91Ó°ÊÓ

Exponential growth is a pattern of data that shows greater increases over time, creating a curve that becomes steeper as it progresses. This occurs when a quantity multiplies by a constant factor greater than 1 in consistent intervals. A real-world example can be seen in populations of organisms or in investments that grow over time. If the number of organisms doubles each year, that's exponential growth. Similarly, if you invest a sum of money and it grows by a fixed percentage every year, that's also exponential growth. In terms of exponential functions, exponential growth is indicated by the base value of the function. If the base of the function, denoted as \( b \) in \( y = a \, b^x \), is greater than 1, it signifies that the function exhibits exponential growth. The larger the base, the faster the growth rate, and the steeper the curve of the function.

Exponential decay is the process in which quantities decrease by a constant proportion over time. It results in a curve that decreases and flattens over time. This pattern is often seen with radioactive substances, which decrease at a consistent rate, or in half-life scenarios. Similarly, depreciation of an asset over time listed in percentage terms also showcases exponential decay.For example, in the function \( y=(0.5)^x \), since the base \( b = 0.5 \) which is less than 1, the function demonstrates exponential decay. Here, the curve starts high and steadily descends as the \( x \) value increases, indicating that the quantity is reducing consistently over time.

The base of an exponential function is a crucial factor in determining the behavior of the function. It's symbolized as \( b \) in the equation \( y = a \, b^x \), with \( a \) representing the initial amount, and \( x \) being the exponent denoting the number of intervals of growth or decay.

• If \( b > 1 \), the function describes exponential growth.
• If \( 0 < b < 1 \), the function describes exponential decay.

Understanding the base allows you to predict whether the function value will increase or decrease as the exponent increases. For instance, a base of \( 1.5 \) would suggest that the quantity expands by 50% every interval, thus growing exponentially. Conversely, a base of \( 0.5 \) implies a decrease by 50% each time, indicating shrinkage through exponential decay. This base component is integral as it defines the entire nature of the exponential process involved.