Find The Equation Of The Exponential Function Represented By The Table Below:$\[ \begin{tabular}{|c|c|} \hline $x$ & $y$ \\ \hline 0 & 0.1 \\ \hline 1 & 0.3 \\ \hline 2 & 0.9 \\ \hline 3 & 2.7 \\ \hline \end{tabular} \\]

Introduction

In mathematics, an exponential function is a function that has the form $y = ab^x$, where $a$ and $b$ are constants. These functions are used to model real-world phenomena, such as population growth, chemical reactions, and financial investments. In this article, we will learn how to find the equation of an exponential function represented by a table.

Understanding Exponential Functions

Before we dive into finding the equation of an exponential function from a table, let's briefly review what exponential functions are. An exponential function is a function that has the form $y = ab^x$, where $a$ and $b$ are constants. The base $b$ is the growth factor, and the exponent $x$ is the variable. The constant $a$ is the initial value of the function.

Analyzing the Table

The table below represents an exponential function.

$x$	$y$
0	0.1
1	0.3
2	0.9
3	2.7

Step 1: Identify the Initial Value

The initial value of the function is the value of $y$ when $x = 0$. In this case, the initial value is $0.1$.

Step 2: Identify the Growth Factor

To find the growth factor, we need to examine the values of $y$ for each value of $x$. We can see that each value of $y$ is obtained by multiplying the previous value by a constant factor. In this case, the growth factor is $3$, since $0.3 = 0.1 \times 3$, $0.9 = 0.3 \times 3$, and $2.7 = 0.9 \times 3$.

Step 3: Write the Equation of the Exponential Function

Now that we have identified the initial value and the growth factor, we can write the equation of the exponential function. The equation is of the form $y = ab^x$, where $a$ is the initial value and $b$ is the growth factor. In this case, the equation is $y = 0.1 \times 3^x$.

Simplifying the Equation

We can simplify the equation by evaluating the expression $0.1 \times 3^x$. This expression can be rewritten as $0.1 \times 3^x = 0.1 \times 3 \times 3^{x-1} = 0.3 \times 3^{x-1}$.

Conclusion

In this article, we learned how to find the equation of an exponential function represented by a table. We identified the initial value and the growth factor from the table and used this information to write the equation of the exponential function. We also simplified the equation by evaluating the expression $0.1 \times 3^x$. This equation can be used to model real-world phenomena, such as population growth, chemical reactions, and financial investments.

Example Problems

Problem 1

Find the equation of the exponential function represented by the table below.

$x$	$y$
0	2
1	6
2	18
3	54

Solution

The initial value of the function is $2$. The growth factor is $3$, since $6 = 2 \times 3$, $18 = 6 \times 3$, and $54 = 18 \times 3$. The equation of the exponential function is $y = 2 \times 3^x$.

Problem 2

Find the equation of the exponential function represented by the table below.

$x$	$y$
0	1
1	2
2	4
3	8

Solution

The initial value of the function is $1$. The growth factor is $2$, since $2 = 1 \times 2$, $4 = 2 \times 2$, and $8 = 4 \times 2$. The equation of the exponential function is $y = 1 \times 2^x$.

Applications of Exponential Functions

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.
• Chemical reactions: Exponential functions can be used to model chemical reactions, where the concentration of a substance grows at a constant rate.
• Financial investments: Exponential functions can be used to model financial investments, where the value of an investment grows at a constant rate.

Conclusion

Q: What is an exponential function?

A: An exponential function is a function that has the form $y = ab^x$, where $a$ and $b$ are constants. The base $b$ is the growth factor, and the exponent $x$ is the variable. The constant $a$ is the initial value of the function.

Q: How do I identify the initial value of an exponential function?

A: The initial value of an exponential function is the value of $y$ when $x = 0$. You can find the initial value by looking at the table and finding the value of $y$ when $x = 0$.

Q: How do I identify the growth factor of an exponential function?

A: To find the growth factor, you need to examine the values of $y$ for each value of $x$. You can see that each value of $y$ is obtained by multiplying the previous value by a constant factor. This constant factor is the growth factor.

Q: How do I write the equation of an exponential function?

A: Once you have identified the initial value and the growth factor, you can write the equation of the exponential function. The equation is of the form $y = ab^x$, where $a$ is the initial value and $b$ is the growth factor.

Q: Can I simplify the equation of an exponential function?

A: Yes, you can simplify the equation of an exponential function by evaluating the expression $ab^x$. This expression can be rewritten as $ab^x = a \times b \times b^{x-1} = a \times b^{x-1}$.

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.
• Chemical reactions: Exponential functions can be used to model chemical reactions, where the concentration of a substance grows at a constant rate.
• Financial investments: Exponential functions can be used to model financial investments, where the value of an investment grows at a constant rate.

Q: How do I use a table to find the equation of an exponential function?

A: To use a table to find the equation of an exponential function, you need to identify the initial value and the growth factor from the table. Once you have identified these values, you can write the equation of the exponential function.

Q: Can I use a calculator to find the equation of an exponential function?

A: Yes, you can use a calculator to find the equation of an exponential function. You can use the calculator to find the initial value and the growth factor from the table, and then use these values to write the equation of the exponential 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:

• Not identifying the initial value and the growth factor correctly: Make sure to identify the initial value and the growth factor correctly from the table.
• Not using the correct equation form: Make sure to use the correct equation form, which is $y = ab^x$.
• Not simplifying the equation correctly: Make sure to simplify the equation correctly by evaluating the expression $ab^x$.

Conclusion

In conclusion, exponential functions are a powerful tool for modeling real-world phenomena. By identifying the initial value and the growth factor from a table, we can write the equation of an exponential function. This equation can be used to model population growth, chemical reactions, and financial investments. We hope that this article has provided a clear understanding of how to find the equation of an exponential function represented by a table.