 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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Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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 inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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).

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

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

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

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.