Find A Formula For The Exponential Function Passing Through The Points { (-2, \frac{4}{9})$}$ And { (3, 108)$} . . . { F(x) =\$} { \square$}$

Introduction

The exponential function is a fundamental concept in mathematics, and it has numerous applications in various fields, including physics, engineering, and economics. In this article, we will focus on finding the formula for the exponential function that passes through the points ${(-2, \frac{4}{9})}$ and ${(3, 108)}$. This problem requires us to use the concept of exponential functions and their properties to find the desired formula.

Understanding Exponential Functions

An exponential function is a function of the form ${f(x) = a \cdot b^x}$, where ${a}$ and ${b}$ are constants, and ${x}$ is the variable. The base ${b}$ is a positive number, and the exponent ${x}$ is a real number. The exponential function has several important properties, including:

• The exponential function is continuous: The exponential function is continuous at all points, which means that it can be drawn without lifting the pencil from the paper.
• The exponential function is one-to-one: The exponential function is a one-to-one function, which means that each value of ${x}$ corresponds to a unique value of ${f(x)}$.
• The exponential function is increasing: The exponential function is an increasing function, which means that as ${x}$ increases, ${f(x)}$ also increases.

Using the Given Points to Find the Formula

We are given two points, ${(-2, \frac{4}{9})}$ and ${(3, 108)}$, that lie on the exponential function. We can use these points to find the formula for the exponential function. To do this, we need to use the concept of the slope of a line.

Finding the Slope of the Line

The slope of a line is a measure of how steep the line is. It is calculated as the ratio of the vertical change (the rise) to the horizontal change (the run). In this case, we can use the two given points to find the slope of the line.

Let's calculate the slope of the line using the two given points:

${m = \frac{y_2 - y_1}{x_2 - x_1}}$

where ${m}$ is the slope, and ${x_1, y_1}$ and ${x_2, y_2}$ are the coordinates of the two points.

Plugging in the values, we get:

${m = \frac{108 - \frac{4}{9}}{3 - (-2)}}$

Simplifying the expression, we get:

${m = \frac{\frac{968}{9}}{5}}$

${m = \frac{968}{45}}$

Finding the Formula for the Exponential Function

Now that we have the slope of the line, we can use it to find the formula for the exponential function. We know that the exponential function has the form ${f(x) = a \cdot b^x}$. We also know that the slope of the line is ${\frac{968}{45}}$. We can use this information to find the values of ${a}$ and ${b}$.

Let's start by plugging in the values of the two given points the formula for the exponential function:

${f(-2) = a \cdot b^{-2}}$

${f(3) = a \cdot b^3}$

We know that the values of ${f(-2)}$ and ${f(3)}$ are ${\frac{4}{9}}$ and ${108}$, respectively. We can use this information to set up a system of equations:

${a \cdot b^{-2} = \frac{4}{9}}$

${a \cdot b^3 = 108}$

Solving the System of Equations

We can solve the system of equations by first multiplying the two equations together:

${a^2 \cdot b = \frac{4}{9} \cdot 108}$

Simplifying the expression, we get:

${a^2 \cdot b = 48}$

Finding the Value of ${a}$

We can use the equation ${a^2 \cdot b = 48}$ to find the value of ${a}$. We know that ${b}$ is a positive number, so we can divide both sides of the equation by ${b}$ to get:

${a^2 = \frac{48}{b}}$

Taking the square root of both sides, we get:

${a = \sqrt{\frac{48}{b}}}$

Finding the Value of ${b}$

We can use the equation ${a \cdot b^{-2} = \frac{4}{9}}$ to find the value of ${b}$. We know that ${a}$ is a positive number, so we can divide both sides of the equation by ${a}$ to get:

${b^{-2} = \frac{\frac{4}{9}}{a}}$

Taking the reciprocal of both sides, we get:

${b^2 = \frac{9a}{4}}$

Finding the Value of ${a}$ and ${b}$

We can use the equations ${a^2 = \frac{48}{b}}$ and ${b^2 = \frac{9a}{4}}$ to find the values of ${a}$ and ${b}$. We can substitute the expression for ${b^2}$ into the equation for ${a^2}$ to get:

${a^2 = \frac{48}{\frac{9a}{4}}}$

Simplifying the expression, we get:

${a^2 = \frac{192}{9a}}$

Cross-multiplying, we get:

${9a^3 = 192}$

Dividing both sides by 9, we get:

${a^3 = \frac{192}{9}}$

Taking the cube root of both sides, we get:

${a = \sqrt[3]{\frac{192}{9}}}$

Simplifying the expression, we get:

${a = \sqrt[3]{\frac{64}{3}}}$

${a = \frac{4}{\sqrt[3]{3}}}$

Finding the Value of ${b}$

We can use the equation ${b^2 = \frac{9a}{4}}$ to find the value of ${b}$. We know that ${a}$ is a positive number, so we can substitute the expression for ${a}$ into the equation for ${b^2}$ to get:

b^2 = \frac{9 \cdot \frac{4}{\sqrt[3]{3}}}{4}]

Simplifying the expression, we get:

${b^2 = \frac{9}{\sqrt[3]{3}}}$

Taking the square root of both sides, we get:

${b = \sqrt{\frac{9}{\sqrt[3]{3}}}}$

Simplifying the expression, we get:

${b = \frac{3}{\sqrt[6]{3}}}$

Finding the Formula for the Exponential Function

Now that we have the values of ${a}$ and ${b}$, we can use them to find the formula for the exponential function. We know that the exponential function has the form ${f(x) = a \cdot b^x}$. We can substitute the values of ${a}$ and ${b}$ into this formula to get:

${f(x) = \frac{4}{\sqrt[3]{3}} \cdot \left(\frac{3}{\sqrt[6]{3}}\right)^x}$

Simplifying the expression, we get:

${f(x) = \frac{4}{\sqrt[3]{3}} \cdot \frac{3^x}{\sqrt[6]{3^x}}}$

Simplifying the expression further, we get:

${f(x) = \frac{4 \cdot 3^x}{\sqrt[3]{3} \cdot \sqrt[6]{3^x}}}$

Simplifying the expression even further, we get:

${f(x) = \frac{4 \cdot 3^x}{3^{\frac{x}{3}}}}$

Simplifying the expression one last time, we get:

${f(x) = 4 \cdot 3^{\frac{2x}{3}}}$

Conclusion

In this article, we used the concept of exponential functions and their properties to find the formula for the exponential function that passes through the points ${(-2, \frac{4}{9})}$ and ${(3, 108)}$. We used the given points to find the slope of the line, and then used the slope to find the values of ${a}$ and ${b}$. We then used these values to find the formula for the exponential function. The final formula for the exponential function is ${f(x) = 4 \cdot 3^{\frac{2x}{3}}}$.

Introduction

In our previous article, we discussed how to find the formula for the exponential function that passes through the points ${(-2, \frac{4}{9})}$ and ${(3, 108)}$. In this article, we will answer some frequently asked questions about exponential functions and their properties.

Q: What is an exponential function?

A: An exponential function is a function of the form ${f(x) = a \cdot b^x}$, where ${a}$ and ${b}$ are constants, and ${x}$ is the variable. The base ${b}$ is a positive number, and the exponent ${x}$ is a real number.

Q: What are the properties of exponential functions?

A: Exponential functions have several important properties, including:

• The exponential function is continuous: The exponential function is continuous at all points, which means that it can be drawn without lifting the pencil from the paper.
• The exponential function is one-to-one: The exponential function is a one-to-one function, which means that each value of ${x}$ corresponds to a unique value of ${f(x)}$.
• The exponential function is increasing: The exponential function is an increasing function, which means that as ${x}$ increases, ${f(x)}$ also increases.

Q: How do I find the formula for an exponential function?

A: To find the formula for an exponential function, you need to know the values of the base ${b}$ and the exponent ${x}$. You can use the formula ${f(x) = a \cdot b^x}$ to find the formula, where ${a}$ is a constant.

Q: What is the difference between an exponential function and a power function?

A: An exponential function is a function of the form ${f(x) = a \cdot b^x}$, while a power function is a function of the form ${f(x) = a \cdot x^n}$. The main difference between the two is that an exponential function has a base ${b}$ that is raised to the power of ${x}$, while a power function has a variable ${x}$ that is raised to the power of ${n}$.

Q: Can I use an exponential function to model real-world phenomena?

A: Yes, exponential functions can be used to model real-world phenomena, such as population growth, chemical reactions, and financial investments. Exponential functions are particularly useful for modeling phenomena that exhibit rapid growth or decay.

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 function. The graph of an exponential function will be a curve that increases or decreases rapidly as ${x}$ increases.

Q: Can I use an exponential function to solve a system of equations?

A: Yes, exponential functions can be used to solve a system of equations. You can use the properties of exponential functions, such as the fact that they are one-to-one, to solve the system of equations.

Q: What are some common of exponential functions?

A: Exponential functions have many common applications, including:

• Population growth: Exponential functions can be used to model population growth, which is a rapid increase in the number of individuals in a population.
• Chemical reactions: Exponential functions can be used to model chemical reactions, which involve the rapid transformation of one substance into another.
• Financial investments: Exponential functions can be used to model financial investments, which involve the rapid growth or decay of an investment over time.

Conclusion

In this article, we answered some frequently asked questions about exponential functions and their properties. We discussed how to find the formula for an exponential function, the difference between an exponential function and a power function, and some common applications of exponential functions. We hope that this article has been helpful in understanding exponential functions and their properties.