Properties of the Vertical Lines
The properties of vertical lines are the mirror image of horizontal line properties in many ways, but with one critical difference: the slope of a vertical line is undefined, not zero. Every point on a vertical line shares the same x-coordinate, and the equation takes the form \(\color{blue}{x = c}\). Understanding these properties is key to working with linear equations and coordinate geometry in Algebra 1.
What Is a Vertical Line?
A vertical line runs straight up and down, parallel to the y-axis. Every point on a vertical line has the same x-coordinate. Because the x-value never changes while the y-value can be anything, vertical lines have a unique set of properties that set them apart from all other lines.
Key Properties of a Vertical Line
1. Slope is Undefined
Slope is rise divided by run. On a vertical line, the run is always 0 no matter how far you travel vertically. Division by zero is undefined, so:
\(\color{blue}{m = \frac{\text{ rise }}{\text{ run }} = \frac{\text{ rise }}{0} = \text{ undefined }}\)
Example: between \(\color{blue}{(3, 1)}\) and \(\color{blue}{(3, 5)}\), slope \(\color{blue}{= \frac{(5 – 1)}{(3 – 3)} = \frac{4}{0} = \text{ undefined }}\).
2. Equation: \(\color{blue}{x = c}\)
The equation of a vertical line is \(\color{blue}{x = c}\), where \(\color{blue}{c}\) is the common x-coordinate of all its points. There is no \(\color{blue}{y}\) term because \(\color{blue}{y}\) can be any value while \(\color{blue}{x}\) stays fixed.
Example: the line through \(\color{blue}{(5, 0)}\), \(\color{blue}{(5, 3)}\), and \(\color{blue}{(5, -7)}\) has equation \(\color{blue}{x = 5}\).
3. X-intercept and y-intercept
A vertical line always has an x-intercept at \(\color{blue}{(c, 0)}\).
A vertical line has a y-intercept only if \(\color{blue}{c = 0}\) (i.e., the line is the y-axis, \(\color{blue}{x = 0}\)). Otherwise, it never crosses the y-axis.
4. Not a Function
A vertical line fails the vertical line test, so it does not represent a function. For any given x-value, there are infinitely many y-values.
5. Parallel to the y-axis
Every vertical line is parallel to the y-axis and perpendicular to every horizontal line.
Step-by-Step Summary
- Identify the common x-coordinate shared by all points on the line.
- Write the equation as \(\color{blue}{x = c}\).
- State that the slope is undefined.
- State the x-intercept as \(\color{blue}{(c, 0)}\).
- Note that there is no y-intercept (unless \(\color{blue}{c = 0}\)).
Watch: Horizontal and Vertical Lines (Video Lesson)
Khan Academy covers both horizontal and vertical lines together, making it easy to compare their properties:
Properties of Vertical Lines – Worked Examples
Example 1: Write the equation of the vertical line through \(\color{blue}{(-4, 2)}\).
All points share \(\color{blue}{x = -4}\). Equation: \(\color{blue}{x = -4}\). Slope: undefined. X-intercept: \(\color{blue}{(-4, 0)}\). No y-intercept.
Example 2: Find the slope between \(\color{blue}{(7, -3)}\) and \(\color{blue}{(7, 9)}\).
\(\color{blue}{m = \frac{(9 – (-3))}{(7 – 7)} = \frac{12}{0}}\) ⇒ undefined. The points lie on the vertical line \(\color{blue}{x = 7}\).
Example 3: Does the line \(\color{blue}{x = 6}\) have a y-intercept?
No. The line \(\color{blue}{x = 6}\) never reaches \(\color{blue}{x = 0}\), so it does not cross the y-axis. Its x-intercept is \(\color{blue}{(6, 0)}\).
Example 4: What is the equation of the y-axis?
The y-axis is the vertical line where \(\color{blue}{x = 0}\) for every point. Its equation is \(\color{blue}{x = 0}\).
More Practice: Horizontal & Vertical Lines Tutorial (Video)
West Explains Best walks through identifying and writing equations for both types of lines with multiple examples:
Exercises for Properties of Vertical Lines
- Write the equation of the vertical line through \(\color{blue}{(2, 9)}\).
- What is the slope of the line \(\color{blue}{x = -5}\)?
- Find the x-intercept of \(\color{blue}{x = 11}\).
- Does the line \(\color{blue}{x = 3}\) have a y-intercept? Explain.
- Two points are \(\color{blue}{(8, -2)}\) and \(\color{blue}{(8, 15)}\). Write the equation and classify the slope.
Answers
- \(\color{blue}{x = 2}\)
- Undefined
- \(\color{blue}{(11, 0)}\)
- No. The line \(\color{blue}{x = 3}\) never reaches \(\color{blue}{x = 0}\), so it does not intersect the y-axis.
- Equation: \(\color{blue}{x = 8}\); slope: undefined.
Want More Practice?
We haven’t published a worksheet built specifically for Properties of the Vertical Lines just yet. In the meantime, the free worksheets below cover closely related skills and concepts. If you’d like extra practice, download any that look helpful, complete the problems, and check your work — they’re a great way to reinforce what you learned on this page and strengthen the foundations this topic builds on:
Frequently Asked Questions
Why is the slope of a vertical line undefined rather than zero?
Slope is rise/run. On a vertical line, the run (horizontal change) is zero. Dividing by zero has no defined result in mathematics, so the slope is undefined, not zero. A slope of zero belongs to horizontal lines.
Is a vertical line a function?
No. A function must have exactly one output for each input. A vertical line assigns infinitely many y-values to one x-value, violating the definition of a function.
Are all vertical lines parallel to each other?
Yes. All vertical lines have undefined slope and run in the same direction, so they are all parallel to one another (and to the y-axis).
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