Exponential Functions
Linera Equations
Simple Trinomials as Products of Binomials
Laws of Exponents and Dividing Monomials
Solving Equations
Multiplying Polynomials
Multiplying and Dividing Rational Expressions
Solving Systems of Linear Inequalities
Mixed-Number Notation
Linear Equations and Inequalities in One Variable
The Quadratic Formula
Fractions and Decimals
Graphing Logarithmic Functions
Multiplication by 111
Solving Systems of Equations - Two Lines
Solving Nonlinear Equations by Factoring
Solving Linear Systems of Equations by Elimination
Rationalizing the Denominator
Simplifying Complex Fractions
Factoring Trinomials
Linear Relations and Functions
Axis of Symmetry and Vertices
Equations Quadratic in Form
The Appearance of a Polynomial Equation
Subtracting Reverses
Non-Linear Equations
Exponents and Order of Operations
Factoring Trinomials by Grouping
Factoring Trinomials of the Type ax 2 + bx + c
The Distance Formula
Invariants Under Rotation
Multiplying and Dividing Monomials
Solving a System of Three Linear Equations by Elimination
Multiplication by 25
Powers of i
Solving Quadratic and Polynomial Equations
Slope-intercept Form for the Equation of a Line
Equations of Lines
Square Roots
Integral Exponents
Product Rule for Radicals
Solving Compound Linear Inequalities
Axis of Symmetry and Vertices
Multiplying Rational Expressions
Reducing Rational Expressions
Properties of Negative Exponents
Numbers, Factors, and Reducing Fractions to Lowest Terms
Solving Quadratic Equations
Factoring Completely General Quadratic Trinomials
Solving a Formula for a Given Variable
Factoring Polynomials
Decimal Numbers and Fractions
Multiplication Properties of Exponents
Multiplying Fractions
Multiplication by 50

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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