Question

Describe how you can tell whether there is a growth or decay rate in an exponential equation. (2pts)

Give an example of an exponential growth equation and an exponential decay equation. (1pt each)

Answer 1

To determine whether there is a growth or decay rate in an exponential equation, you can look at the base of the exponential function:

1. If the base (b) of the exponential equation \(y = ab^x\) is greater than 1 (i.e., \(b > 1\)), the function represents exponential growth. This means that as \(x\) increases, \(y\) also increases rapidly.

2. If the base (b) is between 0 and 1 (i.e., \(0 < b < 1\)), the function represents exponential decay. In this case, as \(x\) increases, \(y\) decreases.

### Examples:
- **Exponential Growth Equation:** \(y = 2(3^x)\)
- **Exponential Decay Equation:** \(y = 5(0.5^x)\)

In the first example, the base 3 is greater than 1, indicating growth. In the second example, the base 0.5 is between 0 and 1, indicating decay.