The Table Of Values Below Represents An Exponential Function. Write An Exponential Equation That Models The Data. \[ \begin{tabular}{|c|c|} \hline X$ & Y Y Y \ \hline -2 & 22 \ \hline -1 & 15.4 \ \hline 0 & 10.78 \ \hline 1 & 7.546 \ \hline 2 &

Introduction

Exponential functions are a fundamental concept in mathematics, and they have numerous applications in various fields, including science, engineering, and economics. These functions describe a relationship between two variables, where one variable grows or decays exponentially with respect to the other variable. In this article, we will explore how to write an exponential equation that models the data presented in a table of values.

Understanding Exponential Functions

An exponential function is a function of the form $f(x) = ab^x$, where $a$ and $b$ are constants, and $b$ is the base of the exponential function. The base $b$ can be any positive real number, but it is usually greater than 1. The constant $a$ is the initial value of the function, and it represents the value of the function when $x = 0$. The variable $x$ is the independent variable, and it represents the input or the variable that is being changed.

The Table of Values

The table of values below represents an exponential function.

$x$	$y$
-2	22
-1	15.4
0	10.78
1	7.546
2	?

Writing the Exponential Equation

To write an exponential equation that models the data, we need to find the values of the constants $a$ and $b$. We can start by looking at the first two rows of the table.

$x$	$y$
-2	22
-1	15.4

We can see that when $x = -2$, $y = 22$, and when $x = -1$, $y = 15.4$. We can use this information to find the value of $b$.

Finding the Value of $b$

We can use the formula $y = ab^x$ to find the value of $b$. We can plug in the values of $x$ and $y$ from the first two rows of the table.

$22 = ab^{-2}$ $15.4 = ab^{-1}$

We can divide the two equations to eliminate the constant $a$.

$\frac{22}{15.4} = \frac{ab^{-2}}{ab^{-1}}$ $\frac{22}{15.4} = b^{-1}$ $1.43 = b^{-1}$

We can take the reciprocal of both sides to find the value of $b$.

$b = \frac{1}{1.43}$ $b = 0.7$

Finding the Value of $a$

Now that we have found the value of $b$, we can use it to find the value of $a$. We can plug in the value of $b$ into one of the original equations.

$22 = ab^{-2}$ $22 = a(0.7)^{-2}$ $22 = a(0.49)$

We can divide both sides by 0.49 to find the value of $a$.

$a = \frac{22}{0.49}$ $a = 44.9$

The Exponential Equation

Now that we have found the values of $a$ and $b$, we can write the exponential equation that models the data.

$f(x) = 44.9(0.7)^x$

Verifying the Equation

To verify the equation, we can plug in the values of $x$ from the table and check if the equation produces the correct values of $y$.

$x$	$y$ (calculated)	$y$ (table)
-2	44.9(0.7)^{-2}	22
-1	44.9(0.7)^{-1}	15.4
0	44.9(0.7)^0	10.78
1	44.9(0.7)^1	7.546
2	44.9(0.7)^2	?

We can see that the equation produces the correct values of $y$ for all the values of $x$ in the table.

Conclusion

$$$$$$$$$$$$$$

Q: What is an exponential function?

A: An exponential function is a function of the form $f(x) = ab^x$, where $a$ and $b$ are constants, and $b$ is the base of the exponential function.

Q: What is the base of an exponential function?

A: The base of an exponential function is the constant $b$ in the equation $f(x) = ab^x$. The base can be any positive real number, but it is usually greater than 1.

Q: What is the initial value of an exponential function?

A: The initial value of an exponential function is the constant $a$ in the equation $f(x) = ab^x$. The initial value represents the value of the function when $x = 0$.

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

A: To find the value of the base $b$, you can use the formula $y = ab^x$ and the values of $x$ and $y$ from the table. You can divide the two equations to eliminate the constant $a$ and then take the reciprocal of both sides to find the value of $b$.

Q: How do I find the value of the initial value $a$ in an exponential function?

A: To find the value of the initial value $a$, you can plug in the value of $b$ into one of the original equations and then divide both sides by the value of $b$ to find the value of $a$.

Q: How do I write an exponential equation that models the data presented in a table of values?

A: To write an exponential equation that models the data, you need to find the values of the constants $a$ and $b$. You can use the formula $y = ab^x$ and the values of $x$ and $y$ from the table to find the values of $a$ and $b$. Then, you can use these values to write the exponential equation.

Q: How do I verify an exponential equation that models the data?

A: To verify an exponential equation, you can plug in the values of $x$ from the table and check if the equation produces the correct values of $y$.

Q: What are some common applications of exponential functions?

A: Exponential functions have numerous applications in various fields, including science, engineering, and economics. Some common applications include modeling population growth, chemical reactions, and financial investments.

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
• Not checking the range of the function
• Not using the correct formula for the function
• Not plugging in the correct values of $x$ and $y$ into the equation

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 plot the points on a coordinate plane and then connect the points to form the graph.

Q: What are some common types of exponential functions?

A: Some common types of exponential functions include:

• Exponential growth functions: $f(x) = ab^x$
• Exponential decay functions: $f(x) = ae^{-bx}$
• Logarithmic functions: $f(x) = \log_b x$

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you can use the formula $y = ab^x$ and the values of $x$ and $y$ from the table. You can also use logarithmic properties to solve the equation.

Q: What are some common applications of logarithmic functions?

A: Logarithmic functions have numerous applications in various fields, including science, engineering, and economics. Some common applications include modeling population growth, chemical reactions, and financial investments.

Q: What are some common mistakes to avoid when working with logarithmic functions?

A: Some common mistakes to avoid when working with logarithmic functions include:

• Not checking the domain of the function
• Not checking the range of the function
• Not using the correct formula for the function
• Not plugging in the correct values of $x$ and $y$ into the equation