Understanding and Graphing Perpendicular Lines to the Y-Axis in Cartesian Coordinates

In this article, we will delve into the concept of lines that are perpendicular to the y-axis and how to determine their equations. We will also explore graphing such lines and find the equation of a line that passes through a specific point. Let's start with a fundamental concept: any line that is perpendicular to the y-axis is a vertical line, and its general equation in Cartesian coordinates is x c, where c is a constant.

Identifying a Line Perpendicular to the Y-Axis

Given a horizontal line such as y -2, we know that this line is parallel to the x-axis. Any line that is perpendicular to a horizontal line will therefore be a vertical line parallel to the y-axis. The general form of the equation for such a line is:

x c, where c is a constant.

Determining the Equation for a Specific Point

Now, let's consider a specific problem: Find the equation of a line that is perpendicular to the y-axis and passes through the point (4, -2).

Step 1: Identify the Vertical Line

Any line perpendicular to y -2 is a vertical line. To find the equation of this vertical line, we need to determine the value of c in the equation x c. Since this vertical line must pass through the point (4, -2), we set x 4.

Step 2: Verify the Line Passes Through the Point (4, -2)

Substitute the x-coordinate of the point (4, -2) into the vertical line equation x c:

x c → c 4

Therefore, the equation of the vertical line that passes through (4, -2) is:

x 4

Graphical Representation

Graphically, the line y -2 is a horizontal line, and the line x 4 is a vertical line crossing the y-axis at x 4. These two lines are perpendicular to each other, as shown below:

Note that the vertical line x 4 is parallel to the y-axis and intersects the x-axis at the point (4, 0).

Additional Considerations and Examples

Let's consider a few more examples to further reinforce the concept:

Example 1: Line Perpendicular to Y-Axis and Passing Through Point (3, 5)

For a line that is perpendicular to the y-axis and passes through the point (3, 5), the equation is:

x 3

Example 2: Line Parallel to X-Axis and Passing Through Point (4, -2)

Another approach to solving this problem is to consider that a line parallel to the x-axis and passing through (4, -2) would have the form y k. Since this line passes through (4, -2), we set k -2. The equation of this line is:

y -2

It is important to note that this line is horizontal and not vertical.

Conclusion

Understanding and graphing lines perpendicular to the y-axis is a fundamental concept in Cartesian coordinates. By recognizing that such lines are vertical and their equations take the form x c, we can determine the specific equation of a line passing through a given point. This knowledge is crucial in various mathematical and practical applications, including graphing, geometry, and calculus.