Which Equation Represents An Exponential Function That Passes Through The Point { (2,80)$}$?A. { F(x) = 4(x)^5$}$ B. { F(x) = 5(x)^4$}$ C. { F(x) = 4(5)^x$}$ D. { F(x) = 5(4)^x$}$
Which Equation Represents an Exponential Function that Passes Through the Point (2,80)?
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 x is the variable. In this article, we will explore which equation represents an exponential function that passes through the point (2,80).
The Point (2,80)
The point (2,80) represents a specific coordinate on the Cartesian plane, where the x-coordinate is 2 and the y-coordinate is 80. To determine which equation represents an exponential function that passes through this point, we need to substitute the values of x and y into each of the given equations and see which one satisfies the equation.
Equation A: f(x) = 4(x)^5
Let's start by substituting the values of x and y into Equation A: f(x) = 4(x)^5. We get:
f(2) = 4(2)^5 f(2) = 4(32) f(2) = 128
However, the y-coordinate of the point (2,80) is 80, not 128. Therefore, Equation A does not represent an exponential function that passes through the point (2,80).
Equation B: f(x) = 5(x)^4
Next, let's substitute the values of x and y into Equation B: f(x) = 5(x)^4. We get:
f(2) = 5(2)^4 f(2) = 5(16) f(2) = 80
The y-coordinate of the point (2,80) is indeed 80, which means that Equation B represents an exponential function that passes through the point (2,80).
Equation C: f(x) = 4(5)^x
Now, let's substitute the values of x and y into Equation C: f(x) = 4(5)^x. We get:
f(2) = 4(5)^2 f(2) = 4(25) f(2) = 100
However, the y-coordinate of the point (2,80) is 80, not 100. Therefore, Equation C does not represent an exponential function that passes through the point (2,80).
Equation D: f(x) = 5(4)^x
Finally, let's substitute the values of x and y into Equation D: f(x) = 5(4)^x. We get:
f(2) = 5(4)^2 f(2) = 5(16) f(2) = 80
The y-coordinate of the point (2,80) is indeed 80, which means that Equation D represents an exponential function that passes through the point (2,80).
Conclusion
In conclusion, the equation that represents an exponential function that passes through the point (2,80) is either Equation B: f(x) = 5(x)^4 or Equation D: f(x) = 5(4)^x. Both equations satisfy the equation and pass through the point (2,80).
Key Takeaways
- Exponential functions are a type of mathematical function that describes a relationship between two variables.
- The general form of an exponential function is f(x) = ab^x, where a and b are constants, and x is the variable.
- To determine which equation represents an exponential function that passes through a specific point, we need to substitute the values of x and y into each of the given equations and see which one satisfies the equation.
Final Answer
The final answer is Equation B: f(x) = 5(x)^4 or Equation D: f(x) = 5(4)^x.
Q&A: Exponential Functions and the Point (2,80)
Frequently Asked Questions
In this article, we will answer some of the most frequently asked questions about exponential functions and the point (2,80).
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 x is the variable.
Q: What is the point (2,80)?
A: The point (2,80) represents a specific coordinate on the Cartesian plane, where the x-coordinate is 2 and the y-coordinate is 80.
Q: How do I determine which equation represents an exponential function that passes through the point (2,80)?
A: To determine which equation represents an exponential function that passes through the point (2,80), you need to substitute the values of x and y into each of the given equations and see which one satisfies the equation.
Q: What are the possible equations that represent an exponential function that passes through the point (2,80)?
A: The possible equations that represent an exponential function that passes through the point (2,80) are Equation B: f(x) = 5(x)^4 and Equation D: f(x) = 5(4)^x.
Q: Why do Equations B and D represent an exponential function that passes through the point (2,80)?
A: Equations B and D represent an exponential function that passes through the point (2,80) because when we substitute the values of x and y into each of the equations, we get the correct y-coordinate of 80.
Q: What is the significance of the point (2,80) in the context of exponential functions?
A: The point (2,80) is significant in the context of exponential functions because it represents a specific coordinate on the Cartesian plane that can be used to determine which equation represents an exponential function.
Q: How can I apply the concept of exponential functions to real-world problems?
A: The concept of exponential functions can be applied to real-world problems such as population growth, financial investments, and chemical reactions. For example, if a population is growing exponentially, we can use an exponential function to model the growth and predict future population sizes.
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 and range of the function
- Not using the correct base and exponent
- Not simplifying the function before solving
- Not checking the solution for extraneous solutions
Conclusion
In conclusion, exponential functions are a powerful tool for modeling real-world problems. By understanding the concept of exponential functions and how to apply them to specific problems, we can gain a deeper understanding of the world around us.