Axis of Symmetry Calculator
Find the axis of symmetry and vertex of quadratic functions in standard, vertex, and factored forms. Our algebra calculator provides step-by-step solutions for parabola analysis and graphing.
Last updated: December 15, 2024
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Symmetry Analysis
Axis of Symmetry:
x = 2.0000
Vertex:
(2.0000, -1.0000)
Parabola Direction:
Opens upward
Analysis:
The parabola has its axis of symmetry at x = 2.0000. The vertex is at (2.0000, -1.0000) and the parabola opens upward.
Calculation Steps:
- Given: f(x) = 1x² + -4x + 3
- Formula: x = -b/(2a)
- Substitute: x = -(-4)/(2×1)
- Calculate: x = 4.0000/2.0000 = 2.0000
- Vertex y-coordinate: k = f(2.0000) = -1.0000
Axis of Symmetry Rules:
- • Standard form: x = -b/(2a)
- • Vertex form: x = h (directly given)
- • Factored form: x = (r₁ + r₂)/2 (midpoint of roots)
- • The axis of symmetry passes through the vertex of the parabola
Quick Example Result
For function f(x) = x² - 4x + 3:
Axis: x = 2, Vertex: (2, -1)
Using formula: x = -b/(2a) = -(-4)/(2×1) = 2
How This Calculator Works
Our axis of symmetry calculator analyzes quadratic functions using fundamental algebraic principles. The calculator applies different formulas based on the quadratic function formto determine the line of symmetry and vertex coordinates.
The Mathematical Formulas
x = -b/(2a)x = hx = (r₁ + r₂)/2These formulas determine the vertical line x = h that divides the parabola into two symmetric halves. The axis of symmetry always passes through the vertex, which is the parabola's turning point.
Shows axis of symmetry line and vertex location for different parabola orientations
Mathematical Foundation
The axis of symmetry is a fundamental property of quadratic functions that stems from their parabolic shape. Every parabola has exactly one line of symmetry that passes through its vertex. This property is essential for understanding quadratic behavior, optimization problems, and graphing techniques in algebra and calculus.
- Standard form uses the derivative approach: vertex occurs where f'(x) = 2ax + b = 0
- Vertex form directly provides the axis as the x-coordinate of the vertex
- Factored form uses the midpoint between roots due to parabola symmetry
- The axis of symmetry helps determine maximum/minimum values and function behavior
Sources & References
- Algebra and Trigonometry - Michael Sullivan (11th Edition)Standard reference for quadratic function analysis
- National Council of Teachers of Mathematics - Algebra Teaching StandardsEducational guidelines for teaching quadratic functions
- Khan Academy - Quadratic Functions and ParabolasComprehensive educational resources on axis of symmetry
Need help with other quadratic calculations? Check out our quadratic formula calculator and vertex calculator.
Get Custom Calculator for Your PlatformExample Analysis
Given Function:
- f(x): 2x² - 8x + 6
- a: 2
- b: -8
- c: 6
Calculation Steps:
- Apply formula: x = -b/(2a)
- Substitute: x = -(-8)/(2×2)
- Calculate: x = 8/4 = 2
- Find vertex: f(2) = 2(2)² - 8(2) + 6 = -2
Result: Axis of symmetry is x = 2, Vertex is (2, -2)
The parabola opens upward (a = 2 > 0) with its minimum point at (2, -2). The axis x = 2 divides the parabola into two symmetric halves.
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