An Exponential Function Follows A Pattern Of Decay Through The Points { (-3,8),(-2,4),$}$ And { (-1,2)$}$. Determine The Base Of The Function.A. { \frac{1}{2}$}$B. { -\frac{1}{2}$}$C. 2D. { -2$}$

Mar 10, 2025 by ADMIN 196 views

An Exponential Function: Unraveling the Mystery of Decay

Exponential functions are a fundamental concept in mathematics, describing growth or decay in various real-world scenarios. In this article, we will delve into the world of exponential functions, focusing on a specific pattern of decay through the points ${(-3,8),(-2,4),}$ and ${(-1,2)}$ . Our objective is to determine the base of the function, which will help us understand the underlying mechanism of this decay.

Understanding Exponential Functions

Exponential functions are of the form ${f(x) = ab^x}$ , where ${a}$ is the initial value, ${b}$ is the base, and ${x}$ is the variable. The base, ${b}$ , determines the rate of growth or decay. If ${b > 1}$ , the function grows exponentially, while if ${0 < b < 1}$ , the function decays exponentially.

The Pattern of Decay

The given points ${(-3,8),(-2,4),}$ and ${(-1,2)}$ suggest a pattern of decay. As we move from left to right, the value of the function decreases. This indicates that the base of the function is less than 1, as the function is decaying.

Determining the Base

To determine the base, we can use the fact that the function passes through the given points. We can write the equation of the function as ${f(x) = ab^x}$ . Substituting the given points, we get:

${8 = ab^{-3}}$ ${4 = ab^{-2}}$ ${2 = ab^{-1}}$

We can simplify these equations by dividing the first equation by the second, and the second equation by the third:

${\frac{8}{4} = \frac{ab^{-3}}{ab^{-2}}}$ ${\frac{4}{2} = \frac{ab^{-2}}{ab^{-1}}}$

Simplifying further, we get:

${2 = b^{-1}}$ ${2 = b^{-1}}$

This shows that the base, ${b}$ , is equal to ${\frac{1}{2}}$ .

In conclusion, the base of the exponential function that follows the pattern of decay through the points ${(-3,8),(-2,4),}$ and ${(-1,2)}$ is ${\frac{1}{2}}$ . This result is consistent with the fact that the function decays as we move from left to right.

The correct answer is:

A. ${\frac{1}{2}}$

Exponential functions have numerous applications in various fields, including finance, biology, and physics. Understanding the concept of exponential decay is crucial in modeling real-world phenomena, such as population growth, radioactive decay, and chemical reactions.

Real-World Applications

Exponential decay is a common phenomenon in various fields, including:

Radioactive decay: The rate of decay of radioactive materials follows an exponential function.
Population growth: The growth of a population can be modeled using an exponential function.
Chemical reactions: The rate of reaction in chemical reactions can be described using an exponential function.

Exponential functions are a fundamental concept in mathematics, describing growth or decay in various real-world scenarios. In our previous article, we explored the concept of exponential functions and determined the base of a function that follows a pattern of decay through the points ${(-3,8),(-2,4),}$ and ${(-1,2)}$ . In this article, we will address some frequently asked questions about exponential functions.

Q: What is an exponential function?

A: An exponential function is a mathematical function of the form ${f(x) = ab^x}$ , where ${a}$ is the initial value, ${b}$ is the base, and ${x}$ is the variable.

Q: What is the base of an exponential function?

A: The base of an exponential function is a constant that determines the rate of growth or decay. If ${b > 1}$ , the function grows exponentially, while if ${0 < b < 1}$ , the function decays exponentially.

Q: How do I determine the base of an exponential function?

A: To determine the base of an exponential function, you can use the fact that the function passes through a given point. You can write the equation of the function as ${f(x) = ab^x}$ and substitute the given point to solve for the base.

Q: What is the difference between exponential growth and decay?

A: Exponential growth occurs when the base of the function is greater than 1, resulting in an increase in the value of the function over time. Exponential decay occurs when the base of the function is between 0 and 1, resulting in a decrease in the value of the function over time.

Q: How do exponential functions apply to real-world scenarios?

A: Exponential functions have numerous applications in various fields, including finance, biology, and physics. They can be used to model population growth, radioactive decay, chemical reactions, and other phenomena.

Q: What are some common applications of exponential functions?

A: Some common applications of exponential functions include:

Radioactive decay: The rate of decay of radioactive materials follows an exponential function.
Population growth: The growth of a population can be modeled using an exponential function.
Chemical reactions: The rate of reaction in chemical reactions can be described using an exponential function.
Finance: Exponential functions can be used to model compound interest and other financial phenomena.

Q: How do I graph an exponential function?

A: To graph an exponential function, you can use a graphing calculator or software. You can also use a table of values to plot the function.

Q: What are some common mistakes to avoid when working with exponential functions?

A: Some common mistakes to avoid when working with exponential functions include:

Confusing the base and the exponent: Make sure to keep track of the base and the exponent when working with exponential functions.
Not checking the domain: Make sure to check the domain of the function to ensure that it is defined for all values of x.
Not considering the asymptotes: Make sure to consider the asymptotes of the function to ensure that it is properly graphed.

In conclusion, exponential functions are a fundamental concept in mathematics, describing growth or decay in various real-world scenarios. By understanding the concept of exponential functions and their applications, you can better model and analyze real-world phenomena. We hope that this Q&A guide has been helpful in addressing some of the most frequently asked questions about exponential functions.