The axis of symmetry of a parabola is the vertical line through its vertex that splits it into two mirror-image halves. For y = ax2 + bx + c, it is the line x = −b2a. Points at equal distances on either side of this line have the same y-value.
The axis x = 1 is a fold line: every point on one side has a twin at the same height on the other side.
Formal definition and notation
A line L is an axis of symmetry of a figure if reflecting the figure across L maps it onto itself. For the graph of f(x) = ax2 + bx + c with a ≠ 0, the axis of symmetry is the line x = h with h = −b2a; then f(h + d) = f(h − d) for every real d. In vertex forma(x − h)2 + k the axis is x = h.
Symbol
Read it as
Meaning
x = h
x equals h
the equation of a vertical line; the axis is always written this way, never as a single number
The vertex is a point, such as (1, −3). The axis of symmetry is a line through it, x = 1.
Axis of Symmetryvs.Line of symmetry of a shape
In geometry, a figure such as a square can have several lines of symmetry in any direction. An up-or-down parabola has exactly one, and it is vertical.
Quick check: 3 questions
Question 1 of 3
What is the axis of symmetry of f(x) = −x2 + 10x − 7?
Show answer and explanation
Answer: B.a = −1, b = 10: x = −102(−1) = 5. x = −5 ignored the negative a, x = 10 forgot to divide by 2a, and y = 5 wrote a horizontal line.
Question 2 of 3
A parabola has axis x = 2 and passes through (−1, 6). Which other point is on it?
Show answer and explanation
Answer: C.(−1, 6) is 3 units left of x = 2, so its mirror is 3 units right: (5, 6). (1, 6) reflects over x = 0, (−1, −6) reflects over the x-axis, and (2, 6) lies on the axis itself.
Question 3 of 3
Which parabola has the y-axis as its axis of symmetry?
Show answer and explanation
Answer: D.9 − 4x2 has b = 0, so x = −02(−4) = 0, the y-axis. The others have axes x = −2, x = 1 and x = 12.
Axis of Symmetry: frequently asked questions
How do you find the axis of symmetry?
For y = ax2 + bx + c, use x = −b2a. For vertex form y = a(x − h)2 + k, the axis is x = h. If you know the two x-intercepts, average them. On a graph, draw the vertical line through the vertex. Always write the answer as an equation, such as x = 4.
Is the axis of symmetry always vertical?
For the graph of a quadratic function, yes: it opens up or down, so the axis is a vertical line x = h. Parabolas that open left or right, such as x = y2, have a horizontal axis y = k, but they are not functions of x. A rotated parabola can even have a slanted axis.
What is the axis of symmetry used for?
It locates the vertex, which gives the maximum or minimum value in problems about height, area or profit. It also halves the work of graphing: every point you plot on one side gives a free mirror point on the other. And it lets you check zeros, which must be equally far from the axis.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSF-IF.B.4, CCSS HSF-IF.C.8.a
Cite this entry
Effortless Math Team. “Axis of Symmetry.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/axis-of-symmetry/
Flashcard
TermAxis of Symmetryx = −b2a
DefinitionThe vertical line through a parabola’s vertex that splits it into mirror halves. y = x2 − 6x + 2: axis x = 3.
VertexA vertex is a corner point where two or more lines, rays, or sides meet. It is the point of an angle, a corner of a polygon such as a triangle, a corner of a solid, and the turning point of a parabola. The plural of vertex is vertices: a square has 4 vertices.Geometry · Grade 4, Grade 5, Grade 6Vertex FormVertex form is a way to write a quadratic function as y = a(x − h)2 + k, so the vertex (h, k) of its parabola can be read directly. The number a sets which way it opens and how wide it is. For example, y = 2(x − 3)2 + 1 has its vertex at (3, 1).Algebra · Algebra 1, Algebra 2NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. 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A shape has line symmetry when a line splits it into two mirror-image halves that match exactly when folded; that fold line is the line of symmetry. It has rotational symmetry when it looks the same after a turn of less than 360°.Geometry · Grade 4, Grade 8, GeometrySideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2ParabolaA parabola is the U-shaped curve that is the graph of a quadratic function such as y = x2. It is symmetric about a line called the axis of symmetry, and its turning point is the vertex. Geometrically, it is every point equally far from a fixed point (the focus) and a fixed line (the directrix).Functions · Algebra 1, Algebra 2, GeometryEqual ToEqual to means having exactly the same value or amount, and it is written with the symbol =. Numbers that look different can still be equal to each other: 12 is equal to 0.5. When two amounts are not the same, one is greater than or less than the other.Symbols and Notation · Kindergarten, Grade 1, Grade 2DenominatorThe denominator is the bottom number in a fraction. It tells how many equal parts make one whole: in 35, the 5 means the whole is cut into 5 equal parts, and the numerator 3 counts the parts you have. The denominator can never be 0.Number System · Grade 3, Grade 4, Grade 5Quadratic FunctionA quadratic function is a function that can be written f(x) = ax2 + bx + c with a ≠ 0. Its graph is a U-shaped curve called a parabola, which opens up when a > 0 and down when a < 0. Its highest or lowest point is the vertex.Functions · Algebra 1, Algebra 2, PrecalculusCompleting the SquareCompleting the square is a method that rewrites x2 + bx as a perfect square by adding (b2)2: x2 + bx + (b2)2 = (x + b2)2. It is used to solve quadratic equations, to write a quadratic in vertex form, and to derive the quadratic formula.Algebra · Algebra 1, Algebra 2, PrecalculusStandard Form of a QuadraticThe standard form of a quadratic is ax2 + bx + c, with the terms in descending powers of x and a ≠ 0. As an equation it is ax2 + bx + c = 0; as a function, y = ax2 + bx + c. It shows a, b and c directly, which the quadratic formula needs, and c is the y-intercept.Algebra · Algebra 1, Algebra 2ReflectionA reflection is a transformation that flips a figure over a line to make a mirror image. That line is the line of reflection: each point and its image are the same distance from the line, on opposite sides. The image is congruent but faces the other way. Across the y-axis, (x, y) goes to (−x, y).Geometry · Grade 8, GeometryZero of a FunctionA zero of a function is an input that makes the output 0: a number a with f(a) = 0. For f(x) = x − 5, the zero is 5. On a graph, the real zeros are the x-values where the curve meets the x-axis. You find zeros by solving f(x) = 0.Functions · Algebra 1, Algebra 2, Precalculus