Mathématiques

Axis of Symmetry of a Parabola

The vertical line x = −b/2a through the turning point of a parabola, worked on y = x² − 6x + 5, with the link to the midpoint of the two roots.

Axis of symmetry

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.

Aussi appelé: Line of symmetry
Où les élèves le rencontrent: 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.

La réponse en bref
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.

Un exemple

y = x² − 6x + 5 → x = −(−6) ÷ (2 × 1) = 3, vertex (3, −4)

Here a = 1 and b = −6, so −b/2a is 6/2 = 3 and the axis of symmetry is the line x = 3. The curve factorises as (x − 1)(x − 5), so it crosses the x-axis at 1 and 5, and the midpoint of those roots is (1 + 5)/2 = 3 — the same line, found a different way. Substituting x = 3 into the original equation gives 9 − 18 + 5 = −4, so the minimum point is (3, −4).

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 fréquentes

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.

Dernière mise à jour 21 août 2026