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Understanding the concept of growth factor is essential when dealing with exponential functions.
The growth factor determines how much a quantity increases over each time period.
In general, for an exponential growth function like \(A = A_{0}(b)^t\), the base \(b\) is the growth factor.
When \(b > 1\), the function represents exponential growth.
For example, in the function \(A = A_{0}(1.0025)^{20t}\), the base is 1.0025, which is greater than 1.
This indicates that the amount is increasing every year.
To find the annual growth factor, perform further calculations, such as raising the base to the power of \(1/n\) where \(n\) is the number of periods per year.
For instance, raising \(1.0025\) to the \(1/20\) power gives the annual growth factor of approximately 1.0512.

Decay factor is pivotal in understanding how quantities decrease over time in exponential decay functions.
The decay factor is the base of the exponential function that's less than 1.
In the function \(A = A_{0}(b)^t\), if \(b < 1\), it signals a decay process.
For example, in the function \(A = A_{0}(0.992)^{t/2}\), the base is 0.992, which is less than 1, indicating decay.
To determine the annual decay factor, you might need to raise the base to the power of \(1/n\), where \(n\) denotes the number of periods assumed per year.
For the function with base 0.992 applied every 2 years, raising 0.992 to the 1/2 power gives the annual decay factor of approximately 0.9960.

Exponential growth occurs when a quantity increases by a consistent percentage each time period.
It can be represented by the function \(A = A_{0}(b)^t\) where \(b > 1\).
This type of growth has various real-world applications such as population growth or compound interest.
In problems involving exponential growth, the base of the function (growth factor) is greater than 1.
For example, in the function \(A = A_{0}e^{0.015t}\), the exponential term's exponent is positive, indicating growth.
Thus, taking \(e^{0.015}\) results in an annual growth factor of approximately 1.0151, showing how the initial amount increases each year.

Exponential decay describes the process by which a quantity decreases by a consistent percentage over equal time intervals.
It is often modeled using the function \(A = A_{0}(b)^t\) where \(0 < b < 1\).
Common examples include radioactive decay or depreciation of assets.
If the base lies between 0 and 1, the function represents decay.
For example, in the function \(A = A_{0}e^{-0.063t}\), the negative exponent indicates decay.
Exponentiating \(e^{-0.063}\) gives an annual decay factor of approximately 0.9389.
This shows the factor by which the quantity reduces each year.