 8.4.1: What effect does the xy term have on the graph of a conic section?
 8.4.2: If the equation of a conic section is written in the form Ax 2 + By...
 8.4.3: If the equation of a conic section is written in the form Ax 2 + Bx...
 8.4.4: Given the equation ax 2 + 4x + 3y 2 12 = 0, what can we conclude if...
 8.4.5: For the equation Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0, the value of ...
 8.4.6: For the following exercises, determine which conic section is repre...
 8.4.7: For the following exercises, determine which conic section is repre...
 8.4.8: For the following exercises, determine which conic section is repre...
 8.4.9: For the following exercises, determine which conic section is repre...
 8.4.10: For the following exercises, determine which conic section is repre...
 8.4.11: For the following exercises, determine which conic section is repre...
 8.4.12: For the following exercises, determine which conic section is repre...
 8.4.13: For the following exercises, determine which conic section is repre...
 8.4.14: For the following exercises, determine which conic section is repre...
 8.4.15: For the following exercises, determine which conic section is repre...
 8.4.16: For the following exercises, determine which conic section is repre...
 8.4.17: For the following exercises, determine which conic section is repre...
 8.4.18: For the following exercises, find a new representation of the given...
 8.4.19: For the following exercises, find a new representation of the given...
 8.4.20: For the following exercises, find a new representation of the given...
 8.4.21: For the following exercises, find a new representation of the given...
 8.4.22: For the following exercises, find a new representation of the given...
 8.4.23: For the following exercises, determine the angle that will eliminat...
 8.4.24: For the following exercises, determine the angle that will eliminat...
 8.4.25: For the following exercises, determine the angle that will eliminat...
 8.4.26: For the following exercises, determine the angle that will eliminat...
 8.4.27: For the following exercises, determine the angle that will eliminat...
 8.4.28: For the following exercises, determine the angle that will eliminat...
 8.4.29: For the following exercises, determine the angle that will eliminat...
 8.4.30: For the following exercises, determine the angle that will eliminat...
 8.4.31: For the following exercises, rotate through the given angle based o...
 8.4.32: For the following exercises, rotate through the given angle based o...
 8.4.33: For the following exercises, rotate through the given angle based o...
 8.4.34: For the following exercises, rotate through the given angle based o...
 8.4.35: For the following exercises, rotate through the given angle based o...
 8.4.36: For the following exercises, rotate through the given angle based o...
 8.4.37: For the following exercises, rotate through the given angle based o...
 8.4.38: For the following exercises, rotate through the given angle based o...
 8.4.39: For the following exercises, graph the equation relative to the x y...
 8.4.40: For the following exercises, graph the equation relative to the x y...
 8.4.41: For the following exercises, graph the equation relative to the x y...
 8.4.42: For the following exercises, graph the equation relative to the x y...
 8.4.43: For the following exercises, graph the equation relative to the x y...
 8.4.44: For the following exercises, graph the equation relative to the x y...
 8.4.45: For the following exercises, graph the equation relative to the x y...
 8.4.46: For the following exercises, graph the equation relative to the x y...
 8.4.47: For the following exercises, graph the equation relative to the x y...
 8.4.48: For the following exercises, graph the equation relative to the x y...
 8.4.49: For the following exercises, graph the equation relative to the x y...
 8.4.50: For the following exercises, determine the angle of rotation in ord...
 8.4.51: For the following exercises, determine the angle of rotation in ord...
 8.4.52: For the following exercises, determine the angle of rotation in ord...
 8.4.53: For the following exercises, determine the angle of rotation in ord...
 8.4.54: For the following exercises, determine the angle of rotation in ord...
 8.4.55: For the following exercises, determine the angle of rotation in ord...
 8.4.56: For the following exercises, determine the value of k based on the ...
 8.4.57: For the following exercises, determine the value of k based on the ...
 8.4.58: For the following exercises, determine the value of k based on the ...
 8.4.59: For the following exercises, determine the value of k based on the ...
 8.4.60: For the following exercises, determine the value of k based on the ...
Solutions for Chapter 8.4: ROTATION OF AXIS
Full solutions for College Algebra  1st Edition
ISBN: 9781938168383
Solutions for Chapter 8.4: ROTATION OF AXIS
Get Full SolutionsSince 60 problems in chapter 8.4: ROTATION OF AXIS have been answered, more than 31842 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: College Algebra, edition: 1. College Algebra was written by and is associated to the ISBN: 9781938168383. Chapter 8.4: ROTATION OF AXIS includes 60 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

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.)

Column space C (A) =
space of all combinations of the columns of A.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib IIĀ· Condition numbers measure the sensitivity of the output to change in the input.

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.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

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.