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A vertical line runs straight up and down, parallel to the y-axis. On a coordinate plane, its x-coordinate stays fixed while its y-coordinate changes, so its equation has the form x = a.
What does a vertical line look like?
It rises and falls without moving left or right. On a coordinate plane, it is parallel to the y-axis. For example, the points (2, −3) and (2, 4) have the same x-coordinate; connecting them gives a vertical segment on the line x = 2.
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A quick way to recognize one is to compare two points: if their x-coordinates match, the segment between them is vertical.
What is the equation of a vertical line?
The equation is x = a, where a is a constant. Every point on the line has coordinates of the form (a, y): x remains a, and y can vary.
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- x = 2 is two units to the right of the y-axis.
- x = −3 is three units to the left of the y-axis.
- x = 0 is the y-axis itself.
So yes, x = 3 is a vertical line: all of its points have x-coordinate 3.
Does a vertical line have a slope?
Its slope is undefined. Using the slope formula, m = (y₂ − y₁)/(x₂ − x₁), the denominator is zero for two distinct points on a vertical line because their x-coordinates are equal. Division by zero is undefined.
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It is not accurate to say the slope is infinity. As the Ohio Department of Education explains in its 2018 Ohio’s Model Curriculum with Instructional Supports: Mathematics, calling it infinite suggests infinity is a number; applying the slope formula instead leads to division by zero. Ohio Department of Education, 2018.
Does a vertical line represent a function?
No, not as a function of x over the whole line. Each point has the same x-value, but that x-value corresponds to many y-values. A function y = f(x) can assign only one output y to each input x, so a vertical line fails the vertical-line test. OpenStax describes a vertical line as indicating a constant input, or x-value. OpenStax, College Algebra, section 4.1.
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Where does a vertical line cross the axes?
For x = a, the line crosses the x-axis at (a, 0). It has a y-intercept only when a = 0; in that case the line is the y-axis.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Vertical and horizontal lines compared
| Line | Orientation | Constant coordinate | Equation | Slope |
|---|---|---|---|---|
| Vertical | Parallel to the y-axis | x | x = a | Undefined |
| Horizontal | Parallel to the x-axis | y | y = b | 0 |
OpenStax states that the slope of a vertical line x = a is undefined. OpenStax, Elementary Algebra, section 4.4.
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