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Axis of Symmetry Calculator

Last updated: June 19, 2026

Written by Blake Boege · Founder, Calculator Answers

An axis of symmetry calculator computes the vertical line dividing a quadratic parabola into symmetric halves. For standard form ax² + bx + c, it solves x = -b/(2a). For vertex form a(x - h)² + k, it extracts x = h. It also computes the vertex coordinates and plots an ASCII curve showing the opening direction.

Find the axis of symmetry and vertex for quadratic functions. Supports both standard form (ax² + bx + c) and vertex form (a(x - h)² + k) with detailed math steps and an ASCII visualization.

Quick Answer

Find the axis of symmetry and vertex for a parabola. Enter the coefficients of standard or vertex form to see the equation and an ASCII graph.

Equation Form

Coefficient values

x² term · e.g. 1

x term · e.g. 4

Constant · e.g. 3

Parabola & Symmetry Axis Visualization

       |     
   \   |   / 
    \  |  /  
     \_|_/   
       | Vertex: (-2, -1)
     x = -2

Parabola Symmetry Details

Axis of Symmetry

x = -2

Vertex: (-2, -1) · Opens Up

Axis Equationx = -2

Vertex (h, k)(-2, -1)

Parabola opensUpward

Coefficient a1

The axis of symmetry is the vertical line that divides a parabola into two symmetric halves.

Step-by-Step Derivations

[1]Identify coefficients for standard form ax² + bx + c:

[2] - a = 1, b = 4, c = 3

[3]Apply the axis of symmetry formula: x = −b / (2a)

[4] - x = −(4) / (2 × 1)

[5] - x = −4 / 2 = -2.0000

[6]Calculate the vertex coordinates (h, k):

[7] - h = -2.0000

[8] - k = f(h) = 1(-2.0000)² + 4(-2.0000) + 3

[9] - k = -1.0000

[10] - Vertex = (-2.0000, -1.0000)

[11]Determine the direction of the parabola:

[12] - Since a = 1 is positive (> 0), the parabola opens UPWARD.

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Examples

Standard Form Equation: y = x² + 4x + 3

Set a = 1, b = 4 · Calculate axis: x = -4 / (2 * 1) = -2 · Vertex x = -2, y = (-2)² + 4(-2) + 3 = -1 · Vertex: (-2, -1) · Parabola opens upward

Vertex Form Equation: y = 2(x - 3)² - 5

Set a = 2, h = 3, k = -5 · Axis of symmetry is x = 3 · Vertex is (3, -5) · Parabola opens upward

How it works

A quadratic equation represents a parabola. The line of symmetry is the vertical fold line where the left and right sides of the curve align.

Standard Form Formula

Given y = ax² + bx + c:

x = -b / (2a)

Once you find the axis value x = h, substitute this value back into the quadratic equation to find the corresponding y-coordinate k = f(h), forming the vertex (h, k).

Comparing Standard Form vs Vertex Form

Standard form is highly common in algebraic expressions, requiring coefficients a and b to compute the line. Vertex form is convenient because it explicitly reveals the vertex coordinates (h, k) without requiring manual evaluation. The axis of symmetry is always vertical for these parabolas.

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Frequently asked questions

The axis of symmetry of a parabola is a vertical line that passes directly through the vertex, dividing the parabola into two matching, mirror-image halves.

For a quadratic equation in standard form y = ax² + bx + c, the axis of symmetry is calculated using the formula x = -b / (2a).

For a quadratic equation in vertex form y = a(x - h)² + k, the axis of symmetry is simply x = h. The vertex is located directly at the coordinate (h, k).

For functions of the form y = f(x), the axis of symmetry is always vertical (x = constant). However, if the parabola is sideways and defined as x = ay² + by + c, the axis of symmetry would be horizontal (y = -b/(2a)). This calculator focuses on vertical parabolas (y as a function of x).

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