Alan Compared The Slope Of The Function { Y = 3x + 2 $}$ To The Slope Of The Linear Function That Fits The Values In This Table.$[ \begin{tabular}{|c|c|} \hline x & Y \ \hline -3 & -2 \ \hline -1 & 2 \ \hline 1 & 6 \ \hline 3 & 10

Introduction

In mathematics, the slope of a linear function is a crucial concept that helps us understand the rate of change of the function with respect to its input variable. When we are given a table of values and asked to find the slope of the linear function that fits these values, we need to use the concept of linear regression. In this article, we will compare the slope of a given function to the slope of a linear function fitted to a table of values.

The Given Function

The given function is $y = 3x + 2$. This is a linear function with a slope of 3 and a y-intercept of 2. The slope of this function represents the rate of change of the function with respect to the input variable x.

The Table of Values

The table of values is given as:

x	y
-3	-2
-1	2
1	6
3	10

Finding the Slope of the Linear Function Fitted to the Table of Values

To find the slope of the linear function fitted to the table of values, we need to use the concept of linear regression. Linear regression is a statistical method that helps us find the best-fitting linear function to a set of data points. The slope of the linear function fitted to the table of values can be found using the following formula:

$$m = \frac{\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n}(x_i - \bar{x})^2}$$

where $m$ is the slope of the linear function, $x_i$ and $y_i$ are the data points, $\bar{x}$ and $\bar{y}$ are the mean of the data points, and $n$ is the number of data points.

Calculating the Slope of the Linear Function Fitted to the Table of Values

Using the formula above, we can calculate the slope of the linear function fitted to the table of values as follows:

x	y
-3	-2
-1	2
1	6
3	10

First, we need to calculate the mean of the data points:

$$\bar{x} = \frac{-3 + (-1) + 1 + 3}{4} = 0$$

$$\bar{y} = \frac{-2 + 2 + 6 + 10}{4} = 4$$

Next, we need to calculate the sum of the products of the deviations from the mean:

$$\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y}) = (-3 - 0)(-2 - 4) + (-1 - 0)(2 - 4) + (1 - 0)(6 - 4) + (3 - 0)(10 - 4)$$

$$= (-3)(-6) + (-1)(-2) + (1)(2) + (3)(6)$$

$$= 18 + 2 + 2 + 18$$

$$= 40$$

Finally, we need to calculate the sum of the squared deviations from the mean:

$$\sum_{i=1}^{n}(x_i - \bar{x})^2 = (-3 - 0)^2 + (-1 - 0)^2 + (1 - 0)^2 + (3 - 0)^2$$

$$= 9 + 1 + 1 + 9$$

$$= 20$$

Now, we can calculate the slope of the linear function fitted to the table of values:

$$m = \frac{\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n}(x_i - \bar{x})^2} = \frac{40}{20} = 2$$

Comparing the Slope of the Given Function to the Slope of the Linear Function Fitted to the Table of Values

The slope of the given function is 3, while the slope of the linear function fitted to the table of values is 2. This means that the rate of change of the given function is different from the rate of change of the linear function fitted to the table of values.

Conclusion

Q: What is the slope of a linear function?

A: The slope of a linear function is a measure of how much the function changes for a one-unit change in the input variable. It is calculated as the ratio of the change in the output variable to the change in the input variable.

Q: How do you calculate the slope of a linear function?

A: To calculate the slope of a linear function, you need to use the formula:

$$m = \frac{\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n}(x_i - \bar{x})^2}$$

where $m$ is the slope of the linear function, $x_i$ and $y_i$ are the data points, $\bar{x}$ and $\bar{y}$ are the mean of the data points, and $n$ is the number of data points.

Q: What is the difference between the slope of a given function and the slope of a linear function fitted to a table of values?

A: The slope of a given function is a fixed value that represents the rate of change of the function with respect to the input variable. The slope of a linear function fitted to a table of values is a calculated value that represents the best-fitting linear function to the data points.

Q: When would you use the slope of a given function versus the slope of a linear function fitted to a table of values?

A: You would use the slope of a given function when you are working with a specific mathematical model or equation, and you need to understand the rate of change of the function with respect to the input variable. You would use the slope of a linear function fitted to a table of values when you are working with a set of data points and you need to find the best-fitting linear function to the data.

Q: Can the slope of a linear function fitted to a table of values be different from the slope of the given function?

A: Yes, the slope of a linear function fitted to a table of values can be different from the slope of the given function. This is because the linear function fitted to the table of values is a calculated value that represents the best-fitting linear function to the data points, whereas the slope of the given function is a fixed value that represents the rate of change of the function with respect to the input variable.

Q: How do you determine if the slope of a linear function fitted to a table of values is a good fit to the data?

A: You can determine if the slope of a linear function fitted to a table of values is a good fit to the data by using statistical measures such as the coefficient of determination (R-squared) or the mean squared error (MSE). These measures can help you evaluate the goodness of fit of the linear function to the data.

Q: Can the slope of a linear function fitted to a table of values be negative?

A: Yes, the slope of a linear function fitted to a table of values can be negative. This means that the linear function is decreasing as the input variable increases.

Q: Can the slope of a linear function fitted to a table of values be zero?

A: Yes, the slope of a linear function fitted to a table of values can be zero. This means that the linear function is a horizontal line, and the input variable does not affect the output variable.

Conclusion

In this article, we answered some frequently asked questions about comparing the slope of a given function to the slope of a linear function fitted to a table of values. We hope that this article has provided you with a better understanding of the concepts and has helped you to answer some of the questions that you may have had.