Determine The Equation Of The Exponential Function Represented By The Following Table:${ \begin{array}{|c|c|} \hline x & Y \ \hline 0 & 1 \ \hline 1 & 2 \ \hline 2 & 4 \ \hline 3 & 8 \ \hline \end{array} }$

Introduction

In mathematics, an exponential function is a function that has the form $f(x) = ab^x$, where $a$ and $b$ are constants. The graph of an exponential function is a curve that rises or falls rapidly, depending on the value of $b$. In this article, we will determine the equation of the exponential function represented by the given table.

Understanding the Table

The given table represents the values of $x$ and $y$ for an exponential function. The table is as follows:

x	y
0	1
1	2
2	4
3	8

Analyzing the Table

From the table, we can see that the value of $y$ is increasing rapidly as the value of $x$ increases. This suggests that the function is an exponential function. We can also see that the value of $y$ is doubling for every increase in the value of $x$ by 1.

Determining the Equation

To determine the equation of the exponential function, we need to find the values of $a$ and $b$. We can use the first two values in the table to find the value of $b$. Since $y = ab^x$, we can write:

$$2 = ab^1$$

We can also write:

$$4 = ab^2$$

We can divide the second equation by the first equation to get:

$$\frac{4}{2} = \frac{ab^2}{ab^1}$$

Simplifying the equation, we get:

$$2 = b$$

Now that we have found the value of $b$, we can substitute it into one of the original equations to find the value of $a$. We can use the first equation:

$$2 = ab^1$$

Substituting $b = 2$, we get:

$$2 = a(2)^1$$

Simplifying the equation, we get:

$$a = 1$$

The Equation of the Exponential Function

Now that we have found the values of $a$ and $b$, we can write the equation of the exponential function:

$$y = 1(2)^x$$

Simplifying the equation, we get:

$$y = 2^x$$

Conclusion

In this article, we determined the equation of the exponential function represented by the given table. We analyzed the table and found that the value of $y$ is increasing rapidly as the value of $x$ increases. We used the first two values in the table to find the value of $b$ and then substituted it into one of the original equations to find the value of $a$. We found that the equation of the exponential function is $y = 2^x$.

Exponential Functions in Real-World Applications

Exponential functions have many real-world applications. Some examples include:

• Population growth: Exponential functions can be used to model population growth. For example, if a population is growing at a rate of 2% per year, the population can be modeled using an exponential function.
• Financial applications: Exponential functions can be used to model financial applications such as compound interest. For example, if a bank account earns a 5% interest rate per year, the balance in the account can be modeled using an exponential function.
• Science and engineering: Exponential functions can be used to model many scientific and engineering applications such as radioactive decay, chemical reactions, and electrical circuits.

Common Types of Exponential Functions

There are several common types of exponential functions, including:

• Exponential growth: This type of function represents a quantity that is increasing at a rate that is proportional to the quantity itself. An example of an exponential growth function is $y = 2^x$.
• Exponential decay: This type of function represents a quantity that is decreasing at a rate that is proportional to the quantity itself. An example of an exponential decay function is $y = 2^{-x}$.
• Logarithmic functions: These functions are the inverse of exponential functions. An example of a logarithmic function is $y = \log_2(x)$.

Solving Exponential Equations

Exponential equations can be solved using logarithms. For example, to solve the equation $2^x = 8$, we can take the logarithm of both sides:

$$\log_2(2^x) = \log_2(8)$$

Using the property of logarithms that $\log_a(a^x) = x$, we can simplify the equation to:

$$x = \log_2(8)$$

Using a calculator, we can find that $\log_2(8) = 3$. Therefore, the solution to the equation is $x = 3$.

Conclusion

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Q: What is an exponential function?

A: An exponential function is a function that has the form $f(x) = ab^x$, where $a$ and $b$ are constants. The graph of an exponential function is a curve that rises or falls rapidly, depending on the value of $b$.

Q: What are some common types of exponential functions?

A: There are several common types of exponential functions, including:

• Exponential growth: This type of function represents a quantity that is increasing at a rate that is proportional to the quantity itself. An example of an exponential growth function is $y = 2^x$.
• Exponential decay: This type of function represents a quantity that is decreasing at a rate that is proportional to the quantity itself. An example of an exponential decay function is $y = 2^{-x}$.
• Logarithmic functions: These functions are the inverse of exponential functions. An example of a logarithmic function is $y = \log_2(x)$.

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

A: To determine the equation of an exponential function, you need to find the values of $a$ and $b$. You can use the first two values in the table to find the value of $b$. Then, you can substitute it into one of the original equations to find the value of $a$.

Q: How do I solve exponential equations?

A: Exponential equations can be solved using logarithms. For example, to solve the equation $2^x = 8$, you can take the logarithm of both sides:

$$\log_2(2^x) = \log_2(8)$$

Using the property of logarithms that $\log_a(a^x) = x$, you can simplify the equation to:

$$x = \log_2(8)$$

Using a calculator, you can find that $\log_2(8) = 3$. Therefore, the solution to the equation is $x = 3$.

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. For example, if a population is growing at a rate of 2% per year, the population can be modeled using an exponential function.
• Financial applications: Exponential functions can be used to model financial applications such as compound interest. For example, if a bank account earns a 5% interest rate per year, the balance in the account can be modeled using an exponential function.
• Science and engineering: Exponential functions can be used to model many scientific and engineering applications such as radioactive decay, chemical reactions, and electrical circuits.

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 checking the domain of the function: Make sure to check the domain of the function to ensure that it is defined for all values of $x$.
• Not using the correct base: Make sure to use the correct base when working with exponential functions. For example, if the function is $y = 2^x$, make sure to use the base 2, not 10.
• Not simplifying the equation: Make sure to simplify the equation before solving it. This can help to avoid errors and make the solution easier to find.

Q: How do I graph an exponential function?

A: To graph an exponential function, you can use a graphing calculator or a computer program. You can also use a table of values to create a graph. For example, if the function is $y = 2^x$, you can create a table of values with $x$ ranging from -5 to 5 and $y$ ranging from 0 to 32.

Q: What are some common exponential functions that I should know?

A: Some common exponential functions that you should know include:

• $y = 2^x$: This function represents exponential growth and is often used to model population growth and financial applications.
• $y = 2^{-x}$: This function represents exponential decay and is often used to model radioactive decay and chemical reactions.
• $y = \log_2(x)$: This function is the inverse of $y = 2^x$ and is often used to model logarithmic growth and decay.

Q: How do I use exponential functions in real-world applications?

A: Exponential functions can be used in many real-world applications, including:

• Modeling population growth: Exponential functions can be used to model population growth and predict future population sizes.
• Modeling financial applications: Exponential functions can be used to model financial applications such as compound interest and predict future account balances.
• Modeling scientific and engineering applications: Exponential functions can be used to model many scientific and engineering applications such as radioactive decay, chemical reactions, and electrical circuits.

Q: What are some common challenges when working with exponential functions?

A: Some common challenges when working with exponential functions include:

• Difficulty in solving equations: Exponential equations can be difficult to solve, especially when the base is not a simple number like 2 or 10.
• Difficulty in graphing functions: Exponential functions can be difficult to graph, especially when the base is not a simple number like 2 or 10.
• Difficulty in interpreting results: Exponential functions can be difficult to interpret, especially when the results are not intuitive or easy to understand.