Plot Two Points To Graph The Function $ Y = 30\left(\frac{5}{6}\right)^x $.
Introduction
Graphing functions is an essential skill in mathematics, particularly in algebra and calculus. It allows us to visualize the behavior of functions, identify patterns, and make predictions about their behavior. In this article, we will focus on plotting the function $ y = 30\left(\frac{5}{6}\right)^x $, which is an exponential function. We will explore the properties of this function, choose two points to plot, and discuss the implications of the graph.
Understanding Exponential Functions
Exponential functions are a type of function that can be written in the form $ y = ab^x $, where $ a $ and $ b $ are constants, and $ x $ is the variable. The base $ b $ determines the rate at which the function grows or decays. In the case of the function $ y = 30\left(\frac{5}{6}\right)^x $, the base is $ \frac{5}{6} $, which is a fraction between 0 and 1. This means that the function will decay exponentially as $ x $ increases.
Choosing Two Points to Plot
To plot the function $ y = 30\left(\frac{5}{6}\right)^x $, we need to choose two points that satisfy the equation. Let's choose $ x = 0 $ and $ x = 1 $ as our two points.
For $ x = 0 $, we have:
So, the first point is $ (0, 30) $.
For $ x = 1 $, we have:
So, the second point is $ (1, 25) $.
Plotting the Points
Now that we have chosen our two points, we can plot them on a coordinate plane. The x-axis represents the value of $ x $, and the y-axis represents the value of $ y $. We can plot the points $ (0, 30) $ and $ (1, 25) $ on the coordinate plane.
Interpreting the Graph
The graph of the function $ y = 30\left(\frac{5}{6}\right)^x $ is a decreasing exponential curve. As $ x $ increases, the value of $ y $ decreases. This is because the base $ \frac{5}{6} $ is a fraction between 0 and 1, which causes the function to decay exponentially.
Conclusion
In this article, we have explored the properties of the exponential function $ y = 30\left(\frac{5}{6}\right)^x $ and chosen two points to plot. We have discussed the implications of the graph and how it can be used to model real-world phenomena. By understanding how to plot functions, we can gain a deeper appreciation for the behavior of mathematical functions and their applications in various fields.
Properties of Exponential Functions
Exponential functions have several important properties that make them useful for modeling real-world phenomena. Some of these properties include:
- Exponential growth: Exponential functions can grow or decay exponentially, depending on the base.
- Asymptotic behavior: Exponential functions can have asymptotic behavior, meaning that they approach a horizontal asymptote as $ x $ increases.
- Domain and range: Exponential functions have a domain of all real numbers and a range of all positive real numbers.
Real-World Applications of Exponential Functions
Exponential functions have numerous real-world applications in fields such as:
- Finance: Exponential functions can be used to model the growth of investments and the decay of debt.
- Biology: Exponential functions can be used to model the growth of populations and the decay of radioactive substances.
- Physics: Exponential functions can be used to model the decay of radioactive substances and the growth of waves.
Conclusion
Introduction
In our previous article, we explored the properties of the exponential function $ y = 30\left(\frac{5}{6}\right)^x $ and chose two points to plot. We also discussed the implications of the graph and how it can be used to model real-world phenomena. In this article, we will answer some frequently asked questions about plotting functions and exponential growth.
Q: What is the difference between exponential growth and exponential decay?
A: Exponential growth occurs when the base of the exponential function is greater than 1, causing the function to increase exponentially as $ x $ increases. Exponential decay occurs when the base of the exponential function is between 0 and 1, causing the function to decrease exponentially as $ x $ increases.
Q: How do I choose two points to plot a function?
A: To choose two points to plot a function, you need to select two values of $ x $ and calculate the corresponding values of $ y $. You can use any two values of $ x $ that you like, but it's often helpful to choose values that are easy to work with, such as $ x = 0 $ and $ x = 1 $.
Q: What is the significance of the base in an exponential function?
A: The base of an exponential function determines the rate at which the function grows or decays. If the base is greater than 1, the function will grow exponentially. If the base is between 0 and 1, the function will decay exponentially.
Q: Can I use any base in an exponential function?
A: No, you cannot use any base in an exponential function. The base must be a positive real number. If the base is 0 or 1, the function will not be exponential.
Q: How do I plot an exponential function on a coordinate plane?
A: To plot an exponential function on a coordinate plane, you need to choose two points that satisfy the equation and plot them on the plane. You can use a ruler or a graphing calculator to help you plot the points.
Q: What are some real-world applications of exponential functions?
A: Exponential functions have numerous real-world applications in fields such as finance, biology, and physics. They can be used to model the growth of investments, the decay of radioactive substances, and the growth of populations.
Q: Can I use exponential functions to model real-world phenomena that are not exponential?
A: While exponential functions can be used to model many real-world phenomena, they may not be the best choice for phenomena that are not exponential. In such cases, you may need to use a different type of function, such as a linear or quadratic function.
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. This is because the base of an exponential function is always positive, and the function will never be zero or negative.
Conclusion
In conclusion, plotting functions and exponential growth are essential skills in mathematics that allow us to visualize the behavior of functions and identify patterns. By understanding how to plot functions and exponential growth, we can gain a deeper appreciation for the behavior of mathematical functions and their applications in various fields. We hope that this Q&A article has been helpful in answering some of your questions about plotting functions and exponential growth.