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Axis of Symmetry Calculator

Axis of Symmetry Calculator

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An online axis of symmetry calculator helps you to determine the position of an axis of symmetry with a standard form equation. This line of symmetry calculator allows you to find the actual position of a parabola. In this article, you can explore how to find axis of symmetry using quadratic equations and much more.

What is the Axis of Symmetry?

The axis of symmetry is a line that divides the object into two equal halves, creating a mirror image like the reflection on each side of the object. The word “symmetric” means balance. Symmetry can be applied to a variety of situations and contexts. Symmetry is the main concept in geometry. This cuts the shape in half, and they are precise reflections of each other, as shown in the image below.

Axis of Symmetry Calculator

Axis of Symmetry Formula:

The axis of symmetry equation comes from the following equations. Here we take the symmetry axis of the parabola. The equation can be expressed in two ways:

1. Standard form

2. Vertex form

Standard Form:

The standard form of the quadratic equation that is used by the axis of symmetry calculator:

$$ Y = ax^2 + bx + c $$

Where, c is the constant form and a, b are the coefficients of “x”.

Here, equation of axis of symmetry is

$$ X = -b / 2a $$

Vertex Form:

The vertex form of the quadratic equation is,

$$ Y = a(x−h)^2 + k $$

(h, k) = vertex of the parabola.

In the vertex form, we can say x = h, because the vertex and the axis of symmetry are on the same straight line.

How to Find the Axis of Symmetry?

Use the following formula to start graphing the quadratic equation

$$ X = -b / 2a $$

Solving for the x provides the axis of symmetry.


Determine the axis of symmetry of the graph of \( y = (x + 5)^2 – 4 \), using the formula.



Axis of Symmetry for \( y = ax^2 + bx + c \text{ is } x = −b / 2a \)


$$ f(x) = (x + 5) 2 − 4 $$
$$ f(x) = x^2 + 10x + 21 $$
$$ a=1,b=10 $$


$$ X = −b / 2a $$

Now, axis of symmetry calculator substitute the values in standard equation

$$ x = −(10) / 2(1) $$
$$ x = −10 / 2 $$
$$ x = −5 / 1 $$
$$ x = − 5 $$

Derive the Axis of Symmetry for Parabola:

The equation for axis of symmetry for a parabola is

$$ x = −b / 2a $$

Why is the axis of symmetry x = −b / 2a?
This is how it works.
By definition, the axis of symmetry of a parabola is a vertical line that divides the parabola into two halves.
The axis of symmetry always goes through the vertex of a parabola. The quadratic equation of parabola:

$$ Y = ax^2 + bx + c $$

where (a, b) is the co-efficient at x and ‘c’ is a constant term.

The quadratic equation \( y = ax^2 + bx + c \) is equal to \( y = ax^2 + bx \), because the constant term ‘c’ does not affect the parabola. Therefore, we take,
$$ Y = ax^2 + bx $$
let y = 0 and x = 0, which implies,
$$ ax^2 + bx = 0 $$
$$ x (ax + b) = 0 $$
$$ (ax + b) = 0 $$
Now, taking x on the other side
$$ X = −b / a $$
Therefore, the two values are 0, −b / a
The mid-point formula is
$$ x = x_1+x_2 / 2 $$
$$ x = 0 + (−b / a) / 2 $$
$$ x = -b / 2a $$

How does Axis of Symmetry Calculator Works?

An online aos calculator find the exact position of axis of symmetric by following steps:


  • Enter an equation to find the axis of symmetry.
  • Hit the calculate button.


  • The axis of symmetry equation calculator displays an equation of axis of symmetry for the given equation and provide step-wise solution.


What are symmetry operations and elements?

Symmetrical elements are geometric objects on which symmetry operations are performed. Symmetry can be a point, a plane, or an axis. Symmetry operation is the movement of an object. After the movement, the object looks the same as before. The existence of symmetric operations means the existence of corresponding symmetric elements, and conversely, the existence of symmetry. The symmetry element means that a certain operation or a set of symmetric operations is possible.

Where is symmetry used?

Symmetry is used when doing calculations that involve circles, spheres, isosceles or equilateral triangles, etc.

What are the main types of Symmetry?

There are four types of symmetry that can be observed in different situations, they are:

  • Translation Symmetry
  • Reflection Symmetry
  • Rotational Symmetry
  • Glide Symmetry


Use this online axis of symmetry calculator to find the axis of symmetry with standard form equation. The axis of symmetry is always perpendicular to the directrix and passes through the focus point of a parabola.


From the source of Wikipedia: Geometrical Symmetry, In social interactions, Pitch structures, In other arts and crafts.

From the source of Lumen Learning: Symmetry Elements and Operations, Identity (E), Proper Rotation and Proper Axis, Reflection and Symmetry Planes, Inversion and Inversion Center, Improper rotation.

From the source of Varsity Tutors: Axis of Symmetry of a Parabola, Equations of Quadratic Functions, vertex of the parabola.