Question

Write an exponential growth function. Explain why your function represents exponential growth.

Answer 1

An example of an exponential growth function is:

\[ f(t) = 5 \cdot e^{0.3t} \]

### Explanation of the Function

1. **Form of an Exponential Function**:
The general form of an exponential function is given by:
\[ f(t) = a \cdot b^{ct} \]
or using the natural base \( e \):
\[ f(t) = a \cdot e^{kt} \]
In the function \( f(t) = 5 \cdot e^{0.3t} \):
- \( a = 5 \) is the initial value or the y-intercept (the value of the function when \( t = 0 \)).
- \( e \) (approximately 2.71828) is the base of the natural logarithm.
- \( k = 0.3 \) is a positive constant that represents the growth rate.

2. **Characteristics of Exponential Growth**:
- **Constant Percentage Growth**: In exponential growth, the quantity increases by a constant percentage over equal intervals of time. In this case, since \( k = 0.3 \), the function grows at a rate of approximately 30% of its current value for each unit increase in \( t \).
- **Doubling or Tripling Time**: Exponential functions have characteristic doubling times, which is the time it takes for the function to double in value. The larger the value of \( k \), the faster the growth.

3. **Behavior Over Time**:
As \( t \) increases, \( e^{0.3t} \) grows rapidly because the exponent \( 0.3t \) increases, causing the function \( f(t) \) to increase very quickly. This rapid increase is a hallmark of exponential growth.

### Summary
The function \( f(t) = 5 \cdot e^{0.3t} \) represents exponential growth because it is in the form of an exponential function with a positive growth constant, signifying that the quantity grows at a percentage rate that is proportional to its current amount.