Quick answer: A horizontal line has a slope of zero because its rise is 0 (0 ÷ run = 0). A vertical line has an undefined slope because its run is 0, and you cannot divide by zero. Horizontal lines have equations like y = 4; vertical lines have equations like x = −2.
"Zero slope or undefined slope?" is one of the most missed questions in Algebra 1, GCSE maths and Grade 9 math. The two ideas sound alike, but they describe lines that point in completely different directions. This short guide explains the difference with the slope formula, a memory trick, worked examples and a 10-question quiz. New to slope? Start with How to Find the Slope of a Line.
Key takeaways
- Horizontal line → y = c → slope 0.
- Vertical line → x = c → slope undefined.
- Memory trick: HOY VUX: Horizontal, 0 slope, Y = … ; Vertical, Undefined, X = …
- Avoid the phrase "no slope": teachers use it for both, so it is ambiguous.
Zero slope vs undefined slope at a glance
| Zero slope | Undefined slope | |
|---|---|---|
| Direction | Horizontal (flat) | Vertical (straight up and down) |
| Equation | y = c (e.g. y = 3) | x = c (e.g. x = −5) |
| Rise / run | 0 / non-zero = 0 | non-zero / 0 = undefined |
| Parallel to | x-axis | y-axis |
| Is it a function? | Yes (a constant function) | No (fails the vertical line test) |
| Intercepts | y-intercept c; no x-intercept unless c = 0 | x-intercept c; no y-intercept unless c = 0 |
| Real-life picture | Walking on flat ground | Climbing a straight wall |
Why a horizontal line has a slope of 0
Take two points on the horizontal line y = 5, for example (−2, 5) and (4, 5). Using the slope formula:
m = (5 − 5) / (4 − (−2)) = 0 / 6 = 0.
The y-value never changes, so the rise is always 0. Zero divided by any non-zero number is 0, so the slope is a real, perfectly good number: zero.
Why a vertical line has an undefined slope
Take two points on the vertical line x = 3, for example (3, −1) and (3, 6):
m = (6 − (−1)) / (3 − 3) = 7 / 0 → undefined.
Division by zero has no answer, so no number can describe the steepness of a vertical line. That is why we say the slope is undefined, not zero. The same idea explains why tan 90° is undefined: slope equals the tangent of the line's angle, and a vertical line makes a 90° angle with the x-axis. See our trigonometric identities guide for more on tan.
Is "undefined slope" the same as "no slope"?
Usually, but not always. Many US textbooks use "no slope" to mean undefined (vertical), while some teachers use it to mean zero (horizontal). Because the phrase is ambiguous, write "slope = 0" or "slope is undefined" on tests and homework so you never lose marks.
Horizontal and vertical lines are perpendicular
Normally two lines are perpendicular when their slopes multiply to −1. Horizontal and vertical lines are the exception: one slope is 0 and the other is undefined, so the rule cannot be applied, yet the lines still meet at 90°. Practise this in the parallel and perpendicular lines quiz.
Zero vs undefined slope quiz (10 questions)
Choose an answer, then tap to check.
1. What is the slope of y = −6? (A) −6 (B) 0 (C) 6 (D) undefined
B, 0. y = c is horizontal.
2. What is the slope of x = 9? (A) 9 (B) 0 (C) undefined (D) 1/9
C, undefined. x = c is vertical.
3. A line passes through (4, −2) and (4, 10). Its slope is: (A) 0 (B) 3 (C) undefined (D) 12
C, undefined. Run = 4 − 4 = 0. Its equation is x = 4.
4. A line passes through (−5, 3) and (7, 3). Its slope is: (A) 0 (B) undefined (C) 1/2 (D) −1
A, 0. Rise = 0. Its equation is y = 3.
5. Which equation has an undefined slope? (A) y = 2x (B) y = 0 (C) x = −1 (D) y = x − 1
C, x = −1.
6. Which line is perpendicular to y = 5 and passes through (2, 8)? (A) y = 8 (B) x = 2 (C) y = 2x (D) x = 8
B, x = 2. A line perpendicular to a horizontal line is vertical, and it must pass through x = 2.
7. The line parallel to the x-axis through (−3, 7) is: (A) x = −3 (B) y = −3 (C) y = 7 (D) x = 7
C, y = 7.
8. What is the slope of the x-axis? (A) 0 (B) 1 (C) undefined (D) −1
A, 0. The x-axis is the line y = 0.
9. What is the slope of the y-axis? (A) 0 (B) 1 (C) undefined (D) infinity
C, undefined. The y-axis is the line x = 0. "Infinity" is not a slope value.
10. Is x = 3 a function? (A) Yes (B) No
B, No. One x-value (3) is paired with every y-value, so it fails the vertical line test.
Scored 8 or more? Challenge yourself with the full 20-question slope quiz.
For teachers: paste the quiz into a document and share it as a handout with Word to PDF. To send several worksheets in one file, use Merge PDF.
Frequently asked questions
What is the slope of a horizontal line?
Zero. The y-value stays the same, so the rise is 0.
What is the slope of a vertical line?
Undefined. The x-value stays the same, so the run is 0 and the slope formula would divide by zero.
Is zero slope positive or negative?
Neither. Zero is neither positive nor negative, which matches a line that neither rises nor falls.
What does HOY VUX stand for?
Horizontal lines have 0 slope and equations Y = c; Vertical lines have Undefined slope and equations X = c.
Can I check a slope with a calculator?
Yes. Enter two points into the Adzita slope calculator. For angle work, use the PakMath angle converter.
Related slope lessons
About the author: Dr. Muhammad Amjad Khadim holds a PhD in Mathematics from COMSATS University Islamabad and founded PakMath, Adzita and PDFWisely.
