Question

Write an exponential growth function. Explain why your function represents exponential growth. (4 points)

Answer 1

An example of an exponential growth function is:

\[ f(t) = a \cdot b^t \]

where:
- \( a \) is the initial amount (the value of the function when \( t = 0 \)),
- \( b \) is the growth factor (a constant greater than 1),
- \( t \) is time.

### Example:
Let's say \( a = 100 \) (the initial value) and \( b = 1.05 \) (which represents a 5% growth rate per time unit). The function becomes:

\[ f(t) = 100 \cdot (1.05)^t \]

### Explanation of Exponential Growth:
1. **Constant Growth Rate**: In this function, the growth is proportional to the current value of \( f(t) \). This means that as \( t \) increases, the amount of change in \( f(t) \) becomes larger because it depends on its current size. In other words, the percentage increase remains constant over time, which is a hallmark of exponential growth.

2. **Diminishing Time to Double**: With exponential growth, a constant percentage of the current amount is added in each time unit. For instance, if \( f(t) \) grows at 5% per time period, it will take less time to reach every subsequent doubling of the initial amount compared to linear growth, where a fixed amount is added.

3. **Rapid Increase**: As time progresses, the output of the function increases very rapidly. For instance, if you plug in integer values for \( t \), you will get results that climb steeply as \( t \) increases, clearly demonstrating how the quantity grows faster and faster.

4. **Graph Behavior**: When graphed, an exponential growth function shows a curve that rises steeply, especially for larger values of \( t \), unlike linear functions which have a constant slope and increase at a steady rate. The growth curve will approach the x-axis but never touch it, indicating that the process can continue indefinitely, characteristic of unbounded growth.

In summary, the function \( f(t) = 100 \cdot (1.05)^t \) represents exponential growth because it demonstrates a constant percentage increase over time, leads to increasingly rapid growth, and displays the typical curve found in exponential functions when plotted on a graph.