Graphing a Line with a Given Point and Slope

Oct 13, 2025 by miodeya.com 39 views

Introduction

Graphing a line is a fundamental concept in mathematics, and it is essential to understand how to graph a line that contains a given point and has a specific slope. In this article, we will discuss how to graph a line that contains the point ${$ (2,0)$}$ and has a slope of ${$ -5$}$.

What is a Line?

A line is a set of points that extend infinitely in two directions. It can be thought of as a straight path that connects two or more points. In mathematics, a line is often represented by an equation in the form of ${$ y = mx + b$}$, where ${$ m$}$ is the slope of the line and ${$ b$}$ is the y-intercept.

What is a Slope?

The slope of a line is a measure of how steep it is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. The slope can be positive, negative, or zero, depending on the direction of the line.

What is a Point?

A point is a location on a coordinate plane that is represented by an ordered pair of numbers, ${$ (x, y)$}$. In this case, the point ${$ (2,0)$}$ represents a location on the coordinate plane where the x-coordinate is 2 and the y-coordinate is 0.

Graphing a Line with a Given Point and Slope

To graph a line that contains the point ${$ (2,0)$}$ and has a slope of ${$ -5$}$, we can use the point-slope form of a line, which is ${$ y - y_1 = m(x - x_1)$}$. In this equation, ${$ m$}$ is the slope of the line, and ${$ (x_1, y_1)$}$ is the given point.

Step 1: Write the Equation of the Line

Using the point-slope form of a line, we can write the equation of the line as ${$ y - 0 = -5(x - 2)$}$.

Step 2: Simplify the Equation

Simplifying the equation, we get ${$ y = -5x + 10$}$.

Step 3: Graph the Line

To graph the line, we can use the equation ${$ y = -5x + 10$}$ and plot the point ${$ (2,0)$}$ on the coordinate plane. We can then draw a line through the point that has a slope of ${$ -5$}$.

Graphing the Line

To graph the line, we can use the following steps:

Plot the point ${$ (2,0)$}$ on the coordinate plane.
Draw a line through the point that has a slope of ${$ -5$}$.
Use a ruler or a straightedge to draw the line.

Example

Let's graph a line that contains the point ${$ (2,0)$}$ and has a slope of ${$ -5$}$.

Step 1: Write the Equation of the Line

Using the point-slope form of a line, we can write the equation of the line as ${$ y - 0 = -5(x - 2)$}$.

Step 2: Simplify the Equation

Simplifying the equation, we get ${$ y = -5x + 10$}$.

Step 3: Graph the Line

To graph the line, we can use the equation ${$ y = -5x + 10$}$ and plot the point ${$ (2,0)$}$ on the coordinate plane. We can then draw a line through the point that has a slope of ${$ -5$}$.

Conclusion

Graphing a line that contains a given point and has a specific slope is a fundamental concept in mathematics. By using the point-slope form of a line and simplifying the equation, we can graph a line that meets the given conditions. In this article, we discussed how to graph a line that contains the point ${$ (2,0)$}$ and has a slope of ${$ -5$}$. We also provided an example of how to graph a line using the point-slope form of a line.

Key Takeaways

A line is a set of points that extend infinitely in two directions.
The slope of a line is a measure of how steep it is.
A point is a location on a coordinate plane that is represented by an ordered pair of numbers.
The point-slope form of a line is ${$ y - y_1 = m(x - x_1)$}$.
To graph a line, we can use the point-slope form of a line and simplify the equation.

References

[1] "Graphing Lines" by Math Open Reference
[2] "Slope of a Line" by Math Is Fun
[3] "Point-Slope Form of a Line" by Purplemath

Additional Resources

[1] "Graphing Lines" by Khan Academy
[2] "Slope of a Line" by IXL
[3] "Point-Slope Form of a Line" by Mathway
Graphing a Line with a Given Point and Slope: Q&A =====================================================

Introduction

Graphing a line is a fundamental concept in mathematics, and it is essential to understand how to graph a line that contains a given point and has a specific slope. In this article, we will answer some frequently asked questions about graphing a line with a given point and slope.

Q: What is the point-slope form of a line?

A: The point-slope form of a line is ${$ y - y_1 = m(x - x_1)$}$, where ${$ m$}$ is the slope of the line and ${$ (x_1, y_1)$}$ is the given point.

Q: How do I find the equation of a line that contains a given point and has a specific slope?

A: To find the equation of a line that contains a given point and has a specific slope, you can use the point-slope form of a line. Simply substitute the given point and slope into the equation and simplify.

Q: What is the slope of a line?

A: The slope of a line is a measure of how steep it is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.

Q: How do I graph a line that contains a given point and has a specific slope?

A: To graph a line that contains a given point and has a specific slope, you can use the point-slope form of a line and simplify the equation. Then, plot the point on the coordinate plane and draw a line through the point that has the specified slope.

Q: What is the difference between the slope-intercept form and the point-slope form of a line?

A: The slope-intercept form of a line is ${$ y = mx + b$}$, where ${$ m$}$ is the slope of the line and ${$ b$}$ is the y-intercept. The point-slope form of a line is ${$ y - y_1 = m(x - x_1)$}$, where ${$ m$}$ is the slope of the line and ${$ (x_1, y_1)$}$ is the given point.

Q: How do I find the y-intercept of a line?

A: To find the y-intercept of a line, you can use the slope-intercept form of a line, ${$ y = mx + b$}$. The y-intercept is the value of ${$ b$}$ in the equation.

Q: What is the x-intercept of a line?

A: The x-intercept of a line is the point where the line crosses the x-axis. To find the x-intercept, you can set ${$ y = 0$}$ in the equation of the line and solve for ${$ x$}$.

Q: How do I find the equation of a line that passes through two points?

A: To find the equation of a line that passes through two points, you can use the slope formula to find the slope of the line, and then use the point-slope form of a line to find the equation.

Q: What is the equation of a horizontal line?

A: The equation of a horizontal line is ${$ y = c$}$, where ${$ c$}$ is a constant.

Q: What is the equation of a vertical line?

A: The equation of a vertical line is ${$ x = c$}$, where ${$ c$}$ is a constant.

Conclusion

Graphing a line with a given point and slope is a fundamental concept in mathematics. By understanding the point-slope form of a line and how to use it to find the equation of a line, you can graph lines with ease. In this article, we answered some frequently asked questions about graphing a line with a given point and slope.

Key Takeaways

The point-slope form of a line is ${$ y - y_1 = m(x - x_1)$}$.
To find the equation of a line that contains a given point and has a specific slope, use the point-slope form of a line.
The slope of a line is a measure of how steep it is.
To graph a line that contains a given point and has a specific slope, use the point-slope form of a line and simplify the equation.

References

[1] "Graphing Lines" by Math Open Reference
[2] "Slope of a Line" by Math Is Fun
[3] "Point-Slope Form of a Line" by Purplemath

Additional Resources

[1] "Graphing Lines" by Khan Academy
[2] "Slope of a Line" by IXL
[3] "Point-Slope Form of a Line" by Mathway