 × (1/2)^2

Now evaluate:

f(6) = 80 × 1/4

Thus:

f(6) = 20

After 6 years, 20 grams of the radioactive substance will remain.

Continuous Growth and Decay

In some cases, growth or decay occurs continuously rather than at fixed intervals. The formula for continuous growth or decay is:

f(t) = a × e^(kt)

In this equation:

• e is the base of the natural logarithm, approximately equal to 2.71828.
• k is the growth (if positive) or decay (if negative) constant.

For example, suppose a population of 200 individuals grows continuously at a rate of 5% per year. The growth constant k is 0.05. The function can be expressed as:

f(t) = 200 × e^(0.05t)

To find the population after 10 years, substitute t = 10 into the function:

f(10) = 200 × e^(0.05 × 10)

Calculate:

f(10) = 200 × e^(0.5)

Using a calculator, we find:

e^(0.5) ≈ 1.64872

Now substitute this value back into the function:

f(10) ≈ 200 × 1.64872

Thus:

f(10) ≈ 329.744

After 10 years, the population will be approximately 330 individuals.

Applications of Exponential Functions

Exponential functions are widely used in various fields:

• Finance: They model compound interest, where the amount of money grows exponentially over time.
• Biology: They describe population growth and the spread of diseases.
• Physics: They represent radioactive decay and cooling processes.
• Economics: They can model inflation rates and economic growth.

Graphing Exponential Functions

When graphing exponential functions, it is important to note the following characteristics:

• The graph of an exponential growth function rises steeply as t increases.
• The graph of an exponential decay function approaches the horizontal axis but never touches it.
• The y-intercept is always at (0, a).
• As t approaches negative infinity, the function approaches zero.

For example, consider the exponential growth function:

f(t) = 2 × 3^t

To graph this function, calculate values for different values of t:

Plot these points on a graph to visualize the exponential growth.

Summary

• Exponential growth occurs when the base of the function is greater than 1.
• Exponential decay occurs when the base of the function is between 0 and 1.
• Continuous growth and decay use the base e in their equations.
• Exponential functions have distinct graph characteristics.

Check your understanding

1. What is the general form of an exponential function?
2. How do you determine the growth factor from a percentage increase?
3. What is the effect of a negative growth constant in a continuous growth model?
4. How does the graph of an exponential decay function behave as time increases?