---
title: "Axis of Symmetry of a Parabola"
url: https://www.learningloftinstitute.com/glossary/axis-of-symmetry
source: Learning Loft Institute
language: en
updated: 2026-08-21
section: "Glossary › Axis of Symmetry of a Parabola"
---
# Axis of Symmetry of a Parabola
> **Answer.** The axis of symmetry of a parabola y = ax² + bx + c is the vertical line x = −b/2a. It passes through the turning point and sits exactly halfway between the two roots, so once you have it you have the x-coordinate of the vertex, and substituting back gives the y-coordinate.
## Key facts
| | |
| --- | --- |
| Definition | The axis of symmetry of a parabola is the vertical line through its turning point, which reflects one half of the curve exactly onto the other. |
| Where students meet it | Grade 9 or 10 when quadratic graphs are first sketched, and again on Edexcel 1MA1 Higher and Cambridge IGCSE 0580 Extended questions asking for a turning point. |
| Also known as | Line of symmetry |
## Where −b/2a comes from
The quadratic formula gives the two roots as (−b + √(b² − 4ac))/2a and (−b − √(b² − 4ac))/2a. They differ only in the sign in front of the square root, which means they sit the same distance either side of −b/2a. Average them and the square root part cancels, leaving −b/2a exactly.
So the formula is not a separate thing to memorise alongside the quadratic formula. It is the middle of the quadratic formula, with the ± part removed. That also explains why it still works when a parabola never touches the x-axis: the axis of symmetry does not care whether the roots are real, because the cancelling happens regardless.
## Three routes to the same line
Which route is quickest depends on the form the quadratic arrives in, and a sensible habit is to use whichever is already available rather than always reaching for the formula. All of these give x = 3 for y = x² − 6x + 5.
The only real hazard is the sign. In y = −2x² + 8x − 3 the value of a is −2 and b is 8, so −b/2a is −8 ÷ −4 = 2, and the axis of symmetry is x = 2 with a maximum point at (2, 5). Two negatives are in play at once, which is where careless working goes wrong, so it is worth writing a and b down separately before substituting them.
- From the coefficients: x = −b/2a = 6/2 = 3
- From the roots: the curve cuts the x-axis at 1 and 5, and (1 + 5)/2 = 3
- From completed square form: y = (x − 3)² − 4, and the bracket is zero at x = 3
- From any two points at the same height: y = 5 at both x = 0 and x = 6, and (0 + 6)/2 = 3
## Two things it is not
It is a line, not a number. The answer to "find the axis of symmetry" is x = 3, and writing just 3 loses the mark on many mark schemes because 3 on its own could be a y-value or a root. The equation of a vertical line always looks like x = something.
It is also not the turning point, though it goes through it. The axis of symmetry is x = 3; the turning point is the single point (3, −4). Questions ask for one or the other and the difference matters, so read whether the answer wanted is a line or a coordinate pair.
## Questions
### Can a parabola have a horizontal axis of symmetry?
Yes, but not one of the form y = ax² + bx + c. A curve like x = y² − 4y opens sideways and its axis of symmetry is horizontal, y = 2. School quadratics are written with y as the subject, so their axis is always vertical, which is why the formula produces an x-value.
### What happens if the quadratic has no real roots?
Nothing changes. y = x² + 2x + 5 never crosses the x-axis, but −b/2a still gives x = −1 and the curve is still symmetrical about that line, with minimum point (−1, 4). The roots are a convenient way to find the axis when they exist; they are not what the axis depends on.
### How do I use the axis of symmetry to sketch the curve?
Find it first, then everything else is cheap. Substitute it back for the turning point, read c for the y-intercept, and use symmetry to place a second point opposite the intercept. For y = x² − 6x + 5 that is (3, −4), (0, 5) and (6, 5) — four features of the sketch from one line of working.
### Is the axis of symmetry the same as the vertex?
No. The vertex, or turning point, is a point with two coordinates; the axis of symmetry is the vertical line through it. They share an x-value, which is why finding one immediately gives most of the other, but an answer written in the wrong form will still be marked wrong.
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