How do you find the initial value in an exponential function?

To find the initial value in an exponential function, identify the coefficient of the base raised to the power of zero.

In an exponential function of the form \( y = ab^x \), the initial value is represented by \( a \). This is because when \( x = 0 \), the term \( b^x \) becomes \( b^0 \), which equals 1. Therefore, the equation simplifies to \( y = a \times 1 \), or simply \( y = a \). This means that the initial value is the value of \( y \) when \( x \) is zero.

For example, consider the exponential function \( y = 3 \times 2^x \). Here, \( a = 3 \) and \( b = 2 \). When \( x = 0 \), the function becomes \( y = 3 \times 2^0 = 3 \times 1 = 3 \). Thus, the initial value is 3.

Understanding the initial value is crucial because it represents the starting point of the function before any growth or decay occurs. In real-world contexts, this could be the initial population of a species, the starting amount of money in a bank account, or the initial quantity of a substance before it begins to change over time.

To summarise, the initial value in an exponential function is the coefficient \( a \) in the equation \( y = ab^x \), and it is the value of \( y \) when \( x \) is zero.