The Table Represents An Exponential Function.${ \begin{tabular}{|c|c|} \hline X X X & Y Y Y \ \hline 1 & 6 \ \hline 2 & 4 \ \hline 3 & 8 3 \frac{8}{3} 3 8 ​ \ \hline 4 & 16 9 \frac{16}{9} 9 16 ​ \ \hline \end{tabular} }$What Is The Multiplicative Rate Of

Introduction

In mathematics, an exponential function is a type of function that exhibits exponential growth or decay. It is characterized by a constant rate of change, which is a key feature that distinguishes it from other types of functions. In this article, we will explore the concept of an exponential function and how to determine the multiplicative rate from a given table.

What is an Exponential Function?

An exponential function is a function of the form $y = ab^x$, where $a$ and $b$ are constants, and $x$ is the variable. The constant $b$ is called the base, and it determines the rate of growth or decay of the function. If $b > 1$, the function grows exponentially, and if $0 < b < 1$, the function decays exponentially.

The Table Represents an Exponential Function

The table given in the problem represents an exponential function. We can see that as the value of $x$ increases, the value of $y$ decreases. This suggests that the function is decaying exponentially.

$x$	$y$
1	6
2	4
3	$\frac{8}{3}$
4	$\frac{16}{9}$

Determining the Multiplicative Rate

To determine the multiplicative rate, we need to find the value of $b$ in the exponential function $y = ab^x$. We can do this by examining the ratio of consecutive values of $y$.

Let's consider the ratio of the second and first values of $y$:

$$\frac{y_2}{y_1} = \frac{4}{6} = \frac{2}{3}$$

This ratio represents the multiplicative rate of the function. We can see that the function decays by a factor of $\frac{2}{3}$ for each increase in $x$ by 1.

Finding the Value of $b$

Now that we have the multiplicative rate, we can find the value of $b$ in the exponential function. We know that the multiplicative rate is equal to $\frac{b}{1}$, so we can set up the equation:

$$\frac{b}{1} = \frac{2}{3}$$

Solving for $b$, we get:

$$b = \frac{2}{3}$$

Conclusion

In this article, we explored the concept of an exponential function and how to determine the multiplicative rate from a given table. We saw that the table represents an exponential function that decays by a factor of $\frac{2}{3}$ for each increase in $x$ by 1. We also found the value of $b$ in the exponential function, which is $\frac{2}{3}$. This value represents the multiplicative rate of the function.

Exponential Functions in Real-World Applications

Exponential functions have many real-world applications, including:

• Population growth: Exponential functions can be used to model population growth, where the population grows at a constant rate.
• Radioactive decay: Exponential functions can be used to model radioactive decay, where the amount of radioactive material decreases at a constant rate.
• Financial applications: Exponential functions can be used to model financial applications, such as compound interest and depreciation.

Examples of Exponential Functions

Here are some examples of exponential functions:

• Population growth: $y = 1000(1.05)^x$
• Radioactive decay: $y = 100(0.95)^x$
• Financial applications: $y = 1000(1.05)^x$

Solving Exponential Equations

Exponential equations can be solved using logarithms. Here are some examples:

• Solving for $x$: $y = 100(0.95)^x \Rightarrow \log(y) = \log(100) + x\log(0.95) \Rightarrow x = \frac{\log(y) - \log(100)}{\log(0.95)}$
• Solving for $y$: $y = 1000(1.05)^x \Rightarrow \log(y) = \log(1000) + x\log(1.05) \Rightarrow y = 1000e^{x\log(1.05)}$

Conclusion

Q: What is an exponential function?

A: An exponential function is a type of function that exhibits exponential growth or decay. It is characterized by a constant rate of change, which is a key feature that distinguishes it from other types of functions.

Q: How do I determine the multiplicative rate from a given table?

A: To determine the multiplicative rate, you need to find the ratio of consecutive values of $y$. This ratio represents the multiplicative rate of the function.

Q: What is the multiplicative rate in the given table?

A: The multiplicative rate in the given table is $\frac{2}{3}$. This means that the function decays by a factor of $\frac{2}{3}$ for each increase in $x$ by 1.

Q: How do I find the value of $b$ in the exponential function?

A: To find the value of $b$, you need to set up the equation $\frac{b}{1} = \frac{2}{3}$ and solve for $b$. This will give you the value of $b$ in the exponential function.

Q: What is the value of $b$ in the exponential function?

A: The value of $b$ in the exponential function is $\frac{2}{3}$.

Q: What are some real-world applications of exponential functions?

A: Exponential functions have many real-world applications, including:

• Population growth: Exponential functions can be used to model population growth, where the population grows at a constant rate.
• Radioactive decay: Exponential functions can be used to model radioactive decay, where the amount of radioactive material decreases at a constant rate.
• Financial applications: Exponential functions can be used to model financial applications, such as compound interest and depreciation.

Q: How do I solve exponential equations?

A: Exponential equations can be solved using logarithms. You can use the following formula to solve for $x$:

$$x = \frac{\log(y) - \log(a)}{\log(b)}$$

Q: What is the formula for solving exponential equations?

A: The formula for solving exponential equations is:

$$x = \frac{\log(y) - \log(a)}{\log(b)}$$

Q: How do I solve for $y$ in an exponential equation?

A: To solve for $y$ in an exponential equation, you can use the following formula:

$$y = a \cdot b^x$$

Q: What is the formula for solving for $y$ in an exponential equation?

A: The formula for solving for $y$ in an exponential equation is:

$$y = a \cdot b^x$$

Q: What are some examples of exponential functions?

A: Here are some examples of exponential functions:

• Population growth: $y = 1000(1.05)^x$
• Radioactive decay: $y = 100(0.95)^x$
• Financial applications: $y = 1000(1.05)^x$

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

A: To determine the base of an exponential function, you need to examine the ratio of consecutive values of $y$. This ratio represents the base of the function.

Q: What is the base of the exponential function in the given table?

A: The base of the exponential function in the given table is $\frac{2}{3}$.

Conclusion

In conclusion, exponential functions are a powerful tool for modeling real-world phenomena. By understanding how to determine the multiplicative rate from a given table, we can gain a deeper understanding of these functions and their applications.