Determining the Multiplicative Rate of Change

Oct 13, 2025 by miodeya.com 114 views

Introduction

In mathematics, exponential functions are a fundamental concept that describe the behavior of various phenomena in the natural world. These functions are characterized by their ability to grow or decay at an exponential rate, which is often represented by a constant factor. In this article, we will explore the concept of multiplicative rate of change, which is a crucial aspect of exponential functions. We will examine a specific function, ${$ f(x) = 10(0.5)^x$}$, and determine its multiplicative rate of change.

What is Multiplicative Rate of Change?

The multiplicative rate of change of a function is a measure of how much the function's output changes when the input changes by a certain amount. In other words, it represents the factor by which the function's output is multiplied when the input is increased or decreased by a unit. For exponential functions, the multiplicative rate of change is a constant factor that determines the rate at which the function grows or decays.

The Function ${$ f(x) = 10(0.5)^x$}$

The given function, ${$ f(x) = 10(0.5)^x$}$, is an exponential function that represents a curve passing through the ordered pairs ${$ (0,10), (1,5),$}$ and ${$ (2,2.5)$}$. To understand the multiplicative rate of change of this function, we need to analyze its behavior as x increases.

Analyzing the Function

Let's examine the function's behavior as x increases from 0 to 1 and then from 1 to 2.

When x increases from 0 to 1, the function's output changes from 10 to 5. This represents a decrease of 50% or a factor of 0.5.
When x increases from 1 to 2, the function's output changes from 5 to 2.5. This represents a decrease of 50% or a factor of 0.5.

Determining the Multiplicative Rate of Change

From the analysis above, we can see that the function's output decreases by a factor of 0.5 when the input increases by 1. This indicates that the multiplicative rate of change of the function is 0.5.

Conclusion

In conclusion, the multiplicative rate of change of the function ${$ f(x) = 10(0.5)^x$}$ is 0.5. This means that for every unit increase in the input, the function's output decreases by 50%. Understanding the multiplicative rate of change of exponential functions is essential in various fields, including mathematics, economics, and physics.

Final Answer

Q&A: Multiplicative Rate of Change

Q: What is the multiplicative rate of change of an exponential function? A: The multiplicative rate of change of an exponential function is a measure of how much the function's output changes when the input changes by a certain amount. It represents the factor by which the function's output is multiplied when the input is increased or decreased by a unit.

Q: How do you determine the multiplicative rate of change of an exponential function? A: To determine the multiplicative rate of change of an exponential function, you need to analyze its behavior as the input increases. You can do this by examining the function's output at different input values and calculating the factor by which the output changes.

Q: What is the multiplicative rate of change of the function ${$ f(x) = 10(0.5)^x$}$? A: The multiplicative rate of change of the function ${$ f(x) = 10(0.5)^x$}$ is 0.5. This means that for every unit increase in the input, the function's output decreases by 50%.

Q: Why is understanding the multiplicative rate of change of exponential functions important? A: Understanding the multiplicative rate of change of exponential functions is essential in various fields, including mathematics, economics, and physics. It helps you analyze and model real-world phenomena, such as population growth, chemical reactions, and financial investments.

Q: Can you give an example of how the multiplicative rate of change of an exponential function is used in real-world applications? A: Yes, the multiplicative rate of change of an exponential function is used in various real-world applications, such as:

Population growth: The multiplicative rate of change of an exponential function can be used to model population growth, where the population increases or decreases at a constant rate.
Chemical reactions: The multiplicative rate of change of an exponential function can be used to model chemical reactions, where the concentration of a substance increases or decreases at a constant rate.
Financial investments: The multiplicative rate of change of an exponential function can be used to model the growth of investments, such as stocks or bonds, where the value increases or decreases at a constant rate.

Q: How do you calculate the multiplicative rate of change of an exponential function? A: To calculate the multiplicative rate of change of an exponential function, you need to:

Analyze the function's behavior: Examine the function's output at different input values and calculate the factor by which the output changes.
Calculate the rate of change: Calculate the rate of change of the function by dividing the change in output by the change in input.
Determine the multiplicative rate of change: Determine the multiplicative rate of change of the function by multiplying the rate of change by the input value.

Conclusion

In conclusion, the multiplicative rate of change of an exponential function is a crucial concept that helps you analyze and model real-world phenomena. Understanding the multiplicative rate of change of exponential functions is essential in various fields, including mathematics, economics, and physics. By following the steps outlined above, you can calculate the multiplicative rate of change of an exponential function and apply it to real-world applications.

Final Answer

The final answer is A. 0.5.