Ohio Assessments for Educators (OAE) Mathematics Practice Exam

What does 'a' represent in the formula f(x) = ae^(rt)?

1. The initial growth rate

2. The rate of change

3. The starting value of the population

4. The time interval

The correct answer is: The starting value of the population

In the formula f(x) = ae^(rt), 'a' represents the starting value of the population or the initial amount before any growth occurs. This is a fundamental aspect of exponential growth models, where 'e' is the base of the natural logarithm, 'r' is the growth rate, and 't' is the time variable. When analyzing the equation, 'a' establishes the baseline from which the population will grow over time based on the growth rate 'r' and the duration 't'. Thus, it serves as the initial condition that reflects the population size or value right at the beginning of the observation period, making it crucial for understanding how the value will change as time progresses. Understanding this context highlights the importance of 'a' in determining the overall behavior of the exponential function, as it sets the stage for how drastic the changes can be as the variables 'r' and 't' interact with 'a'.