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In exponential growth models, the growth factor is pivotal. It determines how rapidly the quantity is increasing over time. The growth factor is found as the base of the exponent in an exponential equation. In the equation provided, \( y = 12(2,155)(1.062)^{x-1} \), the growth factor is \( 1.062 \). This number tells you how much the rent is multiplied each year. A growth factor greater than 1 indicates an increase, while a number less than 1 would suggest a decrease.
To identify the growth factor in such problems, simply look for the base number in the exponential term. Understanding the concept of growth factor helps in predicting future values based on past behavior.
Examples of different growth scenarios include:

• If the growth factor was 1.0, it would imply no change in the amount over time.
• A growth factor of 1.1 would mean a 10% increase in each period.

Recognizing the growth factor allows you to comprehend the dynamics of change in the scenario you are analyzing.

Understanding the percent increase is crucial for interpreting changes in quantities over time. To find it, you must first identify the growth factor from the exponential equation, then convert it into a percentage format. In our equation, the growth factor is \(1.062\).

To convert this to a percent increase, perform the calculation:

• Subtract 1 from the growth factor: \(1.062 - 1 = 0.062\).
• Convert this to a percentage: \(0.062 \times 100 = 6.2\%\).

This result means there is a 6.2% increase per year in the rent. Subtracting 1 in the process represents the original amount, and the difference corresponds to the increase. The multiplication by 100 switches it to percentage form for easy understanding. This transformation is commonly used to express and compare growth rates universally.

The exponential equation is a powerful tool for modeling growth over time. It's particularly useful in fields like finance, biology, and technology. The general form of an exponential equation is \( y = a(b)^{x} \). In this form:

• \( y \) is the output, or the final amount.
• \( a \) represents the initial value or starting point.
• \( b \) is the growth factor.
• \( x \) is the time period or number of times the growth is applied.

In the context of our example, the equation given is \( y = 12(2,155)(1.062)^{x-1} \). Here, 12(2,155) represents the initial monthly rent, and \( 1.062^{x-1} \) models the yearly rent increase.

Exponential equations can depict rapid growth or decay, depending on whether the growth factor is greater than or less than 1. They are distinct for creating smooth curves, reflecting compound effects over time. By understanding exponential equations, one can make predictions about future outcomes based on current data trends.