Which Equation Represents An Exponential Function That Passes Through The Point { (2,36)$}$?A. { F(x)=4(3)^x$}$B. { F(x)=4(x)^3$}$C. { F(x)=6(3)^x$}$D. { F(x)=6(x)^3$}$
Which Equation Represents an Exponential Function that Passes Through the Point (2,36)?
Understanding Exponential Functions
Exponential functions are a type of mathematical function that describes a relationship between two variables, where the dependent variable (y) is a constant raised to the power of the independent variable (x). The general form of an exponential function is f(x) = ab^x, where a and b are constants, and b is the base of the exponent. Exponential functions are commonly used to model population growth, chemical reactions, and financial investments.
Key Characteristics of Exponential Functions
Exponential functions have several key characteristics that distinguish them from other types of functions. These characteristics include:
- Rapid growth: Exponential functions grow rapidly, especially when the base (b) is greater than 1.
- Constant ratio: The ratio of consecutive terms in an exponential function is constant.
- Asymptotic behavior: Exponential functions have asymptotic behavior, meaning that they approach a horizontal asymptote as x approaches infinity.
The Point (2,36)
The point (2,36) is a specific point on the graph of the exponential function. To determine which equation represents the exponential function that passes through this point, we need to substitute x = 2 and y = 36 into each of the given equations and see which one satisfies the equation.
Analyzing the Options
Let's analyze each of the given options:
A. f(x) = 4(3)^x
To determine if this equation represents the exponential function that passes through the point (2,36), we need to substitute x = 2 and y = 36 into the equation:
f(2) = 4(3)^2 f(2) = 4(9) f(2) = 36
This equation satisfies the point (2,36), so it is a possible solution.
B. f(x) = 4(x)^3
To determine if this equation represents the exponential function that passes through the point (2,36), we need to substitute x = 2 and y = 36 into the equation:
f(2) = 4(2)^3 f(2) = 4(8) f(2) = 32
This equation does not satisfy the point (2,36), so it is not a possible solution.
C. f(x) = 6(3)^x
To determine if this equation represents the exponential function that passes through the point (2,36), we need to substitute x = 2 and y = 36 into the equation:
f(2) = 6(3)^2 f(2) = 6(9) f(2) = 54
This equation does not satisfy the point (2,36), so it is not a possible solution.
D. f(x) = 6(x)^3
To determine if this equation represents the exponential function that passes through the point (2,36), we need to substitute x = 2 and y = 36 into the equation:
f(2) = 6(2)^3 f(2) = 6(8) f(2) = 48
This equation does not satisfy the point (2,36), so it is not a possible solution.
Conclusion
Based on the analysis of each option, we can conclude that the equation f(x) = 4(3)^x represents the exponential function that passes through the point (2,36).
Key Takeaways
- Exponential functions are a type of mathematical function that describes a relationship between two variables.
- Exponential functions have several key characteristics, including rapid growth, constant ratio, and asymptotic behavior.
- To determine which equation represents the exponential function that passes through a specific point, we need to substitute the x and y values into each of the given equations and see which one satisfies the equation.
Additional Resources
For more information on exponential functions and how to work with them, check out the following resources:
- Khan Academy: Exponential Functions
- Mathway: Exponential Functions
- Wolfram Alpha: Exponential Functions
Practice Problems
Try working through the following practice problems to test your understanding of exponential functions:
- Find the equation of the exponential function that passes through the points (1,2) and (2,4).
- Determine the value of the exponential function f(x) = 2(3)^x when x = 3.
- Graph the exponential function f(x) = 3(2)^x and identify its key characteristics.
Exponential Function Q&A
Frequently Asked Questions About Exponential Functions
Exponential functions are a fundamental concept in mathematics, and they have numerous applications in various fields, including science, engineering, and finance. However, many students and professionals struggle to understand and work with exponential functions. In this article, we will answer some of the most frequently asked questions about exponential functions.
Q: What is an exponential function?
A: An exponential function is a type of mathematical function that describes a relationship between two variables, where the dependent variable (y) is a constant raised to the power of the independent variable (x). The general form of an exponential function is f(x) = ab^x, where a and b are constants, and b is the base of the exponent.
Q: What are the key characteristics of exponential functions?
A: Exponential functions have several key characteristics, including:
- Rapid growth: Exponential functions grow rapidly, especially when the base (b) is greater than 1.
- Constant ratio: The ratio of consecutive terms in an exponential function is constant.
- Asymptotic behavior: Exponential functions have asymptotic behavior, meaning that they approach a horizontal asymptote as x approaches infinity.
Q: How do I determine the equation of an exponential function that passes through a specific point?
A: To determine the equation of an exponential function that passes through a specific point, you need to substitute the x and y values into the general form of the exponential function, f(x) = ab^x, and solve for a and b.
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) = ab^x, where a and b are constants, and b is the base of the exponent. A power function, on the other hand, is a function of the form f(x) = ax^n, where a and n are constants, and n is the exponent. While both types of functions involve exponentiation, the key difference lies in the base of the exponent.
Q: How do I graph an exponential function?
A: To graph an exponential function, you can use a graphing calculator or a computer program. Alternatively, you can use a table of values to plot the function. To create a table of values, substitute different values of x into the function and calculate the corresponding values of y.
Q: What are some real-world applications of exponential functions?
A: Exponential functions have numerous real-world applications, including:
- Population growth: Exponential functions are used to model population growth, where the population grows at a constant rate.
- Chemical reactions: Exponential functions are used to model chemical reactions, where the concentration of a substance grows or decays at a constant rate.
- Financial investments: Exponential functions are used to model financial investments, where the value of an investment grows or decays at a constant rate.
Q: How do I solve exponential equations?
A: To solve exponential equations, you need to isolate the exponential term and then use logarithms to solve for the variable. For example, to solve the equation 2^x = 8, you can take the logarithm of both sides and then solve for x.
Q: What is the difference between an exponential function and a logarithmic 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 exponent. A logarithmic function, on the other hand, is a function of the form f(x) = log_b(x), where b is the base of the logarithm. While both types of functions involve exponentiation, the key difference lies in the direction of the function.
Conclusion
Exponential functions are a fundamental concept in mathematics, and they have numerous applications in various fields. By understanding the key characteristics of exponential functions and how to work with them, you can solve a wide range of problems and make informed decisions in your personal and professional life.
Additional Resources
For more information on exponential functions and how to work with them, check out the following resources:
- Khan Academy: Exponential Functions
- Mathway: Exponential Functions
- Wolfram Alpha: Exponential Functions
Practice Problems
Try working through the following practice problems to test your understanding of exponential functions:
- Find the equation of the exponential function that passes through the points (1,2) and (2,4).
- Determine the value of the exponential function f(x) = 2(3)^x when x = 3.
- Graph the exponential function f(x) = 3(2)^x and identify its key characteristics.