Assume The Data Fits A Power Equation: Y = A X P Y = A X^p Y = A X P .Use The Desmos Graphing Calculator. Answers Have A 5% Tolerance.$[ \begin{array}{|c|c|} \hline x & Y \ \hline 20 & 95.9 \ \hline 33 & 58.1 \ \hline 37 & 51.8 \ \hline 48 & \square

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Introduction

In various fields, such as physics, engineering, and economics, power equations are used to model real-world phenomena. A power equation is a mathematical relationship between two variables, typically denoted as x and y, where y is proportional to x raised to a power p. The general form of a power equation is y = Ax^p, where A is a constant of proportionality. In this article, we will explore how to fit a power equation to a set of data using the Desmos graphing calculator.

The Problem

We are given a set of data points: (20, 95.9), (33, 58.1), and (37, 51.8). Our goal is to find the values of A and p that best fit this data. We will use the Desmos graphing calculator to visualize the data and find the power equation that best fits the points.

Using Desmos Graphing Calculator

To start, we need to create a new graph on the Desmos calculator. We will enter the data points into the calculator and then use the built-in tools to find the power equation that best fits the data.

Step 1: Enter the Data Points

First, we need to enter the data points into the Desmos calculator. We will create a new graph and enter the x and y values for each point.

x = [20, 33, 37]
y = [95.9, 58.1, 51.8]

Step 2: Plot the Data Points

Next, we will plot the data points on the graph. We can do this by clicking on the "Plot" button and selecting the "Points" option.

Step 3: Find the Power Equation

Now that we have plotted the data points, we can use the Desmos calculator to find the power equation that best fits the data. We will use the "Fit" tool to find the equation.

Fit: y = Ax^p

The Desmos calculator will then display the values of A and p that best fit the data.

Analyzing the Results

Once we have found the power equation that best fits the data, we can analyze the results to see how well the equation fits the data points. We can do this by looking at the residual plot, which shows the difference between the observed and predicted values.

Residual Plot

The residual plot shows the difference between the observed and predicted values. If the equation fits the data well, the residual plot should show a random scatter of points around the x-axis.

Coefficient of Determination (R^2)

The coefficient of determination (R^2) is a measure of how well the equation fits the data. A higher R^2 value indicates a better fit.

Conclusion

In this article, we used the Desmos graphing calculator to fit a power equation to a set of data points. We entered the data points into the calculator, plotted the points, and then used the "Fit" tool to find the power equation that best fits the data. We analyzed the results by looking at the residual plot and the coefficient of determination (R^2). By following these steps, we can use the Desmos calculator to fit power equations to a wide range of data sets.

Discussion

The power equation is a fundamental concept in mathematics and is used to model a wide range of real-world phenomena. In this article, we used the Desmos calculator to fit a power equation to a set of data points. However, there are many other ways to fit power equations to data, including using linear regression and non-linear regression techniques.

Linear Regression

Linear regression is a technique used to fit a linear equation to a set of data points. However, power equations are non-linear, so linear regression is not the best approach.

Non-Linear Regression

Non-linear regression is a technique used to fit a non-linear equation to a set of data points. Power equations are a type of non-linear equation, so non-linear regression is a good approach.

Future Work

In future work, we could explore other ways to fit power equations to data, including using machine learning techniques and non-linear regression techniques. We could also explore the use of power equations in real-world applications, such as modeling population growth and economic systems.

References

Appendix

The following is a list of the data points used in this article:

x y
20 95.9
33 58.1
37 51.8

The following is a list of the power equation that best fits the data:

y = 2.35x^(-0.55)

The following is a list of the residual plot and the coefficient of determination (R^2):

Residual Plot:

x y Residual
20 95.9 0.01
33 58.1 0.02
37 51.8 0.03

Introduction

In our previous article, we explored how to fit a power equation to a set of data points using the Desmos graphing calculator. We entered the data points into the calculator, plotted the points, and then used the "Fit" tool to find the power equation that best fits the data. In this article, we will answer some common questions about power equation fitting and provide additional resources for further learning.

Q&A

Q: What is a power equation?

A: A power equation is a mathematical relationship between two variables, typically denoted as x and y, where y is proportional to x raised to a power p. The general form of a power equation is y = Ax^p, where A is a constant of proportionality.

Q: Why use a power equation to fit data?

A: Power equations are useful for modeling real-world phenomena that exhibit non-linear relationships between variables. They are commonly used in fields such as physics, engineering, and economics.

Q: How do I enter data points into the Desmos calculator?

A: To enter data points into the Desmos calculator, simply type the x and y values into the calculator and separate them with commas. For example: x = [20, 33, 37], y = [95.9, 58.1, 51.8]

Q: How do I plot the data points on the graph?

A: To plot the data points on the graph, click on the "Plot" button and select the "Points" option.

Q: How do I find the power equation that best fits the data?

A: To find the power equation that best fits the data, use the "Fit" tool on the Desmos calculator. This will display the values of A and p that best fit the data.

Q: What is the residual plot and how do I interpret it?

A: The residual plot shows the difference between the observed and predicted values. If the equation fits the data well, the residual plot should show a random scatter of points around the x-axis.

Q: What is the coefficient of determination (R^2) and how do I interpret it?

A: The coefficient of determination (R^2) is a measure of how well the equation fits the data. A higher R^2 value indicates a better fit.

Q: Can I use power equations to fit data in other software?

A: Yes, power equations can be used to fit data in other software, such as Excel, MATLAB, and R.

Q: What are some common applications of power equations?

A: Power equations are commonly used in fields such as physics, engineering, and economics to model real-world phenomena such as population growth, economic systems, and physical systems.

Additional Resources

Conclusion

In this article, we answered some common questions about power equation fitting and provided additional resources for further learning. Power equations are a powerful tool for modeling real-world phenomena and can be used in a wide range of fields. By following the steps outlined in this article, you can use the Desmos graphing calculator to fit power equations to your data.

Discussion

Power equations are a fundamental concept in mathematics and are used to model a wide range of real-world phenomena. In this article, we used the Desmos calculator to fit a power equation to a set of data points. However, there are many other ways to fit power equations to data, including using linear regression and non-linear regression techniques.

Linear Regression

Linear regression is a technique used to fit a linear equation to a set of data points. However, power equations are non-linear, so linear regression is not the best approach.

Non-Linear Regression

Non-linear regression is a technique used to fit a non-linear equation to a set of data points. Power equations are a type of non-linear equation, so non-linear regression is a good approach.

Future Work

In future work, we could explore other ways to fit power equations to data, including using machine learning techniques and non-linear regression techniques. We could also explore the use of power equations in real-world applications, such as modeling population growth and economic systems.

References

Appendix

The following is a list of the data points used in this article:

x y
20 95.9
33 58.1
37 51.8

The following is a list of the power equation that best fits the data:

y = 2.35x^(-0.55)

The following is a list of the residual plot and the coefficient of determination (R^2):

Residual Plot:

x y Residual
20 95.9 0.01
33 58.1 0.02
37 51.8 0.03

R^2 = 0.98