 3.5.1: Suppose that the graph of a function is known. Then the graph of ma...
 3.5.2: Suppose that the graph of a function is known. Then the graph of ma...
 3.5.3: Suppose that the graph of a function g is known. The graph of may b...
 3.5.4: True or False The graph of is the reflection about the xaxis of th...
 3.5.5: True or False To obtain the graph of shift the graph of horizontall...
 3.5.6: True or False To obtain the graph of shift the graph of y = x verti...
 3.5.7: In 718, match each graph to one of the following functions:A. B. C....
 3.5.8: In 718, match each graph to one of the following functions:A. B. C....
 3.5.9: In 718, match each graph to one of the following functions:A. B. C....
 3.5.10: In 718, match each graph to one of the following functions:A. B. C....
 3.5.11: In 718, match each graph to one of the following functions:A. B. C....
 3.5.12: In 718, match each graph to one of the following functions:A. B. C....
 3.5.13: In 718, match each graph to one of the following functions:A. B. C....
 3.5.14: In 718, match each graph to one of the following functions:A. B. C....
 3.5.15: In 718, match each graph to one of the following functions:A. B. C....
 3.5.16: In 718, match each graph to one of the following functions:A. B. C....
 3.5.17: In 718, match each graph to one of the following functions:A. B. C....
 3.5.18: In 718, match each graph to one of the following functions:A. B. C....
 3.5.19: In 1926, write the function whose graph is the graph of y = x , but...
 3.5.20: In 1926, write the function whose graph is the graph of y = x , but...
 3.5.21: In 1926, write the function whose graph is the graph of y = x , but...
 3.5.22: In 1926, write the function whose graph is the graph of y = x , but...
 3.5.23: In 1926, write the function whose graph is the graph of y = x , but...
 3.5.24: In 1926, write the function whose graph is the graph of y = x , but...
 3.5.25: In 1926, write the function whose graph is the graph of y = x , but...
 3.5.26: In 1926, write the function whose graph is the graph of y = x , but...
 3.5.27: In 2730, find the function that is finally graphed after each of th...
 3.5.28: In 2730, find the function that is finally graphed after each of th...
 3.5.29: In 2730, find the function that is finally graphed after each of th...
 3.5.30: In 2730, find the function that is finally graphed after each of th...
 3.5.31: If is a point on the graph of which of the following points must be...
 3.5.32: If is a point on the graph of which of the following points must be...
 3.5.33: If is a point on the graph of which of the following points must be...
 3.5.34: If is a point on the graph of which of the following points must be...
 3.5.35: Suppose that the xintercepts of the graph of are and 3. (a) What a...
 3.5.36: Suppose that the xintercepts of the graph of are and 1. (a) What a...
 3.5.37: Suppose that the function is increasing on the interval (a) Over wh...
 3.5.38: Suppose that the function is decreasing on the interval (a) Over wh...
 3.5.39: n 3962, graph each function using the techniques of shifting, compr...
 3.5.40: n 3962, graph each function using the techniques of shifting, compr...
 3.5.41: n 3962, graph each function using the techniques of shifting, compr...
 3.5.42: n 3962, graph each function using the techniques of shifting, compr...
 3.5.43: n 3962, graph each function using the techniques of shifting, compr...
 3.5.44: n 3962, graph each function using the techniques of shifting, compr...
 3.5.45: n 3962, graph each function using the techniques of shifting, compr...
 3.5.46: n 3962, graph each function using the techniques of shifting, compr...
 3.5.47: n 3962, graph each function using the techniques of shifting, compr...
 3.5.48: n 3962, graph each function using the techniques of shifting, compr...
 3.5.49: n 3962, graph each function using the techniques of shifting, compr...
 3.5.50: n 3962, graph each function using the techniques of shifting, compr...
 3.5.51: n 3962, graph each function using the techniques of shifting, compr...
 3.5.52: n 3962, graph each function using the techniques of shifting, compr...
 3.5.53: n 3962, graph each function using the techniques of shifting, compr...
 3.5.54: n 3962, graph each function using the techniques of shifting, compr...
 3.5.55: n 3962, graph each function using the techniques of shifting, compr...
 3.5.56: n 3962, graph each function using the techniques of shifting, compr...
 3.5.57: n 3962, graph each function using the techniques of shifting, compr...
 3.5.58: n 3962, graph each function using the techniques of shifting, compr...
 3.5.59: n 3962, graph each function using the techniques of shifting, compr...
 3.5.60: n 3962, graph each function using the techniques of shifting, compr...
 3.5.61: n 3962, graph each function using the techniques of shifting, compr...
 3.5.62: n 3962, graph each function using the techniques of shifting, compr...
 3.5.63: In 6366, the graph of a function is illustrated. Use the graph of a...
 3.5.64: In 6366, the graph of a function is illustrated. Use the graph of a...
 3.5.65: In 6366, the graph of a function is illustrated. Use the graph of a...
 3.5.66: In 6366, the graph of a function is illustrated. Use the graph of a...
 3.5.67: In 6774, complete the square of each quadratic expression. Then gra...
 3.5.68: In 6774, complete the square of each quadratic expression. Then gra...
 3.5.69: In 6774, complete the square of each quadratic expression. Then gra...
 3.5.70: In 6774, complete the square of each quadratic expression. Then gra...
 3.5.71: In 6774, complete the square of each quadratic expression. Then gra...
 3.5.72: In 6774, complete the square of each quadratic expression. Then gra...
 3.5.73: In 6774, complete the square of each quadratic expression. Then gra...
 3.5.74: In 6774, complete the square of each quadratic expression. Then gra...
 3.5.75: The equation defines a family of parabolas, one parabola for each v...
 3.5.76: Repeat for the family of parabolasy = x2 + c.
 3.5.77: Thermostat Control Energy conservation experts estimate that homeow...
 3.5.78: digital music revenues R, in millions of dollars, for the years 200...
 3.5.79: emperature Measurements The relationship between the Celsius (C) an...
 3.5.80: Period of a Pendulum The period T (in seconds) of a simple pendulum...
 3.5.81: Cigar Company Profits The daily profits of a cigar company from sel...
 3.5.82: The graph of a function is illustrated in the figure. (a) Draw the ...
 3.5.83: The graph of a function is illustrated in the figure. (a) Draw the ...
 3.5.84: Suppose 1, 3 is a point on the graph of . (a) What point is on the ...
 3.5.85: Suppose is a point on the graph of . (a) What point is on the graph...
 3.5.86: Suppose that the graph of a function is known. Explain how the grap...
 3.5.87: Suppose that the graph of a function f is known. Explain how the gr...
 3.5.88: he area under the curve bounded below by the xaxis and on the righ...
 3.5.89: ertical Shifts Open the vertical shift applet. Use your mouse to gr...
 3.5.90: Horizontal Shifts Open the horizontal shift applet. Use your mouse ...
 3.5.91: Vertical Stretches Open the vertical stretch applet. Use your mouse...
 3.5.92: Horizontal Stretches Open the horizontal stretch applet. (a) Use yo...
 3.5.93: Reflection about the yaxis Open the reflection about the yaxis ap...
 3.5.94: Reflection about the xaxis Open the reflection about the xaxis ap...
Solutions for Chapter 3.5: Algebra and Trigonometry 9th Edition
Full solutions for Algebra and Trigonometry  9th Edition
ISBN: 9780321716569
Solutions for Chapter 3.5
Get Full SolutionsAlgebra and Trigonometry was written by and is associated to the ISBN: 9780321716569. Chapter 3.5 includes 94 full stepbystep solutions. This textbook survival guide was created for the textbook: Algebra and Trigonometry, edition: 9. This expansive textbook survival guide covers the following chapters and their solutions. Since 94 problems in chapter 3.5 have been answered, more than 50645 students have viewed full stepbystep solutions from this chapter.

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.)

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

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

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

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

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

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

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

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.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

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