This **question discusses the annual percent increase or decrease**Â in the given model. To solve questions like this, the reader should know about the exponential growth function. **Exponential growth**Â is a process that **increases the quantity**Â over time. It occurs when the **instantaneous rate of change**Â (i.e., derivative) of an amount with respect to time is **proportional to quantity**Â itself. Described as a function, a**Â quantity undergoing exponential growth**Â represents an exponential **function of time**; that is, variable representing time is an exponent (unlike other types of growth, such as **quadratic growth**).

If **proportionality constant is negative**, then the **quantity decreases**Â over time and is said to undergo **exponential decay**. A discrete definition region with equal intervals is also called **geometric growth**Â or **geometric decrease**Â because the function values form a **geometric progression.**

The formula for the **exponential growth function**Â is

\[ f ( x ) = a ( 1 + r ) ^{ x } \]

Where $ f ( x ) $ is the **initial growth function.**

$ a $ is the**Â initial amount.**

$ r $ is the **growth rate.**

$ x $ is the **number of time intervals.**

Growth like this is seen in**Â real life activities or phenomena**, such as spread of a **viral infection**, growth of debt due to **compound interest**, and spread of viral videos.

**Expert Answer**

Given model

**Equation 1 is:**

\[ y = 0.35 ( 2.3 ) ^ { x } \]

The **exponential growth function**Â is

**Equation 2**Â is

\[ y = A ( 1 + \gamma ) ^ { x } \]

Where $ A $ is the **initial amount.**

$ \gamma $ is the**Â annual percent.**

$ x $ is the **number of years.**

\[ A = 0.35 \]

\[ 1 + \gamma = 2.3 \]

\[ \Rightarrow \gamma = 2.3 – 1 \]

\[ \Rightarrow \gamma = 1.3 \]

\[ \Rightarrow Â \gamma = 1.3 \times 100 \% \]

\[ \gamma = 130 \% \]

The **annual percent increase**Â is $ 130 \% $.

**Numerical Result**

The **annual percent increase**Â of the model $ Â y = 0.35 Â ( 2.3 ) ^ { x } $ is $ 130 \%$.

**Example**

**Find the annual percentage increase or decrease $ y = 0.45 Â ( Â 3.3 Â ) ^ { x } $ models.**

**Solution**

Given model

**Equation 1 is **

\[ Â y = 0.45 ( 2.3 ) ^ { x } \]

The **exponential growth function**Â is

**Equation 2**Â is

\[ Â y = A (1 Â + \gamma ) ^ { x } \]

Where $ A $ is the **initial amount.**

$ \gamma $ is the**Â annual percent.**

$ x $ is the **number of years.**

By using **equation** $ 1 $ and $ 2 $.

\[ A = 0.45 \]

\[ 1 + \gamma = 3.3 \]

\[ \Rightarrow \gamma = 3.3 Â – Â 1 \]

\[ \Rightarrow \gamma = 2.3 \]

\[\Rightarrow Â \gamma = 2.3 \times 100 \% \]

\[ \gamma = 230 \% \]

The **annual percent increase**Â is $ 230 \% $.