For Each Of The Following, Find The Formula For An Exponential Function That Passes Through The Two Points Given.a. \[$(-2, \frac{1}{8})\$\] And \[$(2, 32)\$\]$\[f(x) =\\]

Introduction

Exponential functions are a fundamental concept in mathematics, used to model real-world phenomena that exhibit rapid growth or decay. Given two points on the graph of an exponential function, we can find the formula for the function that passes through those points. In this article, we will explore how to find the formula for an exponential function that passes through the points ${(-2, \frac{1}{8})}$ and ${(2, 32)}$.

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 expressed as a power of a prime number, such as 2, 3, or 5. The constant ${a}$ is the initial value of the function, and it determines the vertical shift of the graph.

The General Form of an Exponential Function

The general form of an exponential function is given by:

${f(x) = ab^x}$

where ${a}$ and ${b}$ are constants, and ${b}$ is the base of the exponential function.

Finding the Formula for an Exponential Function

To find the formula for an exponential function that passes through two points, we can use the following steps:

1. Write the equation of the exponential function: Write the equation of the exponential function in the form ${f(x) = ab^x}$.
2. Substitute the given points: Substitute the given points into the equation to obtain two equations with two unknowns.
3. Solve the system of equations: Solve the system of equations to find the values of ${a}$ and ${b}$.
4. Write the final formula: Write the final formula for the exponential function using the values of ${a}$ and ${b}$ found in step 3.

Finding the Formula for the Given Points

Let's apply the steps outlined above to find the formula for the exponential function that passes through the points ${(-2, \frac{1}{8})}$ and ${(2, 32)}$.

Step 1: Write the equation of the exponential function

The equation of the exponential function is:

${f(x) = ab^x}$

Step 2: Substitute the given points

Substituting the given points into the equation, we obtain:

${\frac{1}{8} = ab^{-2}}$ ${32 = ab^2}$

Step 3: Solve the system of equations

We can solve the system of equations by dividing the second equation by the first equation:

${\frac{32}{\frac{1}{8}} = \frac{ab^2}{ab^{-2}}}$ ${256 = b^4}$

Taking the fourth root of both sides, we obtain:

${b = \pm 4}$

Since the base of an exponential function must be positive, we take the positive value of ${b}$:

${b = 4}$

Now that we have found the value of ${b}$, we can substitute it into one of the original equations to find the value of ${a}$. Let's substitute it into the first equation:

${\frac{1}{8} = a(4)^{-2}}$ ${\frac{1}{8} = \frac{a}{16}}$

Multiplying both sides by 16, we obtain:

${a = \frac{1}{8} \times 16}$ ${a = 2}$

Step 4: Write the final formula

The final formula for the exponential function is:

${f(x) = 2(4)^x}$

Conclusion

In this article, we have shown how to find the formula for an exponential function that passes through two points. We have applied the steps outlined above to find the formula for the exponential function that passes through the points ${(-2, \frac{1}{8})}$ and ${(2, 32)}$. The final formula for the exponential function is:

${f(x) = 2(4)^x}$

This formula can be used to model real-world phenomena that exhibit rapid growth or decay, such as population growth, chemical reactions, or financial investments.

Future Work

In future work, we can explore other applications of exponential functions, such as modeling population growth, chemical reactions, or financial investments. We can also investigate the properties of exponential functions, such as their domain and range, and their behavior as the input variable approaches positive or negative infinity.

References

• [1] "Exponential Functions" by Math Open Reference
• [2] "Exponential Functions" by Khan Academy
• [3] "Exponential Functions" by Wolfram MathWorld

Glossary

• Exponential function: A function of the form ${f(x) = ab^x}$, where ${a}$ and ${b}$ are constants, and ${b}$ is the base of the exponential function.
• Base: The constant ${b}$ in the exponential function ${f(x) = ab^x}$.
• Initial value: The constant ${a}$ in the exponential function ${f(x) = ab^x}$.
• Vertical shift: The shift of the graph of the exponential function up or down by a certain amount.

Appendix

The following is a list of common exponential functions and their properties:

• Exponential function: ${f(x) = 2^x}$
  • Base: 2
  • Initial value: 1
  • Vertical shift: 0
• Exponential function: ${f(x) = 3^x}$
  • Base: 3
  • Initial value: 1
  • Vertical shift: 0
• Exponential function: ${f(x) = 4^x}$
  • Base: 4
  • Initial value: 1
  • Vertical shift: 0

Note: This is not an exhaustive list, and there are many other exponential functions with different bases and initial values.

Introduction

In our previous article, we explored how to find the formula for an exponential function that passes through two points. We applied the steps outlined above to find the formula for the exponential function that passes through the points ${(-2, \frac{1}{8})}$ and ${(2, 32)}$. In this article, we will answer some common questions related to exponential functions and provide additional examples to help solidify your understanding.

Q&A

Q: What is the general form of an exponential function?

A: The general form of an exponential function is given by:

${f(x) = ab^x}$

where ${a}$ and ${b}$ are constants, and ${b}$ is the base of the exponential function.

Q: How do I find the formula for an exponential function that passes through two points?

A: To find the formula for an exponential function that passes through two points, you can use the following steps:

1. Write the equation of the exponential function: Write the equation of the exponential function in the form ${f(x) = ab^x}$.
2. Substitute the given points: Substitute the given points into the equation to obtain two equations with two unknowns.
3. Solve the system of equations: Solve the system of equations to find the values of ${a}$ and ${b}$.
4. Write the final formula: Write the final formula for the exponential function using the values of ${a}$ and ${b}$ found in step 3.

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 determines the rate of growth or decay of the function.

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 determines the vertical shift of the graph.

Q: How do I determine the domain and range of an exponential function?

A: The domain of an exponential function is all real numbers, and the range is all positive real numbers, unless the base is negative, in which case the range is all negative real numbers.

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

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

Examples

Example 1: Finding the formula for an exponential function

Find the formula for the exponential function that passes through the points ${(0, 1)}$ and ${(2, 4)}$.

Solution

Using the steps outlined above, we can find the formula for the exponential function:

1. Write the equation of the exponential function: ${f(x) = ab^x}$
2. Substitute the given points: ${1 = a}$ and ${4 = ab^2}$
3. Solve the system of equations: ${a = 1}$ and ${b^2 = 4}$, so ${b = 2}$
4. Write the final formula: ${f(x) = 1(2)^x}$

Example 2: Finding the domain and range of an exponential function

Find the domain and range of the exponential function ${f(x) = 2^x}$.

Solution

The domain of the function is all real numbers, and the range is all positive real numbers.

Example 3: Using exponential functions to model real-world phenomena

Use the exponential function ${f(x) = 2^x}$ to model the growth of a population over time.

Solution

Assuming the population starts at 100 and grows at a rate of 2% per year, we can use the exponential function to model the growth of the population over time:

${f(x) = 100(2)^x}$

where ${x}$ is the number of years.

Conclusion

In this article, we have answered some common questions related to exponential functions and provided additional examples to help solidify your understanding. We have also explored how to find the formula for an exponential function that passes through two points, and how to determine the domain and range of an exponential function. We hope this article has been helpful in your understanding of exponential functions.

References

• [1] "Exponential Functions" by Math Open Reference
• [2] "Exponential Functions" by Khan Academy
• [3] "Exponential Functions" by Wolfram MathWorld

Glossary

• Exponential function: A function of the form ${f(x) = ab^x}$, where ${a}$ and ${b}$ are constants, and ${b}$ is the base of the exponential function.
• Base: The constant ${b}$ in the exponential function ${f(x) = ab^x}$.
• Initial value: The constant ${a}$ in the exponential function ${f(x) = ab^x}$.
• Vertical shift: The shift of the graph of the exponential function up or down by a certain amount.
• Domain: The set of all input values for which the function is defined.
• Range: The set of all output values for which the function is defined.

Appendix

The following is a list of common exponential functions and their properties:

• Exponential function: ${f(x) = 2^x}$
  • Base: 2
  • Initial value: 1
  • Vertical shift: 0
  • Domain: All real numbers
  • Range: All positive real numbers
• Exponential function: ${f(x) = 3^x}$
  • Base: 3
  • Initial value: 1
  • Vertical shift: 0
  • Domain: All real numbers
  • Range: All positive real numbers
• Exponential function: ${f(x) = 4^x}$
  • Base: 4
  • Initial value: 1
  • Vertical shift: 0
  • Domain: All real numbers
  • Range: All positive real numbers