Systems of Equations
Adding and Subtracting Rational Expressions with Different Denominators
Graphing Linear Equations
Raising an Exponential Expression to a Power
Horizontal Line Test
Quadratic Equations
Mixed Numbers and Improper Fractions
Solving Quadratic Equations by Completing the Square
Solving Exponential Equations
Adding and Subtracting Polynomials
Factorizing simple expressions
Identifying Prime and Composite Numbers
Solving Linear Systems of Equations by Graphing
Complex Conjugates
Graphing Compound Inequalities
Simplified Form of a Square Root
Solving Quadratic Equations Using the Square Root Property
Multiplication Property of Radicals
Determining if a Function has an Inverse
Scientific Notation
Degree of a Polynomial
Factoring Polynomials by Grouping
Solving Linear Systems of Equations
Exponential Functions
Factoring Trinomials by Grouping
The Slope of a Line
Simplifying Complex Fractions That Contain Addition or Subtraction
Solving Absolute Value Equations
Solving Right Triangles
Solving Rational Inequalities with a Sign Graph
Domain and Range of a Function
Multiplying Polynomials
Slope of a Line
Multiplying Rational Expressions
Percent of Change
Equations Involving Fractions or Decimals
Simplifying Expressions Containing only Monomials
Solving Inequalities
Quadratic Equations with Imaginary Solutions
Reducing Fractions to Lowest Terms
Prime and Composite Numbers
Dividing with Exponents
Dividing Rational Expressions
Equivalent Fractions
Graphing Quadratic Functions
Linear Equations and Inequalities in One Variable
Notes on the Difference of 2 Squares
Solving Absolute Value Inequalities
Solving Quadratic Equations
Factoring Polynomials Completely
Using Slopes to Graph Lines
Fractions, Decimals and Percents
Solving Systems of Equations by Substitution
Quotient Rule for Radicals
Prime Polynomials
Solving Nonlinear Equations by Substitution
Simplifying Radical Expressions Containing One Term
Factoring a Sum or Difference of Two Cubes
Finding the Least Common Denominator of Rational Expressions
Multiplying Rational Expressions
Expansion of a Product of Binomials
Solving Equations
Exponential Growth
Factoring by Grouping
Solving One-Step Equations Using Models
Solving Quadratic Equations by Factoring
Adding and Subtracting Polynomials
Rationalizing the Denominator
Rounding Off
The Distributive Property
What is a Quadratic Equation
Laws of Exponents and Multiplying Monomials
The Slope of a Line
Factoring Trinomials by Grouping
Multiplying and Dividing Rational Expressions
Solving Linear Inequalities
Multiplication Property of Exponents
Multiplying and Dividing Fractions 3
Dividing Monomials
Multiplying Polynomials
Adding and Subtracting Functions
Dividing Polynomials
Absolute Value and Distance
Multiplication and Division with Mixed Numbers
Factoring a Polynomial by Finding the GCF
Adding and Subtracting Polynomials
The Rectangular Coordinate System
Polar Form of a Complex Number
Exponents and Order of Operations
Graphing Horizontal and Vertical Lines
Invariants Under Rotation
The Addition Method
Solving Linear Inequalities in One Variable
The Pythagorean Theorem
Try the Free Math Solver or Scroll down to Tutorials!












Please use this form if you would like
to have this math solver on your website,
free of charge.

Invariants Under Rotation

Have a look at the following theorem:


Rotation of Axes

The general equation of the conic

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0

where B ≠ 0, can be rewritten as

A'(x')2 + B'x'y' + C'(y')2 + D'x' + E'y' + F' = 0

by rotating the coordinate axes through an angle θ, where

Note that the constant term is the same in both equations. Because of this, F is said to be invariant under rotation. The following theorem lists some other rotation invariants. The proof of this theorem is left as an exercise.


Theorem 2

Rotation Invariants

The rotation of coordinate axes through an angle θ that transforms the equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 into the form

A'(x')2 + B'x'y' + C'(y')2 + D'x' + E'y' + F' = 0

has the following rotation invariants.

1. F = F'

2. A + C = A' + C'

3. B2 - 4AC = (B')2 - 4A'C'

You can use this theorem to classify the graph of a second-degree equation with an xy-term in much the same way you do for a second-degree equation without an xy-term. Note that because B' = 0, the invariant B2 - 4AC reduces to

B2 - 4AC = - 4A'C' Discriminant

which is called the discriminant of the equation

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0

Because the sign of A'C' determines the type of graph for the equation

A'(x')2 + B'x'y' + C'(y')2 + D'x' + E'y' + F' = 0

the sign of B2 - 4AC must determine the type of graph for the original equation. This result is stated in the following theorem.


Theorem 3

Classification of Conics by the Discriminant

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0

is, except in degenerate cases, determined by its discriminant as follows.

1. Ellipse or circle B2 - 4AC < 0
2. Parabola B2 - 4AC = 0
3. Hyperbola B2 - 4AC > 0



Using the Discriminant

Classify the graph of each of the following equations.

a. 4xy - 9 = 0

b. 2x2 - 3xy + 2y2 - 2x = 0

c. x2 - 6xy + 9y2 - 2y + 1 = 0

d. 3x2 + 8xy + 4y2 - 7 = 0


a. The graph is a hyperbola because

B2 - 4AC = 16 - 0 > 0.

b. The graph is a circle or an ellipse because

B2 - 4AC = 9 - 16 < 0.

c. The graph is a parabola because

B2 - 4AC = 36 - 36 = 0

d. The graph is a hyperbola because

B2 - 4AC = 64 - 48 > 0

All Right Reserved. Copyright 2005-2024