Short Answer 
Exponential functions, represented as f(x) = a^x, show rapid growth or decline depending on the base ‘a’ and are applied in multiple fields. Identifying patterns in data involves looking for multiplicative changes in ‘y’ values relative to ‘x’, while the common ratio, a key feature of these functions, helps define the growth rate.
Step 1: Understanding Exponential Functions
An exponential function is a mathematical function expressed as f(x) = a^x, where ‘a’ is a constant known as the base and ‘x’ is the variable in the exponent. This form leads to rapid growth or decline of the function depending on whether ‘a’ is greater than or less than 1. Exponential functions are prevalent in various fields such as engineering, biology, and finance due to their ability to model real-world phenomena.
Step 2: Identifying Patterns in Data
To determine if the data follows an exponential pattern, examine how the y values change as the x values increase. Notice that the relationship is often multiplicative, meaning that the increase in y values is proportional to the increase in x values. In this case, the observed changes in values suggest that the data indeed fits an exponential model due to its multiplicative nature.
Step 3: Calculating the Common Ratio
The common ratio is a key characteristic of exponential functions, which helps define their growth. For example, if you compare the function’s values at two points, such as f(1) and f(2), the ratio of these values indicates the growth rate. In this instance, the common ratio calculated is approximately 0.993, signifying a consistent pattern in how the function increases as the input ‘x’ changes.