91Ó°ÊÓ

In exponential functions, the initial value is crucial. It represents the starting point of the function when the input value, typically denoted by \( x \), is zero. In the context of the given exercise, the initial value is clearly highlighted by the point at \( x = 0 \). Here, the initial value \( y \) is 500.00. Mathematically, we denote the initial value as \( a \). This means that when \( x = 0 \), \( y = a \). For our function, this gives us \( a = 500.00 \). This initial value sets the stage for how the function behaves and gives us a reference point from which the exponential changes start. Knowing the initial value allows us to construct the complete exponential model, as it is one of the essential constants in the general form of the function.

The base of an exponential function, denoted as \( b \), is another key component. It determines the rate at which the function grows or decays. Think of \( b \) as the factor by which the function's value is multiplied at each step. To find the base, you can use consecutive points from the data. For example, you can consider \( (0, 500.00) \) and \( (1, 425.00) \). By setting up the equation: \( 500.00 \times b = 425.00 \), we solve for \( b \) by dividing both sides by 500.00:
\[ b = \frac{425.00}{500.00} = 0.85 \]
This tells us that the value of the function decreases to 85% of its previous value with each step from one \( x \) value to the next. Therefore, our base \( b = 0.85 \), plays a pivotal role in defining the multiplier effect of the exponential function from one input value to the next.

The decay rate in an exponential function indicates how quickly the function’s value decreases over time. When the base \( b \) of an exponential function is less than 1, it signifies a decay. The decay rate can be calculated using the formula:
\[ \text{Decay Rate} = (1 - b) \times 100\text{\text{\text{\%}}} \]
For the given example, with \( b = 0.85 \):
\[ \text{Decay Rate} = (1 - 0.85) \times 100\text{\text{\text{\%}}} = 0.15 \times 100\text{\text{\text{\%}}} = 15\text{\text{\text{\%}}} \]
This tells us the function’s value decreases by 15% with each step. Understanding the decay rate helps in predicting how quickly values fall over time and can be essential for real-life scenarios where exponential decay is observed, such as radioactive decay, cooling of substances, or depreciating assets.