Symmetry - Algebra - Pauls Online Math Notes

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• Home
• Classes
• Algebra
  • 1. Preliminaries
    • 1.1 Integer Exponents
    • 1.2 Rational Exponents
    • 1.3 Radicals
    • 1.4 Polynomials
    • 1.5 Factoring Polynomials
    • 1.6 Rational Expressions
    • 1.7 Complex Numbers
  • 2. Solving Equations and Inequalities
    • 2.1 Solutions and Solution Sets
    • 2.2 Linear Equations
    • 2.3 Applications of Linear Equations
    • 2.4 Equations With More Than One Variable
    • 2.5 Quadratic Equations - Part I
    • 2.6 Quadratic Equations - Part II
    • 2.7 Quadratic Equations : A Summary
    • 2.8 Applications of Quadratic Equations
    • 2.9 Equations Reducible to Quadratic in Form
    • 2.10 Equations with Radicals
    • 2.11 Linear Inequalities
    • 2.12 Polynomial Inequalities
    • 2.13 Rational Inequalities
    • 2.14 Absolute Value Equations
    • 2.15 Absolute Value Inequalities
  • 3. Graphing and Functions
    • 3.1 Graphing
    • 3.2 Lines
    • 3.3 Circles
    • 3.4 The Definition of a Function
    • 3.5 Graphing Functions
    • 3.6 Combining Functions
    • 3.7 Inverse Functions
  • 4. Common Graphs
    • 4.1 Lines, Circles and Piecewise Functions
    • 4.2 Parabolas
    • 4.3 Ellipses
    • 4.4 Hyperbolas
    • 4.5 Miscellaneous Functions
    • 4.6 Transformations
    • 4.7 Symmetry
    • 4.8 Rational Functions
  • 5. Polynomial Functions
    • 5.1 Dividing Polynomials
    • 5.2 Zeroes/Roots of Polynomials
    • 5.3 Graphing Polynomials
    • 5.4 Finding Zeroes of Polynomials
    • 5.5 Partial Fractions
  • 6. Exponential and Logarithm Functions
    • 6.1 Exponential Functions
    • 6.2 Logarithm Functions
    • 6.3 Solving Exponential Equations
    • 6.4 Solving Logarithm Equations
    • 6.5 Applications
  • 7. Systems of Equations
    • 7.1 Linear Systems with Two Variables
    • 7.2 Linear Systems with Three Variables
    • 7.3 Augmented Matrices
    • 7.4 More on the Augmented Matrix
    • 7.5 Nonlinear Systems
• Calculus I
  • 1. Review
    • 1.1 Functions
    • 1.2 Inverse Functions
    • 1.3 Trig Functions
    • 1.4 Solving Trig Equations
    • 1.5 Trig Equations with Calculators, Part I
    • 1.6 Trig Equations with Calculators, Part II
    • 1.7 Exponential Functions
    • 1.8 Logarithm Functions
    • 1.9 Exponential and Logarithm Equations
    • 1.10 Common Graphs
  • 2. Limits
    • 2.1 Tangent Lines and Rates of Change
    • 2.2 The Limit
    • 2.3 One-Sided Limits
    • 2.4 Limit Properties
    • 2.5 Computing Limits
    • 2.6 Infinite Limits
    • 2.7 Limits At Infinity, Part I
    • 2.8 Limits At Infinity, Part II
    • 2.9 Continuity
    • 2.10 The Definition of the Limit
  • 3. Derivatives
    • 3.1 The Definition of the Derivative
    • 3.2 Interpretation of the Derivative
    • 3.3 Differentiation Formulas
    • 3.4 Product and Quotient Rule
    • 3.5 Derivatives of Trig Functions
    • 3.6 Derivatives of Exponential and Logarithm Functions
    • 3.7 Derivatives of Inverse Trig Functions
    • 3.8 Derivatives of Hyperbolic Functions
    • 3.9 Chain Rule
    • 3.10 Implicit Differentiation
    • 3.11 Related Rates
    • 3.12 Higher Order Derivatives
    • 3.13 Logarithmic Differentiation
  • 4. Applications of Derivatives
    • 4.1 Rates of Change
    • 4.2 Critical Points
    • 4.3 Minimum and Maximum Values
    • 4.4 Finding Absolute Extrema
    • 4.5 The Shape of a Graph, Part I
    • 4.6 The Shape of a Graph, Part II
    • 4.7 The Mean Value Theorem
    • 4.8 Optimization
    • 4.9 More Optimization Problems
    • 4.10 L'Hospital's Rule and Indeterminate Forms
    • 4.11 Linear Approximations
    • 4.12 Differentials
    • 4.13 Newton's Method
    • 4.14 Business Applications
  • 5. Integrals
    • 5.1 Indefinite Integrals
    • 5.2 Computing Indefinite Integrals
    • 5.3 Substitution Rule for Indefinite Integrals
    • 5.4 More Substitution Rule
    • 5.5 Area Problem
    • 5.6 Definition of the Definite Integral
    • 5.7 Computing Definite Integrals
    • 5.8 Substitution Rule for Definite Integrals
  • 6. Applications of Integrals
    • 6.1 Average Function Value
    • 6.2 Area Between Curves
    • 6.3 Volumes of Solids of Revolution / Method of Rings
    • 6.4 Volumes of Solids of Revolution/Method of Cylinders
    • 6.5 More Volume Problems
    • 6.6 Work
  • Appendix A. Extras
    • A.1 Proof of Various Limit Properties
    • A.2 Proof of Various Derivative Properties
    • A.3 Proof of Trig Limits
    • A.4 Proofs of Derivative Applications Facts
    • A.5 Proof of Various Integral Properties
    • A.6 Area and Volume Formulas
    • A.7 Types of Infinity
    • A.8 Summation Notation
    • A.9 Constant of Integration
• Calculus II
  • 7. Integration Techniques
    • 7.1 Integration by Parts
    • 7.2 Integrals Involving Trig Functions
    • 7.3 Trig Substitutions
    • 7.4 Partial Fractions
    • 7.5 Integrals Involving Roots
    • 7.6 Integrals Involving Quadratics
    • 7.7 Integration Strategy
    • 7.8 Improper Integrals
    • 7.9 Comparison Test for Improper Integrals
    • 7.10 Approximating Definite Integrals
  • 8. Applications of Integrals
    • 8.1 Arc Length
    • 8.2 Surface Area
    • 8.3 Center of Mass
    • 8.4 Hydrostatic Pressure
    • 8.5 Probability
  • 9. Parametric Equations and Polar Coordinates
    • 9.1 Parametric Equations and Curves
    • 9.2 Tangents with Parametric Equations
    • 9.3 Area with Parametric Equations
    • 9.4 Arc Length with Parametric Equations
    • 9.5 Surface Area with Parametric Equations
    • 9.6 Polar Coordinates
    • 9.7 Tangents with Polar Coordinates
    • 9.8 Area with Polar Coordinates
    • 9.9 Arc Length with Polar Coordinates
    • 9.10 Surface Area with Polar Coordinates
    • 9.11 Arc Length and Surface Area Revisited
  • 10. Series & Sequences
    • 10.1 Sequences
    • 10.2 More on Sequences
    • 10.3 Series - The Basics
    • 10.4 Convergence/Divergence of Series
    • 10.5 Special Series
    • 10.6 Integral Test
    • 10.7 Comparison Test/Limit Comparison Test
    • 10.8 Alternating Series Test
    • 10.9 Absolute Convergence
    • 10.10 Ratio Test
    • 10.11 Root Test
    • 10.12 Strategy for Series
    • 10.13 Estimating the Value of a Series
    • 10.14 Power Series
    • 10.15 Power Series and Functions
    • 10.16 Taylor Series
    • 10.17 Applications of Series
    • 10.18 Binomial Series
  • 11. Vectors
    • 11.1 Vectors - The Basics
    • 11.2 Vector Arithmetic
    • 11.3 Dot Product
    • 11.4 Cross Product
  • 12. 3-Dimensional Space
    • 12.1 The 3-D Coordinate System
    • 12.2 Equations of Lines
    • 12.3 Equations of Planes
    • 12.4 Quadric Surfaces
    • 12.5 Functions of Several Variables
    • 12.6 Vector Functions
    • 12.7 Calculus with Vector Functions
    • 12.8 Tangent, Normal and Binormal Vectors
    • 12.9 Arc Length with Vector Functions
    • 12.10 Curvature
    • 12.11 Velocity and Acceleration
    • 12.12 Cylindrical Coordinates
    • 12.13 Spherical Coordinates
• Calculus III
  • 12. 3-Dimensional Space
    • 12.1 The 3-D Coordinate System
    • 12.2 Equations of Lines
    • 12.3 Equations of Planes
    • 12.4 Quadric Surfaces
    • 12.5 Functions of Several Variables
    • 12.6 Vector Functions
    • 12.7 Calculus with Vector Functions
    • 12.8 Tangent, Normal and Binormal Vectors
    • 12.9 Arc Length with Vector Functions
    • 12.10 Curvature
    • 12.11 Velocity and Acceleration
    • 12.12 Cylindrical Coordinates
    • 12.13 Spherical Coordinates
  • 13. Partial Derivatives
    • 13.1 Limits
    • 13.2 Partial Derivatives
    • 13.3 Interpretations of Partial Derivatives
    • 13.4 Higher Order Partial Derivatives
    • 13.5 Differentials
    • 13.6 Chain Rule
    • 13.7 Directional Derivatives
  • 14. Applications of Partial Derivatives
    • 14.1 Tangent Planes and Linear Approximations
    • 14.2 Gradient Vector, Tangent Planes and Normal Lines
    • 14.3 Relative Minimums and Maximums
    • 14.4 Absolute Minimums and Maximums
    • 14.5 Lagrange Multipliers
  • 15. Multiple Integrals
    • 15.1 Double Integrals
    • 15.2 Iterated Integrals
    • 15.3 Double Integrals over General Regions
    • 15.4 Double Integrals in Polar Coordinates
    • 15.5 Triple Integrals
    • 15.6 Triple Integrals in Cylindrical Coordinates
    • 15.7 Triple Integrals in Spherical Coordinates
    • 15.8 Change of Variables
    • 15.9 Surface Area
    • 15.10 Area and Volume Revisited
  • 16. Line Integrals
    • 16.1 Vector Fields
    • 16.2 Line Integrals - Part I
    • 16.3 Line Integrals - Part II
    • 16.4 Line Integrals of Vector Fields
    • 16.5 Fundamental Theorem for Line Integrals
    • 16.6 Conservative Vector Fields
    • 16.7 Green's Theorem
  • 17.Surface Integrals
    • 17.1 Curl and Divergence
    • 17.2 Parametric Surfaces
    • 17.3 Surface Integrals
    • 17.4 Surface Integrals of Vector Fields
    • 17.5 Stokes' Theorem
    • 17.6 Divergence Theorem
• Differential Equations
  • 1. Basic Concepts
    • 1.1 Definitions
    • 1.2 Direction Fields
    • 1.3 Final Thoughts
  • 2. First Order DE's
    • 2.1 Linear Equations
    • 2.2 Separable Equations
    • 2.3 Exact Equations
    • 2.4 Bernoulli Differential Equations
    • 2.5 Substitutions
    • 2.6 Intervals of Validity
    • 2.7 Modeling with First Order DE's
    • 2.8 Equilibrium Solutions
    • 2.9 Euler's Method
  • 3. Second Order DE's
    • 3.1 Basic Concepts
    • 3.2 Real & Distinct Roots
    • 3.3 Complex Roots
    • 3.4 Repeated Roots
    • 3.5 Reduction of Order
    • 3.6 Fundamental Sets of Solutions
    • 3.7 More on the Wronskian
    • 3.8 Nonhomogeneous Differential Equations
    • 3.9 Undetermined Coefficients
    • 3.10 Variation of Parameters
    • 3.11 Mechanical Vibrations
  • 4. Laplace Transforms
    • 4.1 The Definition
    • 4.2 Laplace Transforms
    • 4.3 Inverse Laplace Transforms
    • 4.4 Step Functions
    • 4.5 Solving IVP's with Laplace Transforms
    • 4.6 Nonconstant Coefficient IVP's
    • 4.7 IVP's With Step Functions
    • 4.8 Dirac Delta Function
    • 4.9 Convolution Integrals
    • 4.10 Table Of Laplace Transforms
  • 5. Systems of DE's
    • 5.1 Review : Systems of Equations
    • 5.2 Review : Matrices & Vectors
    • 5.3 Review : Eigenvalues & Eigenvectors
    • 5.4 Systems of Differential Equations
    • 5.5 Solutions to Systems
    • 5.6 Phase Plane
    • 5.7 Real Eigenvalues
    • 5.8 Complex Eigenvalues
    • 5.9 Repeated Eigenvalues
    • 5.10 Nonhomogeneous Systems
    • 5.11 Laplace Transforms
    • 5.12 Modeling
  • 6. Series Solutions to DE's
    • 6.1 Review : Power Series
    • 6.2 Review : Taylor Series
    • 6.3 Series Solutions
    • 6.4 Euler Equations
  • 7. Higher Order Differential Equations
    • 7.1 Basic Concepts for nth Order Linear Equations
    • 7.2 Linear Homogeneous Differential Equations
    • 7.3 Undetermined Coefficients
    • 7.4 Variation of Parameters
    • 7.5 Laplace Transforms
    • 7.6 Systems of Differential Equations
    • 7.7 Series Solutions
  • 8. Boundary Value Problems & Fourier Series
    • 8.1 Boundary Value Problems
    • 8.2 Eigenvalues and Eigenfunctions
    • 8.3 Periodic Functions & Orthogonal Functions
    • 8.4 Fourier Sine Series
    • 8.5 Fourier Cosine Series
    • 8.6 Fourier Series
    • 8.7 Convergence of Fourier Series
  • 9. Partial Differential Equations
    • 9.1 The Heat Equation
    • 9.2 The Wave Equation
    • 9.3 Terminology
    • 9.4 Separation of Variables
    • 9.5 Solving the Heat Equation
    • 9.6 Heat Equation with Non-Zero Temperature Boundaries
    • 9.7 Laplace's Equation
    • 9.8 Vibrating String
    • 9.9 Summary of Separation of Variables
• Extras
• Algebra & Trig Review
  • 1. Algebra
    • 1.1 Exponents
    • 1.2 Absolute Value
    • 1.3 Radicals
    • 1.4 Rationalizing
    • 1.5 Functions
    • 1.6 Multiplying Polynomials
    • 1.7 Factoring
    • 1.8 Simplifying Rational Expressions
    • 1.9 Graphing and Common Graphs
    • 1.10 Solving Equations, Part I
    • 1.11 Solving Equations, Part II
    • 1.12 Solving Systems of Equations
    • 1.13 Solving Inequalities
    • 1.14 Absolute Value Equations and Inequalities
  • 2. Trigonometry
    • 2.1 Trig Function Evaluation
    • 2.2 Graphs of Trig Functions
    • 2.3 Trig Formulas
    • 2.4 Solving Trig Equations
    • 2.5 Inverse Trig Functions
  • 3. Exponentials & Logarithms
    • 3.1 Basic Exponential Functions
    • 3.2 Basic Logarithm Functions
    • 3.3 Logarithm Properties
    • 3.4 Simplifying Logarithms
    • 3.5 Solving Exponential Equations
    • 3.6 Solving Logarithm Equations
• Common Math Errors
  • 1. General Errors
  • 2. Algebra Errors
  • 3. Trig Errors
  • 4. Common Errors
  • 5. Calculus Errors
• Complex Number Primer
  • 1. The Definition
  • 2. Arithmetic
  • 3. Conjugate and Modulus
  • 4. Polar and Exponential Forms
  • 5. Powers and Roots
• How To Study Math
  • 1. General Tips
  • 2. Taking Notes
  • 3. Getting Help
  • 4. Doing Homework
  • 5. Problem Solving
  • 6. Studying For an Exam
  • 7. Taking an Exam
  • 8. Learn From Your Errors
• Cheat Sheets & Tables
• Misc Links
• Contact Me
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Paul's Online Notes Home / Algebra / Common Graphs / Symmetry Prev. Section Notes Practice Problems Assignment Problems Next Section Show Mobile Notice Show All Notes Hide All Notes Mobile Notice You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best viewed in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (you should be able to scroll/swipe to see them) and some of the menu items will be cut off due to the narrow screen width.

Section 4.7 : Symmetry

In this section we are going to take a look at something that we used back when we were graphing parabolas. However, we’re going to take a more general view of it this section. Many graphs have symmetry to them.

Symmetry can be useful in graphing an equation since it says that if we know one portion of the graph then we will also know the remaining (and symmetric) portion of the graph as well. We used this fact when we were graphing parabolas to get an extra point of some of the graphs.

In this section we want to look at three types of symmetry.

1. A graph is said to be symmetric about the \(x\)-axis if whenever \(\left( {a,b} \right)\) is on the graph then so is \(\left( {a, - b} \right)\). Here is a sketch of a graph that is symmetric about the \(x\)-axis.
2. A graph is said to be symmetric about the \(y\)-axis if whenever \(\left( {a,b} \right)\) is on the graph then so is \(\left( { - a,b} \right)\). Here is a sketch of a graph that is symmetric about the \(y\)-axis.
3. A graph is said to be symmetric about the origin if whenever \(\left( {a,b} \right)\) is on the graph then so is \(\left( { - a, - b} \right)\). Here is a sketch of a graph that is symmetric about the origin.

Note that most graphs don’t have any kind of symmetry. Also, it is possible for a graph to have more than one kind of symmetry. For example, the graph of a circle centered at the origin exhibits all three symmetries.

Tests for Symmetry

We’ve some fairly simply tests for each of the different types of symmetry.