Directions: Write An Exponential Function For A Graph That Passes Through The Points (1, 12) And (3, 192).

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Introduction

Exponential functions are a fundamental concept in mathematics, used to model real-world phenomena that exhibit rapid growth or decay. In this article, we will explore how to write an exponential function for a graph that passes through specific points. We will use the points (1, 12) and (3, 192) as examples to demonstrate the process.

Understanding Exponential Functions

An exponential function is a mathematical function of the form f(x) = ab^x, where a and b are constants, and x is the variable. The base b determines the rate of growth or decay of the function. If b > 1, the function grows rapidly, while if 0 < b < 1, the function decays rapidly.

Step 1: Identify the General Form of the Exponential Function

The general form of an exponential function is f(x) = ab^x. To write an exponential function for a graph that passes through the points (1, 12) and (3, 192), we need to find the values of a and b.

Step 2: Use the Given Points to Create a System of Equations

We can use the given points to create a system of equations. Let's substitute the points (1, 12) and (3, 192) into the general form of the exponential function:

f(1) = 12 => a * b^1 = 12 f(3) = 192 => a * b^3 = 192

Step 3: Simplify the System of Equations

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

(a * b^3) / (a * b^1) = 192 / 12 b^2 = 16

Step 4: Solve for b

Taking the square root of both sides of the equation, we get:

b = ±4

Since the base b must be positive, we take the positive square root:

b = 4

Step 5: Solve for a

Now that we have the value of b, we can substitute it back into one of the original equations to solve for a. Let's use the first equation:

a * b^1 = 12 a * 4^1 = 12 a * 4 = 12 a = 3

Step 6: Write the Exponential Function

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

f(x) = 3 * 4^x

Conclusion

In this article, we demonstrated how to write an exponential function for a graph that passes through the points (1, 12) and (3, 192). We used the general form of the exponential function, created a system of equations using the given points, simplified the system of equations, solved for b, solved for a, and finally wrote the exponential function. This process can be applied to any set of points to find the corresponding exponential function.

Example Use Cases

Exponential functions have many real-world applications, including:

  • Modeling population growth or decay
  • Describing chemical reactions
  • Analyzing financial data
  • Predicting weather patterns

Tips and Tricks

  • When working with exponential functions, it's essential to remember that the base b determines the rate of growth or decay.
  • Use the given points to create a system of equations, and then simplify the system to solve for the values of a and b.
  • Make sure to check your work by plugging the values of a and b back into the original equations.

Further Reading

For more information on exponential functions, including their properties and applications, see the following resources:

  • Khan Academy: Exponential Functions
  • Mathway: Exponential Functions
  • Wolfram MathWorld: Exponential Function

References

  • Larson, R., & Hostetler, R. P. (2013). Calculus: Early Transcendentals. Cengage Learning.
  • Stewart, J. (2015). Calculus: Early Transcendentals. Cengage Learning.
  • Anton, H. (2017). Calculus: Early Transcendentals. John Wiley & Sons.
    Directions: Write an Exponential Function for a Graph =====================================================

Q&A: Exponential Functions

Q: What is an exponential function?

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

Q: What is the base b in an exponential function?

A: The base b determines the rate of growth or decay of the function. If b > 1, the function grows rapidly, while if 0 < b < 1, the function decays rapidly.

Q: How do I find the values of a and b in an exponential function?

A: To find the values of a and b, you can use the given points to create a system of equations. Let's say you have the points (1, 12) and (3, 192). You can substitute these points into the general form of the exponential function to create a system of equations.

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

A: The general form of an exponential function is f(x) = ab^x.

Q: How do I simplify the system of equations?

A: To simplify the system of equations, you can divide the second equation by the first equation. This will help you eliminate the variable a and solve for b.

Q: What if I get a negative value for b?

A: If you get a negative value for b, it means that the function is decaying rapidly. However, in most cases, the base b is positive, so you can take the positive square root of the equation to solve for b.

Q: Can I use any value for a and b?

A: No, you cannot use any value for a and b. The values of a and b must be chosen such that the function passes through the given points.

Q: How do I check my work?

A: To check your work, you can plug the values of a and b back into the original equations to make sure that they satisfy the system of equations.

Q: What are some real-world applications of exponential functions?

A: Exponential functions have many real-world applications, including modeling population growth or decay, describing chemical reactions, analyzing financial data, and predicting weather patterns.

Q: What are some tips and tricks for working with exponential functions?

A: Some tips and tricks for working with exponential functions include:

  • Remembering that the base b determines the rate of growth or decay
  • Using the given points to create a system of equations
  • Simplifying the system of equations to solve for b
  • Checking your work by plugging the values of a and b back into the original equations

Q: Where can I find more information on exponential functions?

A: You can find more information on exponential functions in the following resources:

  • Khan Academy: Exponential Functions
  • Mathway: Exponential Functions
  • Wolfram MathWorld: 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 checking your work
  • Not using the correct values for a and b
  • Not simplifying the system of equations
  • Not remembering that the base b determines the rate of growth or decay

Conclusion

In this article, we have provided a comprehensive guide to exponential functions, including their definition, properties, and applications. We have also answered some common questions about exponential functions and provided some tips and tricks for working with them. Whether you are a student or a professional, this guide will help you understand and work with exponential functions with confidence.