 142.1: y = sin (  _ 2 )
 142.2: y = tan ( + 60)
 142.3: y = cos (  45)
 142.4: y = sec ( + _ 3 )
 142.5: y = cos + _1 4
 142.6: y = sec  5
 142.7: y = tan + 4
 142.8: y = sin + 0.25
 142.9: y = 3 sin [2(  30)] + 10
 142.10: y = 2 cot (3 + 135)  6
 142.11: y = _1 2 sec [4 (  _ 4 )] + 1
 142.12: y = _2 3 cos[_1 2( + _ 6 )]  2
 142.13: Determine the vertical shift, amplitu de, and period of a function ...
 142.14: Write the equation for the height h of the weight above the floor a...
 142.15: Draw a graph of the function you wrote in Exercise 14.
 142.16: y = cos ( + 90)
 142.17: y = cot (  30)
 142.18: y = sin (  _ 4 )
 142.19: y = cos ( + _ 3 )
 142.20: y = _1 4 tan ( + 22.5)
 142.21: y = 3 sin (  75)
 142.22: y = sin  1
 142.23: y = sec + 2
 142.24: y = cos  5
 142.25: y = csc  _3 4
 142.26: y = _1 2 sin + _1 2
 142.27: y = 6 cos + 1.5
 142.28: y = 2 sin [3(  45)] + 1
 142.29: y = 4 cos [2( + 30)]  5
 142.30: y = 3 csc [ _1 2 ( + 60) ]  3.5
 142.31: y = 6 cot [ _2 3 (  90)] + 0.75
 142.32: y = _1 4 cos (2  150) + 1
 142.33: y = _2 5 tan (6 + 135) 4
 142.34: y = 3 + 2 sin [(2 + _ 4 )]
 142.35: y = 4 + 5 sec [ _1 3( + _2 3 )]
 142.36: Find the maximum number of owls. After how many years does this occur?
 142.37: What is the minimum number of mice? How long does it take for the p...
 142.38: Why would the maximum owl population follow behind the population o...
 142.39: Graph y = 3  _1 2 cos and y = 3 + _1 2 cos ( + ). How do the graph...
 142.40: Compare the graphs of y = sin [ _1 4(  _ 2 )] and y = cos [ _1 4(...
 142.41: Graph y = 5 + tan ( + _ 4 ). Describe the transformation to the par...
 142.42: Draw a graph of the function y = _2 3 cos (  50) + 2. How does thi...
 142.43: MUSIC When represented on oscilloscope, the note A above middle C h...
 142.44: TIDES The height of the water in a harbor rose to a maximum height ...
 142.45: OPEN ENDED Write the equation of a trigonometric function with a ph...
 142.46: CHALLENGE The graph of y = cot is a transformation of the graph of ...
 142.47: Writing in Math Use the information on page 829 to explain how tran...
 142.48: ACT/SAT Which equation is represented by the graph? A y = cot ( + 4...
 142.49: REVIEW Refer to the figure below. If c = 14, find the value of b. F...
 142.50: y 3 csc
 142.51: y sin _ 2
 142.52: y 3 tan _2 3
 142.53: sin Cos1 _2 3 5
 142.54: cos Cos1 _4 7 55
 142.55: Sin1 sin _5 6 5
 142.56: cos Tan1 _3 4
 142.57: GEOMETRY Find the total number of diagonals that can be drawn in a ...
 142.58: 4x 24 5
 142.59: 4.33x 1 78.5 6
 142.60: 7x 2 53x
 142.61: 3 a 2 _2 a 3 6
 142.62: w + 12 4w 16 _ w + 4 2w 8
 142.63: 3y + 1 2y 10 _1 y2 2y 15
 142.64: cos 150
 142.65: tan 135 6
 142.66: sin _3 2
 142.67: cos 3
 142.68: sin () 6
 142.69: tan _5 6 70
 142.70: cos 225 7
 142.71: tan 405
Solutions for Chapter 142: Translations of Trigonometric Graphs
Full solutions for Algebra 2, Student Edition (MERRILL ALGEBRA 2)  1st Edition
ISBN: 9780078738302
Solutions for Chapter 142: Translations of Trigonometric Graphs
Get Full SolutionsAlgebra 2, Student Edition (MERRILL ALGEBRA 2) was written by and is associated to the ISBN: 9780078738302. This textbook survival guide was created for the textbook: Algebra 2, Student Edition (MERRILL ALGEBRA 2), edition: 1. Since 71 problems in chapter 142: Translations of Trigonometric Graphs have been answered, more than 60719 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 142: Translations of Trigonometric Graphs includes 71 full stepbystep solutions.

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).

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.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

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

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

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.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

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

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).