How do you determine if a function represents exponential growth?

A function represents exponential growth if it increases by a constant percentage or factor over equal intervals.

To determine if a function shows exponential growth, you first need to look at its general form. An exponential growth function can be written as \( f(x) = a \cdot b^x \), where \( a \) is a constant, \( b \) is the base of the exponential (and must be greater than 1), and \( x \) is the exponent. The key characteristic of exponential growth is that the function increases by a consistent percentage or factor over equal intervals of \( x \).

For example, consider the function \( f(x) = 2 \cdot 3^x \). Here, \( a = 2 \) and \( b = 3 \). As \( x \) increases, the value of \( f(x) \) grows rapidly because it is being multiplied by 3 for each unit increase in \( x \). This consistent multiplication by a factor greater than 1 indicates exponential growth.

Another way to identify exponential growth is by examining the rate of change. In exponential growth, the rate of change itself increases over time. If you plot the function on a graph, an exponential growth function will produce a curve that gets steeper and steeper as \( x \) increases.

You can also compare the function's values at different points. If the ratio of the function's values at equally spaced intervals is constant, then the function is likely exponential. For instance, if \( f(1) = 6 \), \( f(2) = 18 \), and \( f(3) = 54 \), the ratio \( \frac{f(2)}{f(1)} = 3 \) and \( \frac{f(3)}{f(2)} = 3 \), confirming exponential growth with a base of 3.