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Introduction

In this article, we will explore the concept of modeling population growth using linear and exponential functions. We will examine a table of estimated population data for a certain city over time and determine which type of function would better model the data.

Understanding Linear and Exponential Functions

Before we dive into the data, let's briefly review the concepts of linear and exponential functions.

Linear Functions

A linear function is a polynomial function of degree one, which means it has the form:

f(x) = mx + b

where m is the slope and b is the y-intercept. Linear functions have a constant rate of change, which means that for every unit increase in x, the value of f(x) increases by the same amount.

Exponential Functions

An exponential function is a function of the form:

f(x) = ab^x

where a is the initial value and b is the growth factor. Exponential functions have a constant growth rate, which means that the value of f(x) increases by a fixed percentage for every unit increase in x.

The Data

The estimated population of the city over time is given in the table below:

Analyzing the Data

To determine which type of function would better model the data, let's examine the pattern of population growth.

• The population starts at 1000 and increases by 200 every 5 years.
• The population increases by 300 every 5 years from 10 to 15 years.
• The population increases by 400 every 5 years from 15 to 20 years.

As we can see, the population is increasing at an accelerating rate, which suggests that an exponential function would be a better fit.

Why Exponential Functions are a Better Fit

There are several reasons why exponential functions are a better fit for this data:

• Accelerating growth: The population is increasing at an accelerating rate, which is a characteristic of exponential growth.
• Constant growth rate: The population is increasing by a fixed percentage for every unit increase in time, which is a characteristic of exponential functions.
• Non-linear relationship: The relationship between population and time is non-linear, which means that a linear function would not be able to capture the accelerating growth.

Conclusion

In conclusion, the estimated population of the city over time is best modeled by an exponential function. The accelerating growth and constant growth rate of the population suggest that an exponential function would be a better fit than a linear function.

Key Takeaways

• Exponential functions are a better fit for data that exhibits accelerating growth.
• Exponential functions have a constant growth rate, which means that the value of f(x) increases by a fixed percentage for every unit increase in x.
• Linear functions have a constant rate of change, which means that for every unit increase in x, the value of f(x) increases by the same amount.

Real-World Applications

Exponential functions have many real-world applications, including:

• Population growth: Exponential functions can be used to model population growth in cities, countries, and even the world.
• Financial modeling: Exponential functions can be used to model the growth of investments, such as stocks and bonds.
• Science and engineering: Exponential functions can be used to model the growth of chemical reactions, the spread of diseases, and the behavior of electrical circuits.

Final Thoughts

In conclusion, the estimated population of the city over time is best modeled by an exponential function. The accelerating growth and constant growth rate of the population suggest that an exponential function would be a better fit than a linear function. Exponential functions have many real-world applications, including population growth, financial modeling, and science and engineering.