Drawing the Line of Best Fit

Oct 13, 2025 by magicalceylontours.com 114 views

Understanding the Problem

In this problem, we are given a table of data showing the total earnings ( $y$ ) of a food server who works ( $x$ ) hours. Our task is to create a scatterplot by hand using the provided data, draw the line of best fit, and write its equation.

Step 1: Organize the Data

Before we can create a scatterplot, we need to organize the data into a table. The table should have two columns: one for the number of hours worked ( $x$ ) and one for the total earnings ( $y$ ).

$x$ (hours)	$y$ (earnings)
1	10
2	20
3	30
4	40
5	50
6	60
7	70
8	80
9	90
10	100

Step 2: Create a Scatterplot

A scatterplot is a type of graph that shows the relationship between two variables. In this case, we want to show the relationship between the number of hours worked ( $x$ ) and the total earnings ( $y$ ). To create a scatterplot, we need to plot the points on a coordinate plane.

Plotting the Points

To plot the points, we need to find the coordinates of each data point. The coordinates are given by the values of $x$ and $y$ in the table.

For example, the first data point has $x = 1$ and $y = 10$ , so we plot the point (1, 10) on the coordinate plane.

We repeat this process for each data point, plotting the points (2, 20), (3, 30), (4, 40), (5, 50), (6, 60), (7, 70), (8, 80), (9, 90), and (10, 100).

Drawing the Line of Best Fit

The line of best fit is a line that best represents the relationship between the two variables. To draw the line of best fit, we need to find the equation of the line that best fits the data.

Finding the Equation of the Line

To find the equation of the line, we need to find the slope and the y-intercept. The slope is the ratio of the change in $y$ to the change in $x$ . The y-intercept is the value of $y$ when $x = 0$ .

Calculating the Slope

To calculate the slope, we need to find the change in $y$ and the change in $x$ between two points on the line.

Let's use the points (1, 10) and (2, 20) to calculate the slope.

The change in $y$ is $20 - 10 = 10$ .

The change in $x$ is $2 - 1 = 1$ .

The slope is the ratio of the change in $y$ to the change in $x$ , which is $10/1 = 10$ .

Calculating the Y-Intercept

To calculate the y-intercept, we need to find the value of $y$ when $x = 0$ .

Let's use the point (0, 0) to calculate the y-intercept.

The y-intercept is the value of $y$ when $x = 0$ , which is 0.

Writing the Equation of the Line

Now that we have the slope and the y-intercept, we can write the equation of the line.

The equation of the line is $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

In this case, the slope is 10 and the y-intercept is 0, so the equation of the line is $y = 10x$ .

Conclusion

In this problem, we created a scatterplot by hand using the provided data, drew the line of best fit, and wrote its equation. We found the equation of the line to be $y = 10x$ , which represents the relationship between the number of hours worked and the total earnings.

Tips and Variations

To create a scatterplot, you can use a graphing calculator or a computer program such as Excel or Python.
To draw the line of best fit, you can use a ruler or a straightedge to draw a line that best represents the relationship between the two variables.
To find the equation of the line, you can use the slope-intercept form of a linear equation, which is $y = mx + b$ .
To calculate the slope, you can use the formula $m = (y_2 - y_1) / (x_2 - x_1)$ , where $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the line.
To calculate the y-intercept, you can use the formula $b = y_1 - mx_1$ , where $(x_1, y_1)$ is a point on the line and $m$ is the slope.

Real-World Applications

Scatterplots are used in many real-world applications, such as finance, economics, and science.
The line of best fit is used to make predictions and forecasts in many fields, such as business, medicine, and engineering.
The equation of the line is used to model the relationship between two variables in many fields, such as physics, chemistry, and biology.

Common Mistakes

Not organizing the data into a table before creating a scatterplot.
Not plotting the points on a coordinate plane before drawing the line of best fit.
Not using the slope-intercept form of a linear equation to find the equation of the line.
Not calculating the slope and y-intercept correctly.
Not using a ruler or straightedge to draw the line of best fit.

Conclusion

Q: What is a scatterplot?

A: A scatterplot is a type of graph that shows the relationship between two variables. It is used to visualize the data and identify patterns or trends.

Q: Why is it important to create a scatterplot?

A: Creating a scatterplot is important because it helps to identify the relationship between two variables. It can also be used to make predictions and forecasts.

Q: How do I create a scatterplot?

A: To create a scatterplot, you need to organize the data into a table, plot the points on a coordinate plane, and draw the line of best fit.

Q: What is the line of best fit?

A: The line of best fit is a line that best represents the relationship between the two variables. It is used to make predictions and forecasts.

Q: How do I draw the line of best fit?

A: To draw the line of best fit, you need to find the equation of the line. This can be done by using the slope-intercept form of a linear equation, which is y = mx + b.

Q: What is the slope-intercept form of a linear equation?

A: The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept.

Q: How do I calculate the slope?

A: To calculate the slope, you need to find the change in y and the change in x between two points on the line. The slope is then calculated as the ratio of the change in y to the change in x.

Q: How do I calculate the y-intercept?

A: To calculate the y-intercept, you need to find the value of y when x = 0. This can be done by using the equation y = mx + b and substituting x = 0.

Q: What are some common mistakes to avoid when creating a scatterplot and line of best fit?

A: Some common mistakes to avoid when creating a scatterplot and line of best fit include not organizing the data into a table, not plotting the points on a coordinate plane, not using the slope-intercept form of a linear equation, and not calculating the slope and y-intercept correctly.

Q: What are some real-world applications of scatterplots and lines of best fit?

A: Scatterplots and lines of best fit are used in many real-world applications, including finance, economics, science, business, medicine, and engineering.

Q: How can I use a scatterplot and line of best fit to make predictions and forecasts?

A: To use a scatterplot and line of best fit to make predictions and forecasts, you need to identify the relationship between the two variables and use the equation of the line to make predictions.

Q: What are some tips for creating a scatterplot and line of best fit?

A: Some tips for creating a scatterplot and line of best fit include using a ruler or straightedge to draw the line of best fit, using a graphing calculator or computer program to create the scatterplot, and checking your work to ensure that the equation of the line is correct.

Q: How can I practice creating scatterplots and lines of best fit?

A: You can practice creating scatterplots and lines of best fit by using online resources, such as graphing calculators or computer programs, or by working with a teacher or tutor to create scatterplots and lines of best fit.