Graph The Line.$y = -\frac{1}{3}x + 2$

Feb 27, 2025 by ADMIN 39 views

**Graphing the Line: A Comprehensive Guide**

Introduction

Graphing a line is a fundamental concept in mathematics, and it's essential to understand how to do it correctly. In this article, we will focus on graphing the line represented by the equation $y = -\frac{1}{3}x + 2$ . We will break down the process into manageable steps, and by the end of this article, you will be able to graph the line with ease.

Understanding the Equation

The equation $y = -\frac{1}{3}x + 2$ is a linear equation in the slope-intercept form, which is $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept. In this equation, the slope is $-\frac{1}{3}$ , and the y-intercept is $2$ .

What is the Slope?

The slope of a line is a measure of how steep it is. It's calculated as the ratio of the vertical change (rise) to the horizontal change (run). In this case, the slope is $-\frac{1}{3}$ , which means that for every one unit we move to the right, we move down by three units.

What is the Y-Intercept?

The y-intercept is the point where the line intersects the y-axis. In this case, the y-intercept is $2$ , which means that the line intersects the y-axis at the point $(0, 2)$ .

Graphing the Line

To graph the line, we need to find two points on the line. We can do this by substituting different values of $x$ into the equation and solving for $y$ .

Finding the First Point

Let's substitute $x = 0$ into the equation:

$y = -\frac{1}{3}(0) + 2$ $y = 2$

So, the first point on the line is $(0, 2)$ .

Finding the Second Point

Let's substitute $x = 3$ into the equation:

$y = -\frac{1}{3}(3) + 2$ $y = -1 + 2$ $y = 1$

So, the second point on the line is $(3, 1)$ .

Plotting the Points

Now that we have two points on the line, we can plot them on a coordinate plane.

Plotting the First Point

To plot the first point $(0, 2)$ , we need to move two units up from the origin (0, 0) and then move one unit to the right.

Plotting the Second Point

To plot the second point $(3, 1)$ , we need to move three units to the right from the origin (0, 0) and then move one unit down.

Drawing the Line

Now that we have plotted the two points, we can draw the line by connecting the points with a straight line.

The Final Graph

Here is the final graph of the line $y = -\frac{1}{3}x + 2$ :

Conclusion

Graphing a line is a straightforward process that involves finding two points on the line and plotting them on a coordinate plane. By following the steps outlined in this article, you should be able to graph the line $y = -\frac{1}{3}x + 2$ with ease. Remember to always substitute different values of $x$ into the equation to find the corresponding values of $y$ , and then plot the points on a coordinate plane.

Tips and Variations

To graph a line with a positive slope, simply change the sign of the slope in the equation.
To graph a line with a negative slope, keep the sign of the slope in the equation.
To graph a line with a zero slope, set the slope to zero in the equation.
To graph a line with a vertical slope, set the slope to infinity in the equation.

Common Mistakes

Failing to substitute different values of $x$ into the equation to find the corresponding values of $y$ .
Plotting the points incorrectly on the coordinate plane.
Drawing the line incorrectly by not connecting the points with a straight line.

Real-World Applications

Graphing lines has many real-world applications, including:

Modeling population growth
Predicting stock prices
Calculating the trajectory of a projectile
Designing electrical circuits

Conclusion

Q&A: Frequently Asked Questions

Q: What is the slope of the line $y = -\frac{1}{3}x + 2$ ? A: The slope of the line $y = -\frac{1}{3}x + 2$ is $-\frac{1}{3}$ .

Q: What is the y-intercept of the line $y = -\frac{1}{3}x + 2$ ? A: The y-intercept of the line $y = -\frac{1}{3}x + 2$ is $2$ .

Q: How do I graph the line $y = -\frac{1}{3}x + 2$ ? A: To graph the line $y = -\frac{1}{3}x + 2$ , you need to find two points on the line and plot them on a coordinate plane. You can do this by substituting different values of $x$ into the equation and solving for $y$ .

Q: What are the two points on the line $y = -\frac{1}{3}x + 2$ ? A: The two points on the line $y = -\frac{1}{3}x + 2$ are $(0, 2)$ and $(3, 1)$ .

Q: How do I plot the points on a coordinate plane? A: To plot the points on a coordinate plane, you need to move the corresponding number of units up or down from the origin (0, 0) and then move the corresponding number of units to the right or left.

Q: How do I draw the line on a coordinate plane? A: To draw the line on a coordinate plane, you need to connect the two points with a straight line.

Q: What are some common mistakes when graphing a line? A: Some common mistakes when graphing a line include failing to substitute different values of $x$ into the equation to find the corresponding values of $y$ , plotting the points incorrectly on the coordinate plane, and drawing the line incorrectly by not connecting the points with a straight line.

Q: What are some real-world applications of graphing lines? A: Some real-world applications of graphing lines include modeling population growth, predicting stock prices, calculating the trajectory of a projectile, and designing electrical circuits.

Q: How do I graph a line with a positive slope? A: To graph a line with a positive slope, simply change the sign of the slope in the equation.

Q: How do I graph a line with a negative slope? A: To graph a line with a negative slope, keep the sign of the slope in the equation.

Q: How do I graph a line with a zero slope? A: To graph a line with a zero slope, set the slope to zero in the equation.

Q: How do I graph a line with a vertical slope? A: To graph a line with a vertical slope, set the slope to infinity in the equation.

Conclusion

Graphing a line is a fundamental concept in mathematics that has many real-world applications. By following the steps outlined in this article, you should be able to graph the line $y = -\frac{1}{3}x + 2$ with ease. Remember to always substitute different values of $x$ into the equation to find the corresponding values of $y$ , and then plot the points on a coordinate plane.