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When you look at an exponential function, such as \( K(r) = 33(0.92)^r \), the base is a crucial component. The base of an exponential function determines the direction and rate of the change. In our function, the base is \( 0.92 \). This base is what you multiply by itself \( r \) times as the input \( r \) changes.
In exponential functions:

• If the base is greater than 1, then you have exponential growth. This means that as \( r \) increases, the function's output grows larger.
• If the base is between 0 and 1, like our example, you encounter exponential decay. Here, as \( r \) increases, the function's output gets smaller.

Understanding the base of an exponential function is the first step in identifying the nature of the function's behavior.

In exponential functions, the constant percentage change is derived from the base value. It's the percentage by which the function grows or decays with each step of \( r \). To find this change, you do a simple calculation that reveals how rapidly the quantity is changing.
Here's how to calculate the percentage change:

• Subtract the base from 1 (i.e., \( 1 - ext{base} \)) to find out how much the value changes per increment.
• Multiply the result by 100 to convert it to a percentage.

For the function \( K(r) = 33(0.92)^r \), the base is \( 0.92 \). By calculating \( 1 - 0.92 = 0.08 \), we see the function changes by 8%. Then, \( 0.08 \times 100 = 8\% \), which tells us there's an 8% decay for each increase in \( r \). This regular decay rate is what we call the constant percentage change.

Exponential growth and decay describe how quantities change step-by-step in a way defined by their exponential bases. These terms show whether a function increases or decreases as time or another variable progresses. Let's delve deeper into these phenomena:
Exponential growth occurs when the base of the exponential function is greater than 1, causing the quantity to increase as the power increases. With each step, the amount grows by a fixed percentage relative to its current size.
Exponential decay, on the other hand, takes place when the base is between 0 and 1, leading to a decrease. The overall quantity reduces by a specific percentage with every increase in the power.
Both growth and decay have their applications in real life:

• Exponential growth is seen in populations, investments, and technology adoption, where there is a consistent rate of increase.
• Exponential decay models scenarios like radioactive decay, depreciation of assets, or cooling processes, where there is a steady rate of decline.

Recognizing these patterns helps us understand and predict how different phenomena will behave over time.