 4.1: f x 3x x 10
 4.2: f x 4x3 2 5x
 4.3: f x 8 x2 10x 24
 4.4: f x x2 x 2 x2 4 f
 4.5: f x 4 x 3
 4.6: f x 2x2 5x 3 x2 2 f
 4.7: g x x2 x2 4
 4.8: g x 1 x 32 g
 4.9: h x 5x 20 x2 2x 24 g
 4.10: h x x3 4x2 x2 3x 2 h
 4.11: AVERAGE COST A business has a production cost of for producing unit...
 4.12: SEIZURE OF ILLEGAL DRUGS The cost (in millions of dollars) for the ...
 4.13: f x 3 2x 2
 4.14: f x 4 x
 4.15: g x 2 x 1 x
 4.16: h x x 4 x 7 g
 4.17: p x 5x2 4x2 1
 4.18: f x 2x x 2 4
 4.19: f x x x 2 1
 4.20: h x 9 x 32 f
 4.21: f x 6x2 x 2 1
 4.22: y 2x 2 x 2 4 f
 4.23: f x 6x2 11x 3 3x2 x y
 4.24: f x 6x2 7x 2 4x2 1 f x
 4.25: f x 2x3 x2 1
 4.26: f x x2 1 x 1
 4.27: f x x2 3x 10 x 2
 4.28: f x x3 x2 25 f
 4.29: f x 3x3 2x2 3x 2 3x2 x 4 f x
 4.30: f x 3x3 4x2 12x 16 3x2 5x 2 f
 4.31: AVERAGE COST The cost of producing units of a product is and the av...
 4.32: PAGE DESIGN A page that is inches wide and inches high contains 30 ...
 4.33: PHOTOSYNTHESIS The amount of uptake (in milligrams per square decim...
 4.34: MEDICINE The concentration of a medication in the bloodstream hours...
 4.35: y 2 16 2 16x
 4.36: 16x2 y y 2 16
 4.37: x2 64 y2 4 1
 4.38: x2 1 y2 36 1 x
 4.39: x 2 400 2 20y 0
 4.40: x2 y x 2 400
 4.41: y2 49 x2 144 1
 4.42: x2 49 y2 144 1 y
 4.43: Passes through the point horizontal axis
 4.44: Passes through the point vertical axis
 4.45: Focus:
 4.46: Focus
 4.47: Directrix:
 4.48: Directrix:
 4.49: SATELLITE ANTENNA A cross section of a large parabolic antenna (see...
 4.50: SUSPENSION BRIDGE Each cable of a suspension bridge is suspended (i...
 4.51: Vertices: ; minor axis of length 6
 4.52: Vertices: ; minor axis of length 2
 4.53: Vertices: ; passes through the point
 4.54: Vertices: ; foci:
 4.55: Foci: ; minor axis of length 10
 4.56: Foci: ; major axis of length 12
 4.57: ARCHITECTURE A semielliptical archway is to be formed over the entr...
 4.58: WADING POOL You are building a wading pool that is in the shape of ...
 4.59: Vertices: ; foci:
 4.60: Vertices: ; foci
 4.61: Vertices: ; asymptotes
 4.62: Vertices: ; asymptotes:
 4.63: Vertex: directrix:
 4.64: Focus: directrix:
 4.65: Vertex: focus:
 4.66: Vertex: focus:
 4.67: Vertices: passes through the point
 4.68: Center: vertices:
 4.69: Vertices: foci:
 4.70: Vertices: foci:
 4.71: Vertices: asymptotes:
 4.72: Vertices: passes through the point
 4.73: Vertices: foci:
 4.74: Vertices: foci:
 4.75: Foci: asymptotes:
 4.76: Foci: asymptotes:
 4.77: 2 6x 2y 9 0
 4.78: y 2 12y 8x 20 0 x
 4.79: x 2 y 2 2x 4y 5 0 y
 4.80: 16x 2 16y 2 16x 24y 3 0 x
 4.81: x 2 9y 2 10x 18y 25 0
 4.82: 4x 2 y 2 16x 15 0
 4.83: 9x2 y2 72x 8y 119 0 4
 4.84: x 2 9y 2 10x 18y 7 0
 4.85: ARCHITECTURE A parabolic archway is 12 meters high at the vertex. A...
 4.86: ARCHITECTURE A church window (see figure) is bounded above by a par...
 4.87: RUNNING PATH Let represent a water fountain located in a city park....
 4.88: The domain of a rational function can never be the set of all real ...
 4.89: The graph of the equation can be a single point. Ax Ey F 0
Solutions for Chapter 4: Rational Functions and Conics
Full solutions for Algebra and Trigonometry  8th Edition
ISBN: 9781439048474
Solutions for Chapter 4: Rational Functions and Conics
Get Full SolutionsAlgebra and Trigonometry was written by and is associated to the ISBN: 9781439048474. This expansive textbook survival guide covers the following chapters and their solutions. Since 89 problems in chapter 4: Rational Functions and Conics have been answered, more than 47711 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Algebra and Trigonometry, edition: 8. Chapter 4: Rational Functions and Conics includes 89 full stepbystep solutions.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

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.

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

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

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

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

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).

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.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

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.

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

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