The Table Represents An Exponential Function.$\[ \begin{tabular}{|c|c|} \hline $x$ & $y$ \\ \hline 1 & 0.25 \\ \hline 2 & 0.125 \\ \hline 3 & 0.0625 \\ \hline 4 & 0.03125 \\ \hline \end{tabular} \\]What Is The Multiplicative Rate Of Change Of

Introduction

In mathematics, an exponential function is a 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 exponential functions and how to find the multiplicative rate of change 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 of the exponential function, and it determines the rate of change of the function. If $b > 1$, the function exhibits exponential growth, and if $0 < b < 1$, the function exhibits exponential decay.

The Table Represents an Exponential Function

The table below represents an exponential function.

Finding the Multiplicative Rate of Change

To find the multiplicative rate of change of the exponential function represented by the table, we need to find the constant $b$ that determines the rate of change. We can do this by examining the relationship between the values of $x$ and $y$ in the table.

Notice that each value of $y$ is obtained by multiplying the previous value of $y$ by a constant factor. Specifically, we have:

$$\begin{align*} y_2 &= y_1 \times \frac{1}{2} \\ y_3 &= y_2 \times \frac{1}{2} \\ y_4 &= y_3 \times \frac{1}{2} \end{align*}$$

This suggests that the multiplicative rate of change of the function is $\frac{1}{2}$.

Proof of the Multiplicative Rate of Change

To prove that the multiplicative rate of change of the function is indeed $\frac{1}{2}$, we can use the following argument.

Suppose that the multiplicative rate of change of the function is $r$. Then, we have:

$$\begin{align*} y_2 &= y_1 \times r \\ y_3 &= y_2 \times r \\ y_4 &= y_3 \times r \end{align*}$$

Substituting the values of $y_1$, $y_2$, $y_3$, and $y_4$ from the table, we get:

$$\begin{align*} 0.125 &= 0.25 \times r \\ 0.0625 &= 0.125 \times r \\ 0.03125 &= 0.0625 \times r \end{align*}$$

Solving for $r$ in each of these equations, we get:

$$\begin{align*} r &= \frac{0.125}{0.25} = \frac{1}{2} \\ r &= \frac{0.0625}{0.125} = \frac{1}{2} \\ r &= \frac{0.03125}{0.0625} = \frac{1}{2} \end{align*}$$

This shows that the multiplicative rate of change of the function is indeed $\frac{1}{2}$.

Conclusion

In this article, we have explored the concept of exponential functions and how to find the multiplicative rate of change from a given table. We have shown that the multiplicative rate of change of the function represented by the table is $\frac{1}{2}$. This result is consistent with the fact that the values of $y$ in the table are obtained by multiplying the previous value of $y$ by a constant factor of $\frac{1}{2}$. We hope that this article has provided a clear and concise explanation of the concept of exponential functions and how to find the multiplicative rate of change.

References

• [1] "Exponential Functions" by Math Open Reference
• [2] "Multiplicative Rate of Change" by Khan Academy

Frequently Asked Questions

• Q: What is an exponential function? A: An exponential function is a function of the form $y = ab^x$, where $a$ and $b$ are constants, and $x$ is the variable.
• Q: What is the multiplicative rate of change of an exponential function? A: The multiplicative rate of change of an exponential function is the constant factor by which the value of the function changes at each step.
• Q: How do I find the multiplicative rate of change of an exponential function from a given table? A: To find the multiplicative rate of change of an exponential function from a given table, you can examine the relationship between the values of $x$ and $y$ in the table and find the constant factor by which the value of $y$ changes at each step.
  Frequently Asked Questions: Exponential Functions and Multiplicative Rate of Change ====================================================================================

Q: What is an exponential function?

A: 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 of the exponential function, and it determines the rate of change of the function.

Q: What is the multiplicative rate of change of an exponential function?

A: The multiplicative rate of change of an exponential function is the constant factor by which the value of the function changes at each step. It is a key feature of exponential functions that distinguishes them from other types of functions.

Q: How do I find the multiplicative rate of change of an exponential function from a given table?

A: To find the multiplicative rate of change of an exponential function from a given table, you can examine the relationship between the values of $x$ and $y$ in the table and find the constant factor by which the value of $y$ changes at each step. This can be done by dividing each value of $y$ by the previous value of $y$.

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

A: Exponential growth occurs when the base of the exponential function is greater than 1, and the value of the function increases rapidly over time. Exponential decay occurs when the base of the exponential function is less than 1, and the value of the function decreases rapidly over time.

Q: How do I determine whether an exponential function is growing or decaying?

A: To determine whether an exponential function is growing or decaying, you can examine the value of the base of the function. If the base is greater than 1, the function is growing. If the base is less than 1, the function is decaying.

Q: What is the significance of the multiplicative rate of change in exponential functions?

A: The multiplicative rate of change is a key feature of exponential functions that determines the rate at which the value of the function changes over time. It is a critical component of many real-world applications, including population growth, chemical reactions, and financial modeling.

Q: How do I use the multiplicative rate of change to model real-world phenomena?

A: To use the multiplicative rate of change to model real-world phenomena, you can start by identifying the key factors that influence the phenomenon. Then, you can use the multiplicative rate of change to determine the rate at which the phenomenon changes over time. This can be done by using the formula $y = ab^x$, where $a$ is the initial value of the phenomenon, $b$ is the multiplicative rate of change, and $x$ is the time.

Q: What are some common applications of exponential functions and multiplicative rate of change?

A: Exponential functions and multiplicative rate of change have many real-world applications, including:

• Population growth and decline
• Chemical reactions and decay
• Financial modeling and investment
• Epidemiology and disease spread
• Climate modeling and weather forecasting

Q: How do I calculate the multiplicative rate of change of an exponential function?

A: To calculate the multiplicative rate of change of an exponential function, you can use the formula $r = \frac{y_{n+1}}{y_n}$, where $r$ is the multiplicative rate of change, $y_n$ is the value of the function at time $n$, and $y_{n+1}$ is the value of the function at time $n+1$.

Q: What is the relationship between the multiplicative rate of change and the base of an exponential function?

A: The multiplicative rate of change is directly related to the base of an exponential function. Specifically, the multiplicative rate of change is equal to the base of the function raised to the power of 1. This can be expressed mathematically as $r = b^1$, where $r$ is the multiplicative rate of change and $b$ is the base of the function.