How do you find the growth rate in an exponential function?

To find the growth rate in an exponential function, identify the base of the exponential expression.

In an exponential function, the general form is \( y = a \cdot b^x \), where \( y \) is the output, \( a \) is the initial value, \( b \) is the base, and \( x \) is the exponent. The growth rate is determined by the base \( b \). If \( b > 1 \), the function represents exponential growth, and if \( 0 < b < 1 \), it represents exponential decay.

To find the growth rate specifically, you need to look at the base \( b \). The growth rate \( r \) can be found using the formula \( r = b - 1 \). For example, if the exponential function is \( y = 2 \cdot 1.05^x \), the base \( b \) is 1.05. Therefore, the growth rate \( r \) is \( 1.05 - 1 = 0.05 \), or 5%.

Understanding the growth rate helps you analyse how quickly the function's value increases over time. In real-world contexts, such as population growth or interest rates, knowing the growth rate allows you to make predictions and informed decisions. Remember, the initial value \( a \) affects the starting point of the function, but the base \( b \) determines the rate at which the function grows or decays.