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An exponential function is a mathematical expression of the form \( y = a \cdot b^x \), where \( a \) and \( b \) are constants, and \( x \) is an exponent. These functions are characterized by their rapid rate of growth or decay.
Exponential growth occurs when the base \( b \) is greater than 1, which means that as \( x \) increases, \( y \) increases exponentially. Conversely, exponential decay happens when \( 0 < b < 1 \), leading to a decrease in \( y \) as \( x \) increases.
In our solution, the exponential function used is \( y = 0.068 \cdot (1.57)^x \), where each year corresponds to a point \( x \). The base \( 1.57 \) is greater than 1, indicating that sales are increasing exponentially over time.

Data modeling involves creating a mathematical model—or equation—that represents a real-world scenario using data. In this context, we use exponential regression to create a model for sales data.
Exponential regression helps identify patterns by fitting an exponential curve to the given data points. This process typically requires statistical software or a calculator to compute the best-fit line that minimizes the difference between the actual data points and the values predicted by the model.
The model \( y = 0.068 \cdot (1.57)^x \) was derived using exponential regression. This model effectively summarizes the data trends over the years 1988 to 1993, allowing us to make future estimations.

Sales estimation is the process of predicting future sales based on historical data. With a well-fitted model, businesses can plan better by anticipating market demands.
Using the exponential model \( y = 0.068 \cdot (1.57)^x \), we estimated sales in future years by substituting \( x \) with the corresponding year value. For instance, to estimate sales for the year 1997, we set \( x = 9 \). Calculating this gives \( y \approx 39.2 \) million units, suggesting a significant increase in sales based on past trends.
This method helps businesses make informed decisions by considering expected future growth or decline.

Graphing functions is an essential skill in understanding mathematical models. It visually represents how closely a model fits the data.
To graph the exponential function \( y = 0.068 \cdot (1.57)^x \), you start by plotting the actual data points: \((0, 0.075), (2, 0.24), (4, 1.5), (5, 4.8)\). Next, you draw the curve that represents the model function across these points.
This graph serves as a visual confirmation that the model accurately reflects the data trends. You can easily spot whether the curve passes near all data points, indicating a good fit. This visual tool also helps in predicting future values by extending the curve.