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 quantities, where one quantity (the output or dependent variable) grows or decays at a constant rate with respect to the other quantity (the input or independent variable). The general form of an exponential function is f(x) = ab^x, where a is the initial value, b is the growth or decay factor, and x is the input variable.
Key Characteristics of Exponential Functions
Exponential functions have several key characteristics that distinguish them from other types of functions. These characteristics include:
- Exponential growth or decay: Exponential functions can exhibit either exponential growth or decay, depending on the value of the growth or decay factor (b).
- Constant rate of change: Exponential functions have a constant rate of change, which means that the output changes at a constant rate with respect to the input.
- Non-linear: Exponential functions are non-linear, meaning that the output does not change at a constant rate with respect to the input.
The Problem: Finding the Equation of an Exponential Function that Passes Through a Given Point
In this problem, we are given a point (2, 80) and asked to find the equation of an exponential function that passes through this point. To solve this problem, we need to use the general form of an exponential function and substitute the given point into the equation.
Substituting the Given Point into the Equation
Let's substitute the given point (2, 80) into the general form of an exponential function:
f(x) = ab^x
We know that f(2) = 80, so we can substitute this value into the equation:
80 = ab^2
Evaluating the Options
Now, let's evaluate the options to see which one satisfies the equation:
A. f(x) = 4(x)^5
This option is not an exponential function, as it has a variable base (x) and a constant exponent (5).
B. f(x) = 5(x)^4
This option is also not an exponential function, as it has a variable base (x) and a constant exponent (4).
C. f(x) = 4(5)^x
This option is an exponential function, as it has a constant base (5) and a variable exponent (x). Let's substitute x = 2 into this equation to see if it satisfies the given point:
f(2) = 4(5)^2 = 4(25) = 100
This option does not satisfy the given point (2, 80).
D. f(x) = 5(4)^x
This option is also an exponential function, as it has a constant base (4) and a variable exponent (x). Let's substitute x = 2 into this equation to see if it satisfies the given point:
f(2) = 5(4)^2 = 5(16) = 80
This option satisfies the given point (2, 80).
Conclusion
Based on the analysis above, the equation that represents an exponential function that passes through the point (2, 80) is:
f(x) = 5(4)^x
This equation satisfies the given point and has the characteristics of an exponential function, including exponential growth or decay and a constant rate of change.
Q&A: Exponential Functions and the Equation that Passes Through the Point (2, 80)
Frequently Asked Questions
In this article, we will answer some frequently asked questions about exponential functions and the equation that passes through 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 quantities, where one quantity (the output or dependent variable) grows or decays at a constant rate with respect to the other quantity (the input or independent variable).
Q: What are the key characteristics of exponential functions?
A: Exponential functions have several key characteristics, including:
- Exponential growth or decay: Exponential functions can exhibit either exponential growth or decay, depending on the value of the growth or decay factor (b).
- Constant rate of change: Exponential functions have a constant rate of change, which means that the output changes at a constant rate with respect to the input.
- Non-linear: Exponential functions are non-linear, meaning that the output does not change at a constant rate with respect to the input.
Q: How do I determine if an equation is an exponential function?
A: To determine if an equation is an exponential function, look for the following characteristics:
- Constant base: The base of the equation should be a constant value, such as 2 or 5.
- Variable exponent: The exponent of the equation should be a variable, such as x or 2x.
- Exponential growth or decay: The equation should exhibit exponential growth or decay, depending on the value of the growth or decay factor (b).
Q: How do I find the equation of an exponential function that passes through a given point?
A: To find the equation of an exponential function that passes through a given point, use the general form of an exponential function:
f(x) = ab^x
Substitute the given point into the equation and solve for the unknown values (a and b).
Q: What is the equation of an exponential function that passes through the point (2, 80)?
A: The equation of an exponential function that passes through the point (2, 80) is:
f(x) = 5(4)^x
This equation satisfies the given point and has the characteristics of an exponential function, including exponential growth or decay and a constant rate of change.
Q: How do I evaluate the options to find the correct equation?
A: To evaluate the options, substitute the given point into each equation and check if it satisfies the point. If the equation satisfies the point, it is a possible solution. If the equation does not satisfy the point, it is not a possible solution.
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:
- Confusing exponential functions with linear functions: Exponential functions have a constant rate of change, while linear functions have a constant slope.
- Using the wrong base or exponent: Make sure to use the correct base and exponent when working with exponential functions.
- Not checking the equation against the given point: Always check the equation against the given point to ensure that it satisfies the point.
Conclusion
In this article, we have answered some frequently asked questions about exponential functions and the equation that passes through the point (2, 80). We have also provided some tips and common mistakes to avoid when working with exponential functions. By following these tips and avoiding common mistakes, you can become more confident and proficient in working with exponential functions.