Which Table Represents An Exponential Function Of The Form $y=b^x$ When $0\ \textless \ B\ \textless \ 1$? \[ \begin{tabular}{|c|c|} \hline X$ & Y Y Y \ \hline -3 & 1 27 \frac{1}{27} 27 1 ​ \ \hline -2 & 1 9 \frac{1}{9} 9 1 ​ \ \hline -1 &

Introduction

Exponential functions are a fundamental concept in mathematics, and they play a crucial role in various fields, including science, engineering, and economics. In this article, we will focus on identifying the correct table representation of an exponential function of the form $y=b^x$ when $0\ \textless \ b\ \textless \ 1$. This type of function is characterized by a base $b$ that is greater than 0 and less than 1, and it is used to model situations where a quantity decreases exponentially over time.

What is an Exponential Function?

An exponential function is a mathematical function of the form $y=b^x$, where $b$ is the base and $x$ is the exponent. The base $b$ can be any positive real number, and the exponent $x$ can be any real number. When $b$ is greater than 1, the function represents an exponential growth, where the value of $y$ increases rapidly as $x$ increases. On the other hand, when $b$ is less than 1, the function represents an exponential decay, where the value of $y$ decreases rapidly as $x$ increases.

The Given Tables

We are given four tables, each representing a different exponential function. Our task is to identify the table that represents an exponential function of the form $y=b^x$ when $0\ \textless \ b\ \textless \ 1$. Let's examine each table carefully and determine which one meets the given criteria.

Table 1

Table 2

Table 3

Table 4

Analyzing the Tables

To determine which table represents an exponential function of the form $y=b^x$ when $0\ \textless \ b\ \textless \ 1$, we need to examine the values of $y$ for each table. We are looking for a table where the value of $y$ decreases as $x$ increases, and the ratio of consecutive values of $y$ is constant.

Let's analyze each table:

• Table 1: The value of $y$ increases as $x$ increases, which is not consistent with an exponential decay.
• Table 2: The value of $y$ decreases as $x$ increases, and the ratio of consecutive values of $y$ is constant. This suggests that Table 2 represents an exponential function of the form $y=b^x$ when $0\ \textless \ b\ \textless \ 1$.
• Table 3: The value of $y$ decreases as $x$ increases, but the ratio of consecutive values of $y$ is not constant. This suggests that Table 3 does not represent an exponential function of the form $y=b^x$ when $0\ \textless \ b\ \textless \ 1$.
• Table 4: The value of $y$ increases as $x$ increases, which is not consistent with an exponential decay.

Conclusion

Based on our analysis, we can conclude that Table 2 represents an exponential function of the form $y=b^x$ when $0\ \textless \ b\ \textless \ 1$. This table meets the given criteria, where the value of $y$ decreases as $x$ increases, and the ratio of consecutive values of $y$ is constant.

Final Thoughts

Introduction

Exponential functions are a fundamental concept in mathematics, and they play a crucial role in various fields, including science, engineering, and economics. In our previous article, we discussed the concept of exponential functions and identified the correct table representation of an exponential function of the form $y=b^x$ when $0\ \textless \ b\ \textless \ 1$. In this article, we will provide a Q&A guide to help you better understand exponential functions and their applications.

Q: What is an exponential function?

A: An exponential function is a mathematical function of the form $y=b^x$, where $b$ is the base and $x$ is the exponent. The base $b$ can be any positive real number, and the exponent $x$ can be any real number.

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

A: Exponential growth occurs when the value of $y$ increases rapidly as $x$ increases, and the base $b$ is greater than 1. Exponential decay occurs when the value of $y$ decreases rapidly as $x$ increases, and the base $b$ is less than 1.

Q: How do you determine if a function is exponential?

A: To determine if a function is exponential, you need to examine the values of $y$ for each input value of $x$. If the value of $y$ increases or decreases rapidly as $x$ increases, and the ratio of consecutive values of $y$ is constant, then the function is likely exponential.

Q: What are some common applications of exponential functions?

A: Exponential functions have many applications in various fields, including:

• Population growth: Exponential functions can be used to model population growth, where the population increases rapidly over time.
• Radioactive decay: Exponential functions can be used to model radioactive decay, where the amount of radioactive material decreases rapidly over time.
• Financial modeling: Exponential functions can be used to model financial situations, such as compound interest and depreciation.
• Biology: Exponential functions can be used to model biological processes, such as the growth of bacteria and the spread of diseases.

Q: How do you graph an exponential function?

A: To graph an exponential function, you need to plot the values of $y$ for each input value of $x$. You can use a graphing calculator or a computer program to graph 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 exponential growth and exponential decay: Make sure to identify whether the function is growing or decaying.
• Not checking the base: Make sure to check the base of the function to determine whether it is exponential.
• Not using the correct formula: Make sure to use the correct formula for the exponential function, which is $y=b^x$.

Conclusion

In this article, we have provided a Q&A guide to help you better understand exponential functions and their applications. We have discussed the concept of exponential functions, identified the correct table representation of an exponential function of the form $y=b^x$ when $0\ \textless \ b\ \textless \ 1$, and provided answers to common questions about exponential functions. We hope this guide has been helpful in your understanding of exponential functions.

Final Thoughts

Exponential functions are a fundamental concept in mathematics, and they have many applications in various fields. By understanding exponential functions, you can better model and analyze real-world situations, and make more informed decisions. We hope this guide has been helpful in your journey to understanding exponential functions.