 Chapter 9: Circles and Parabolas
 Chapter 9: Circles and Parabolas
 Chapter 1: Lines in the Plane
 Chapter 1.1: Lines in the Plane
 Chapter 1.2: Functions
 Chapter 1.3: Graphs of Functions
 Chapter 1.4: Shifting, Reflecting, and Stretching Graphs
 Chapter 1.5: Combinations of Functions
 Chapter 1.6: Inverse Functions
 Chapter 1.7: Linear Models and Scatter Plots
 Chapter 10: Analytic Geometry in Three Dimensions
 Chapter 10.1: The ThreeDimensional Coordinate System
 Chapter 10.2: Vectors in Space
 Chapter 10.3: The Cross Product of Two Vectors
 Chapter 10.4: Lines and Planes in Space
 Chapter 11: Limits and an Introduction to Calculus
 Chapter 11.1: Introduction to Limits
 Chapter 11.2: Techniques for Evaluating Limits
 Chapter 11.3: The Tangent Line Problem
 Chapter 11.4: Limits at Infinity and Limits of Sequences
 Chapter 11.5: The Area Problem
 Chapter 2: Polynomial and Rational Functions
 Chapter 2.1: Quadratic Functions
 Chapter 2.2: Polynomial Functions of Higher Degree
 Chapter 2.3: Real Zeros of Polynomial Functions
 Chapter 2.4: Complex Numbers
 Chapter 2.5: The Fundamental Theorem of Algebra
 Chapter 2.6: Rational Functions and Asymptotes
 Chapter 2.7: Graphs of Rational Functions
 Chapter 2.8: Quadratic Models
 Chapter 3: Exponential and Logarithmic Functions
 Chapter 3.1: Exponential Functions and Their Graphs
 Chapter 3.2: Logarithmic Functions and Their Graphs
 Chapter 3.3: Properties of Logarithms
 Chapter 3.4: Solving Exponential and Logarithmic Equations
 Chapter 3.5: Exponential and Logarithmic Models
 Chapter 3.6: Nonlinear Models
 Chapter 4: Trigonometric Functions
 Chapter 4.1: Radian and Degree Measure
 Chapter 4.2: Trigonometric Functions: The Unit Circle
 Chapter 4.3: Right Triangle Trigonometry
 Chapter 4.4: Trigonometric Functions of Any Angle
 Chapter 4.5: Graphs of Sine and Cosine Functions
 Chapter 4.6: Graphs of Other Trigonometric Functions
 Chapter 4.7: Inverse Trigonometric Functions
 Chapter 4.8: Applications and Models
 Chapter 5: Analytic Trigonometry
 Chapter 5.1: Using Fundamental Identities
 Chapter 5.2: Verifying Trigonometric Identities
 Chapter 5.3: Solving Trigonometric Equations
 Chapter 5.4: Sum and Difference Formulas
 Chapter 5.5: MultipleAngle and ProducttoSum Formulas
 Chapter 6: Additional Topics in Trigonometry
 Chapter 6.1: Law of Sines
 Chapter 6.2: Law of Cosines
 Chapter 6.3: Vectors in the Plane
 Chapter 6.4: Vectors and Dot Products
 Chapter 6.5: Trigonometric Form of a Complex Number
 Chapter 7: Linear Systems and Matrices
 Chapter 7.1: Solving Systems of Equations
 Chapter 7.2: Systems of Linear Equations in Two Variables
 Chapter 7.3: Multivariable Linear Systems
 Chapter 7.4: Matrices and Systems of Equations
 Chapter 7.5: Operations with Matrices
 Chapter 7.6: The Inverse of a Square Matrix
 Chapter 7.7: The Determinant of a Square Matrix
 Chapter 7.8: Phase Shift; Sinusoidal Curve Fitting
 Chapter 8: Sequences, Series, and Probability
 Chapter 8.1: Sequences, Series, and Probability
 Chapter 8.2: Arithmetic Sequences and Partial Sums
 Chapter 8.3: Geometric Sequences and Series
 Chapter 8.4: Mathematical Induction
 Chapter 8.5: The Binomial Theorem
 Chapter 8.6: Counting Principles
 Chapter 8.7: Probability
 Chapter 9: Circles and Parabolas
 Chapter 9.1: Circles and Parabolas
 Chapter 9.2: Ellipses
 Chapter 9.3: Hyperbolas
 Chapter 9.4: Rotation and Systems of Quadratic Equations
 Chapter 9.5: Parametric Equations
 Chapter 9.6: Polar Coordinates
 Chapter 9.7: Graphs of Polar Equations
 Chapter 9.8: Polar Equations of Conics
Precalculus With Limits A Graphing Approach 5th Edition  Solutions by Chapter
Full solutions for Precalculus With Limits A Graphing Approach  5th Edition
ISBN: 9780618851522
Precalculus With Limits A Graphing Approach  5th Edition  Solutions by Chapter
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Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Iterative method.
A sequence of steps intended to approach the desired solution.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.