maths classes
2D Shapes2cosacosb Formula30-60-90 Formulas3D ShapesAbsolute Value FormulaAcute AngleAcute Angle triangleAdditionAlgebra FormulasAlgebra of MatricesAlgebraic EquationsAlgebraic ExpressionsAngle FormulaAnnulusAnova FormulaAnti-derivative FormulaAntiderivative FormulaApplication of DerivativesApplications of IntegrationArc Length FormulaArccot FormulaArctan FormulaArea Formula for QuadrilateralsArea FormulasArea Of A Sector Of A Circle FormulaArea Of An Octagon FormulaArea Of Isosceles TriangleArea Of ShapesArea Under the Curve FormulaArea of RectangleArea of Regular Polygon FormulaArea of TriangleArea of a Circle FormulaArea of a Pentagon FormulaArea of a Square FormulaArea of a Trapezoid FormulaArithmetic Mean FormulaArithmetic ProgressionsArithmetic Sequence Recursive FormulaArithmetic and Geometric ProgressionAscending OrderAssociative Property FormulaAsymptote FormulaAverage Deviation FormulaAverage Rate of Change FormulaAveragesAxioms Of ProbabilityAxis of Symmetry FormulaBasic Math FormulasBasics Of AlgebraBinary FormulaBinomial Probability FormulaBinomial Theorem FormulaBinomial distributionBodmas RuleBoolean AlgebraBusiness MathematicsCalculusCelsius FormulaCentral Angle of a Circle FormulaCentral Limit Theorem FormulaCentroid of a Trapezoid FormulaChain RuleChain Rule FormulaChange of Base FormulaChi Square FormulaCirclesCircumference FormulaCoefficient of Determination FormulaCoefficient of Variation FormulaCofactor FormulaComplete the square formulaComplex numbersCompound Interest FormulaConditional Probability FormulaConeConfidence Interval FormulaCongruence of TrianglesCorrelation Coefficient FormulaCos Double Angle FormulaCos Square theta FormulaCos Theta FormulaCosec Cot FormulaCosecant FormulaCosine FormulaCovariance FormulaCubeCurated Maths Resources for Teachers – EdulyteCylinderDecimalsDifferential calculusDiscover the world of MathsEllipseEquilateral triangleEuler’s formulaEven numbersExponentsFractionFraction to decimalGeometric sequenceHeptagonHyperbolaIntegersIntegrationIntegration by partsLinesLocusMatricesNatural numbersNumber lineOdd numbersParallelogramPercentage formulaPerimeterPolygonPolynomialsPrismProbabilityPyramidPythagoras theoremRoman NumeralsScalene triangleSetsShapes NamesSimple interest formulaSlope formulaSolid shapesSphereSquareStandard deviation formulaSubtractionSymmetryTimeTrianglesTrigonometry formulaTypes of anglesValue of PiVariance formulaVectorVolume formulasVolume of a coneVolume of sphere formulaWhole numbers

Axis of Symmetry Formula

Edulyte Maths Lessons

Learn Maths anytime, anywhere

Sign Up

Unlocking Elegance: The Enigmatic Axis of Symmetry Formula

Comprehensive Definition, Description, Examples & Rules 

Edulyte Maths Lessons

Learn Maths anytime, anywhere

Sign Up
What will you learn

Understanding the Axis of Symmetry

The symmetric axis or the axis of symmetry refers to an imaginary line conceived in geometry and mathematics that can divide an image or a shape into two parts that are completely identical to each other. The concept of symmetry is very important in various fields of mathematics as it helps us understand the different kinds of shapes better, ease up calculations and theories, along with helping us know and understand quadratic equations more efficiently. The axis of symmetry can also be seen in concepts apart from mathematics and geometry, and in the real world like architecture where it is needed to design symmetrical structures for maintaining the right order and foundation, and engineering to develop structures like roads and bridges that are fundamentally symmetrical and hence, strong. 

The Formula for the Axis of Symmetry

Here is a representation of the axis of symmetry formula that helps us find the symmetrical axis in various situations:

For a quadratic equation of the form f(x) = ax^2 + bx + cx, the axis of symmetry can be found using:

x = -b / 2a

Description of the variables employed in the formula:

  • a: Coefficient of the quadratic term
  • b: Coefficient of the linear term
  • c: Constant term
  • x: Axis of symmetry

Equation for the Axis of Symmetry: Derivation and Application

The given axis of symmetry equation can be employed to figure out the symmetrical axis in different scenarios:

x = -b / 2a

This equation can be used for a variety of applications like:

  • Computing quadratic equations
  • Locating the symmetrical axis in different types of geometrical shapes.
  • Discerning the various calculus functions and curves. 
  • Designing symmetrical buildings, bridges, arches, etc.
  • Achieving a balance in various statistical reports

How to Determine the Axis of Symmetry

Here is a detailed process of steps you need to take to find out the axis of symmetry in different shapes and functions:

  • Quadratic Functions

For a quadratic equation of the form f(x) = ax^2 + bx + cx, the axis of symmetry can be found using: x = -b / 2a. To find the value of x,

Note the values for a, b, and c from the provided information.

Employ these values in the formula.

Compute and simplify the equation for x. 

what is the equation for the axis of symmetry

  • Rectangles

For rectangles, the symmetrical axis or the line that passes through the center of the rectangle will always coincide with the longer side of the rectangle.

determine axis of symmetry

  • Parabolas or Conic Sections

For a parabolic equation of the form y = ax^2 + bx + cx, the axis of symmetry can be found using: x = -b / 2a. To find the value of x,

Note the values for a, b, and c from the provided information.

Employ these values in the formula.

Compute and simplify the equation for x.

symmetric axis

The axis of symmetry for conic shapes or parabolas will always be vertical and pass through the vertex. 

  • Curves and Functions in Calculus

For functions of calculus like y = sin (x) or y = cos (x), the y-axis acts as the symmetrical axis itself because these functions are symmetrical along the y-axis.

find axis of symmetry

Symmetric Axis in Geometry

The concept of the symmetric axis is crucial in the field of geometry as it helps us determine the line that divides any geometrical shape into two equal parts. The equally sectioned two parts, as a result, are completely identical to each other and will coincide or one would fit perfectly over the other if you overlap them. 

Here is a list of different types of symmetric shapes with their axes of symmetry:

  • Squares

A square has four lines of symmetry, amongst which two are diagonals of the square and the other two are perpendicular bisectors.

  • Circles

A circle has an endless or infinite number of axes of symmetry passing through its center.

  • Rectangles

A rectangle has two lines of symmetry, one along the longer side and the other along the shorter one. 

  • Equilateral Triangles

An equilateral triangle has three lines of symmetry, amongst which one is a median, one is an altitude and one is an angle bisector. 

Practical Methods for Finding the Axis of Symmetry

For real-world scenarios like finding the axis of symmetry in fields like architecture, physics, etc. follow the given steps and tips carefully:

Architecture

Reduce the various architectural symmetrical shapes like domes, arches, roofs, etc. into basic geometric shapes like circles, semicircles, squares, etc. to find the axis of symmetry of various architectural elements.

Physics

Make use of the principles and rules of symmetry and finding the symmetrical axis and incorporate them in the various applications of physics to carry out accurate calculations and results.

Engineering

Visualize the various structures employed in the field of engineering as fundamental shapes found in geometry and estimate their axis of symmetry for safer and stronger engineered structures.

Statistics

Make use of various types of graphs to pictorially represent the different trends and changes seen in statistical data and reports and then determine a common point of symmetry to ensure balance and uniformity. 

Axis of Symmetry in Quadratic Functions: An Example

For a quadratic equation of the form f(x) = ax^2 + bx + cx, the axis of symmetry can be found using:

x = -b / 2a

Description of the variables employed in the formula:

a: Coefficient of the quadratic term

b: Coefficient of the linear term

c: Constant term

x: Axis of symmetry

Axis of symmetry example

Find axis of symmetry for a quadratic equation: f(x) = 5x^2 – 4x + 2

Solution

According to the information provided in the question, we can determine axis of symmetry with: 

a= 5

b= -4

c= 2

Using the axis of symmetry formula: x = -b / 2a

x= – (-4) / 2*5

x= 4 / 10 

x= 0.4

Hence, the axis of symmetry for the quadratic equation f(x) = 5x^2 – 4x + 2 is 0.4. 

Applications of the Axis of Symmetry

The axis of symmetry has applications in a variety of fields we see in our everyday lives, for instance:

Architecture

Architecture makes use of the concept of the axis of symmetry all the time. It helps build safer and more robust architectural structures from symmetrical bridges and buildings to domes, arches, windows and more. 

Art

The concept of the axis of symmetry is used in art to create visually appealing and beautiful patterns and designs in a variety of applications such as cloth designs, the artwork of exquisite flooring tiles, and so on. 

Engineering

The axis of symmetry is commonly used in engineering practices such as maintaining the security and sturdiness of huge constructions such as towers, skyscrapers, and dams by ensuring that the weight of the material is split evenly along an axis. 

Step Up Your Math Game Today!

Free sign-up for a personalised dashboard, learning tools, and unlimited possibilities!

learn maths with edulyte

Key Takeaways

  1. The symmetric axis or the axis of symmetry refers to an imaginary line conceived in geometry and mathematics that can divide an image or a shape into two parts that are completely identical to each other.

  2. For a quadratic equation of the form f(x) = ax^2 + bx + cx, the axis of symmetry can be found using: x = -b / 2a

  3. The concept of the axis of symmetry is used in a variety of real-world scenarios like art, architecture, engineering, and more.

Quiz

Check your score in the end

Quiz

Check your score in the end
Question of

Question comes here

Frequently Asked Questions

The formula for calculating the axis of symmetry in quadratic equations is x = -b / 2a.

Axis of symmetry plays a crucial role in various real-world applications like art, engineering, statistics, architecture, and more. 

Symmetry is an umbrella term that denotes balance and equal division of the two parts of any shape or structure to ensure an organized distribution of elements. The symmetric axis or the axis of symmetry refers to an imaginary line conceived in geometry and mathematics that can divide an image or a shape into two parts that are completely identical to each other, or in other words, symmetrical to each other. 

The axis of symmetry for a parabola denotes its vertical symmetry along with determining the x-coordinate of the vertex. 

Some common misconceptions or mistakes when working with the axis of symmetry are the belief that the axis of symmetry always passes through the origin of a shape and applying the axis of symmetry formula in non-quadratic equations. Some ways to avoid these mistakes and misconceptions are always checking if the equation is quadratic before applying the axis of symmetry formula, and remembering that the axis of symmetry can fall anywhere on the x-axis, not necessarily on x=0.

Like
Share it with your friends

Learn Maths

2D Shapes2cosacosb Formula30-60-90 Formulas3D ShapesAbsolute Value FormulaAcute AngleAcute Angle triangleAdditionAlgebra FormulasAlgebra of MatricesAlgebraic EquationsAlgebraic ExpressionsAngle FormulaAnnulusAnova FormulaAnti-derivative FormulaAntiderivative FormulaApplication of DerivativesApplications of IntegrationArc Length FormulaArccot FormulaArctan FormulaArea Formula for QuadrilateralsArea FormulasArea Of A Sector Of A Circle FormulaArea Of An Octagon FormulaArea Of Isosceles TriangleArea Of ShapesArea Under the Curve FormulaArea of RectangleArea of Regular Polygon FormulaArea of TriangleArea of a Circle FormulaArea of a Pentagon FormulaArea of a Square FormulaArea of a Trapezoid FormulaArithmetic Mean FormulaArithmetic ProgressionsArithmetic Sequence Recursive FormulaArithmetic and Geometric ProgressionAscending OrderAssociative Property FormulaAsymptote FormulaAverage Deviation FormulaAverage Rate of Change FormulaAveragesAxioms Of ProbabilityAxis of Symmetry FormulaBasic Math FormulasBasics Of AlgebraBinary FormulaBinomial Probability FormulaBinomial Theorem FormulaBinomial distributionBodmas RuleBoolean AlgebraBusiness MathematicsCalculusCelsius FormulaCentral Angle of a Circle FormulaCentral Limit Theorem FormulaCentroid of a Trapezoid FormulaChain RuleChain Rule FormulaChange of Base FormulaChi Square FormulaCirclesCircumference FormulaCoefficient of Determination FormulaCoefficient of Variation FormulaCofactor FormulaComplete the square formulaComplex numbersCompound Interest FormulaConditional Probability FormulaConeConfidence Interval FormulaCongruence of TrianglesCorrelation Coefficient FormulaCos Double Angle FormulaCos Square theta FormulaCos Theta FormulaCosec Cot FormulaCosecant FormulaCosine FormulaCovariance FormulaCubeCurated Maths Resources for Teachers – EdulyteCylinderDecimalsDifferential calculusDiscover the world of MathsEllipseEquilateral triangleEuler’s formulaEven numbersExponentsFractionFraction to decimalGeometric sequenceHeptagonHyperbolaIntegersIntegrationIntegration by partsLinesLocusMatricesNatural numbersNumber lineOdd numbersParallelogramPercentage formulaPerimeterPolygonPolynomialsPrismProbabilityPyramidPythagoras theoremRoman NumeralsScalene triangleSetsShapes NamesSimple interest formulaSlope formulaSolid shapesSphereSquareStandard deviation formulaSubtractionSymmetryTimeTrianglesTrigonometry formulaTypes of anglesValue of PiVariance formulaVectorVolume formulasVolume of a coneVolume of sphere formulaWhole numbers
Newsletters

Sign up for Learner Newsletter

Subscribe now for discounts, learning resources, blogs and guides. We do not spam.

We won’t pass your details on to anyone else. By clicking the subscribe button you agree to our Terms of Use and Privacy Policy.

Newsletters

Sign up for Tutor Newsletter

Subscribe now for discounts, learning resources, blogs and guides. We do not spam.

We won’t pass your details on to anyone else. By clicking the subscribe button you agree to our Terms of Use and Privacy Policy.

PTE Tutorials: Customised Packages for Every Learner

Standard

$75 AUD

One time

popular

Premium

$275 AUD

One time

Elite

$575 AUD

One time