 7.8.7.8.1: Fill in the blanks. Three points are _______ if they lie on the sam...
 7.8.7.8.2: Fill in the blanks. The method of using determinants to solve a sys...
 7.8.7.8.3: Fill in the blanks. A message written according to a secret code is...
 7.8.7.8.4: Fill in the blanks. A message written according to a secret code is...
 7.8.7.8.5: In Exercises 16, use a determinant to find the area of the figure w...
 7.8.7.8.6: In Exercises 16, use a determinant to find the area of the figure w...
 7.8.7.8.7: In Exercises 7 and 8, find x such that the triangle has an area of ...
 7.8.7.8.8: In Exercises 7 and 8, find x such that the triangle has an area of ...
 7.8.7.8.9: In Exercises 912, use a determinant to determine whether the points...
 7.8.7.8.10: In Exercises 912, use a determinant to determine whether the points...
 7.8.7.8.11: In Exercises 912, use a determinant to determine whether the points...
 7.8.7.8.12: In Exercises 912, use a determinant to determine whether the points...
 7.8.7.8.13: In Exercises 13 and 14, find x such that the points are collinear
 7.8.7.8.14: In Exercises 13 and 14, find x such that the points are collinear
 7.8.7.8.15: In Exercises 1522, use Cramers Rule to solve (if possible) the syst...
 7.8.7.8.16: In Exercises 1522, use Cramers Rule to solve (if possible) the syst...
 7.8.7.8.17: In Exercises 1522, use Cramers Rule to solve (if possible) the syst...
 7.8.7.8.18: In Exercises 1522, use Cramers Rule to solve (if possible) the syst...
 7.8.7.8.19: In Exercises 1522, use Cramers Rule to solve (if possible) the syst...
 7.8.7.8.20: In Exercises 1522, use Cramers Rule to solve (if possible) the syst...
 7.8.7.8.21: In Exercises 1522, use Cramers Rule to solve (if possible) the syst...
 7.8.7.8.22: In Exercises 1522, use Cramers Rule to solve (if possible) the syst...
 7.8.7.8.23: In Exercises 2326, solve the system of equations using (a) Gaussian...
 7.8.7.8.24: In Exercises 2326, solve the system of equations using (a) Gaussian...
 7.8.7.8.25: In Exercises 2326, solve the system of equations using (a) Gaussian...
 7.8.7.8.26: In Exercises 2326, solve the system of equations using (a) Gaussian...
 7.8.7.8.27: Sports The average salaries (in thousands of dollars) for football ...
 7.8.7.8.28: Retail Sales The retail sales (in millions of dollars) for lawn car...
 7.8.7.8.29: In Exercises 29 and 30, write the uncoded row matrices for the mess...
 7.8.7.8.30: In Exercises 29 and 30, write the uncoded row matrices for the mess...
 7.8.7.8.31: In Exercises 31 and 32, write a cryptogram for the message using th...
 7.8.7.8.32: In Exercises 31 and 32, write a cryptogram for the message using th...
 7.8.7.8.33: In Exercises 3335, use to decode the cryptogram.
 7.8.7.8.34: In Exercises 3335, use to decode the cryptogram.
 7.8.7.8.35: In Exercises 3335, use to decode the cryptogram.
 7.8.7.8.36: . The following cryptogram was encoded with a matrix. 8 21 5 10 5 2...
 7.8.7.8.37: True or False? In Exercises 37 and 38, determine whether the statem...
 7.8.7.8.38: True or False? In Exercises 37 and 38, determine whether the statem...
 7.8.7.8.39: Writing At this point in the book, you have learned several methods...
 7.8.7.8.40: Writing Use your schools library, the Internet, or some other refer...
 7.8.7.8.41: In Exercises 4144, find the general form of the equation of the lin...
 7.8.7.8.42: In Exercises 4144, find the general form of the equation of the lin...
 7.8.7.8.43: In Exercises 4144, find the general form of the equation of the lin...
 7.8.7.8.44: In Exercises 4144, find the general form of the equation of the lin...
 7.8.7.8.45: In Exercises 45 and 46, sketch the graph of the rational function. ...
 7.8.7.8.46: In Exercises 45 and 46, sketch the graph of the rational function. ...
Solutions for Chapter 7.8: Phase Shift; Sinusoidal Curve Fitting
Full solutions for Precalculus With Limits A Graphing Approach  5th Edition
ISBN: 9780618851522
Solutions for Chapter 7.8: Phase Shift; Sinusoidal Curve Fitting
Get Full SolutionsThis textbook survival guide was created for the textbook: Precalculus With Limits A Graphing Approach, edition: 5. This expansive textbook survival guide covers the following chapters and their solutions. Since 46 problems in chapter 7.8: Phase Shift; Sinusoidal Curve Fitting have been answered, more than 36110 students have viewed full stepbystep solutions from this chapter. Precalculus With Limits A Graphing Approach was written by and is associated to the ISBN: 9780618851522. Chapter 7.8: Phase Shift; Sinusoidal Curve Fitting includes 46 full stepbystep solutions.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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 II· 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.

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

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

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

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Solvable system Ax = b.
The right side b is in the column space of A.

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

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