- 4.3.Problem 1: Graph y 4 3 cos x.
- 4.3.a: How does the term affect the graph of y sin x ?
- 4.3.1: The value k in y k sin x represents a __________ translation. If k ...
- 4.3.Problem 2: Graph y 120 110 cos 1 2 t . 2
- 4.3.b: How does the term affect the graph of y sin x ?
- 4.3.2: The value h in y sin (x h) represents a _________ translation. If h...
- 4.3.Problem 3:
- 4.3.c: How do you find the period for the graph of y A sin (Bx C)?
- 4.3.3: The value C in y sin (Bx C) is called the _________.
- 4.3.Problem 4: Graph one cycle of y sin x 2 3
- 4.3.d: How do you find the horizontal translation for the graph of y A sin...
- 4.3.4: The graph of y sin (Bx C) will have a horizontal shift of _________.
- 4.3.Problem 5: Graph y 4 sin 3x 2 for 0 x 2. x
- 4.3.5: y 5 sin x
- 4.3.Problem 6: Graph one cycle of y 2 4 cos 2x 3 .
- 4.3.6: y 2 cos x
- 4.3.7: y 1 4 cos x 8
- 4.3.8: y 1 2 sin x G
- 4.3.9: y 2 sin x
- 4.3.10: y 4 sin x
- 4.3.11: y 5 cos x 1
- 4.3.12: y 1 cos x
- 4.3.13: y 3 sin x 1
- 4.3.14: y 6 sin x
- 4.3.15: y 1 2 cos x 1
- 4.3.16: y 3 2 cos x G
- 4.3.17: y 4 4 sin 2x
- 4.3.18: y 2 2 sin 4x
- 4.3.19: y 1 cos 3x 20
- 4.3.20: y 1 sin 3x
- 4.3.21: y cos (x ) 2
- 4.3.22: y cos (x ) 2
- 4.3.23: sin x 2 3 24.
- 4.3.24: y sin x 3 4 G
- 4.3.25: y sin x 2
- 4.3.26: y sin x
- 4.3.27: y sin x 28
- 4.3.28: y sin x
- 4.3.29: y sin x
- 4.3.30: y sin x
- 4.3.31: y cos x
- 4.3.32: y cos x
- 4.3.33: y cos x
- 4.3.34: y cos x
- 4.3.35: y cos 3 x 3
- 4.3.36: y cos 2 x 2 37
- 4.3.37: y sin (6x ) 38
- 4.3.38: y sin (4x )
- 4.3.39: y 3 sin 1 2 x 6 40
- 4.3.40: y 2 cos 1 3 x 6 F
- 4.3.41: y sin (2x ) 42
- 4.3.42: y sin (2x )
- 4.3.43: y sin x 44
- 4.3.44: y sin x
- 4.3.45: cos 2x 46
- 4.3.46: y cos 2x 4
- 4.3.47: 2 sin x 4
- 4.3.48: y 3 cos x
- 4.3.49: y cos 3x
- 4.3.50: y cos
- 4.3.51: y 3 sin
- 4.3.52: y 3 cos
- 4.3.53: y 1 sin (2x ) 54
- 4.3.54: y 1 sin (2x ) 5
- 4.3.55: 3 sin x 56
- 4.3.56: y 3 sin x
- 4.3.57: y 2 cos 2x 58
- 4.3.58: y 2 cos 2x G
- 4.3.59: y 2 3 cos x 60.
- 4.3.60: y 3 2 sin x 6
- 4.3.61: y sin (3x ) 62
- 4.3.62: y cos (3x ) Gr
- 4.3.63: y 4 cos 2x , x 6
- 4.3.64: y sin 3x , x 65
- 4.3.65: y 3 cos x , 2 x 2 66.
- 4.3.66: y 2 cos x , x
- 4.3.67: Oscillating Spring A mass attached to a spring oscillates upward an...
- 4.3.68: Oscillating Spring A mass attached to a spring oscillates upward an...
- 4.3.69: Sound Wave The oscillations in air pressure representing the sound ...
- 4.3.70: RLC Circuit The electric current in an RLC circuit can be modeled b...
- 4.3.71: Arc Length Find the length of arc cut off by a central angle of /6 ...
- 4.3.72: Arc Length How long is the arc cut off by a central angle of 90 in ...
- 4.3.73: Radius of a Circle Find the radius of a circle if a central angle o...
- 4.3.74: Radius of a Circle In a circle, a central angle of 135 cuts off an ...
- 4.3.75: Sketch the graph of y 2 cos 2 x 2 . Which of the following matches ...
- 4.3.76: Identify the vertical translation for the graph of y 1 4 cos 3x 2 ....
- 4.3.77: Identify the horizontal translation for the graph of y 1 4 cos 3x 2...
- 4.3.78: Identify the phase for the graph of y 1 4 cos 3x 2 . a. 2 b. 6 c. 2...
Solutions for Chapter 4.3: Vertical and Horizontal Translations
Full solutions for Trigonometry | 7th Edition
ISBN: 9781111826857
Solutions for Chapter 4.3
20
5
This textbook survival guide was created for the textbook: Trigonometry, edition: 7. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 4.3: Vertical and Horizontal Translations includes 88 full step-by-step solutions. Trigonometry was written by and is associated to the ISBN: 9781111826857. Since 88 problems in chapter 4.3: Vertical and Horizontal Translations have been answered, more than 41606 students have viewed full step-by-step solutions from this chapter.
Key Math Terms and definitions covered in this textbook
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Affine transformation
Tv = Av + Vo = linear transformation plus shift.
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Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
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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.)
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Cofactor Cij.
Remove row i and column j; multiply the determinant by (-I)i + j •
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Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
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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.
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Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
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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.
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Gram-Schmidt 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.
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Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
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Length II x II.
Square root of x T x (Pythagoras in n dimensions).
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Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.
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Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.
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Pseudoinverse A+ (Moore-Penrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).
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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.
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Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.
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Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
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Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
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Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.
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Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.