 4.2.1: For the rational function given by if the degree of is exactly one ...
 4.2.2: The graph of has a ________ asymptote at
 4.2.3: g x 2 x 4
 4.2.4: x 2 x 5 g
 4.2.5: g x 2 x
 4.2.6: g x 1 x 2
 4.2.7: g x 3 x 2 1 2
 4.2.8: g x 3 x 2 g
 4.2.9: g x 3 x 12
 4.2.10: g x 1 x
 4.2.11: g x 3 2 4 x 23
 4.2.12: g x 4 x g x 3 2
 4.2.13: g x 4 x 3
 4.2.14: g x 2 x 3
 4.2.15: f x 1 x 2
 4.2.16: f x 1 x 3 f
 4.2.17: h x 1 x 4
 4.2.18: g x 1 6 x
 4.2.19: C x 7 2x 2 x
 4.2.20: P x 1 3x 1 x C
 4.2.21: g x 1 x 2 2
 4.2.22: f x 2 3 x 2 g
 4.2.23: f x x 2 x 2 9
 4.2.24: f t 1 2t t f
 4.2.25: h x x 2 x 2 9 f
 4.2.26: g x x x 2 9
 4.2.27: s 4s s 2 4
 4.2.28: f x 1 x 22 g
 4.2.29: g x 4 x 1 x x 4
 4.2.30: h x 2 x 2 x 2 g
 4.2.31: f x 2x x 2 3x 4 h
 4.2.32: f x 3x x 2 2x 3 f
 4.2.33: h x x2 5x 4 x2 4 f
 4.2.34: g x x2 2x 8 x2 9 h x
 4.2.35: f x 6x x2 5x 14 g
 4.2.36: f x 3 x2 1 x2 2x 15
 4.2.37: f x 2x2 5x 3 x 3 2x2 x 2 f x
 4.2.38: f x x2 x 2 x 3 2x2 5x 6 f x
 4.2.39: f x x2 3x x2 x 6
 4.2.40: f x 5 x 4 x2 x 12 f
 4.2.41: f x 2x2 5x 2 2x2 x 6 f
 4.2.42: f x 3x2 8x 4 2x2 3x 2 f x
 4.2.43: f t t2 1 t 1 f
 4.2.44: f x x2 36 x 6 f
 4.2.45: f x g x x 1 x
 4.2.46: f x g x x x 2 x 2 x 2 2x , f x g x x 1 x 2 1 x 1 , f g. gf f. f f g...
 4.2.47: f x x 2 x 2 2x , f
 4.2.48: f x x 2 x 2 2x , f
 4.2.49: h x x2 9 x
 4.2.50: g x x2 5 x
 4.2.51: f x 2x2 1 x
 4.2.52: f x 1 x2 x
 4.2.53: g x x2 1 x
 4.2.54: h x x2 x 1 g
 4.2.55: f t t 2 1 t 5
 4.2.56: f x x2 3x 1
 4.2.57: f x x 3 x 2 4
 4.2.58: g x x 3 2x 2 8 f
 4.2.59: f x x3 1 x2 x g
 4.2.60: f x x4 x x3
 4.2.61: f x x 2 x 1 x 1 f
 4.2.62: f x 2x 2 5x 5 x 2 f
 4.2.63: f x 2x3 x2 2x 1 x2 3x 2 f
 4.2.64: f x 2x3 x2 8x 4 x2 3x 2 f
 4.2.65: f x x 2 5x 8 x 3
 4.2.66: f x 2x 2 x x 1
 4.2.67: g x 1 3x 2 x 3 x 2
 4.2.68: h x 12 2x x 2 2 4 x g
 4.2.69: 2 8 64 4 2 4 6
 4.2.70: y 2x x 3 y
 4.2.71: y 1 x x
 4.2.72: y x 3 2 x
 4.2.73: y 1 x 5 4 x
 4.2.74: y 20 2 x 1 3 x
 4.2.75: y x 6 x 1 y
 4.2.76: y x 9 x
 4.2.77: CONCENTRATION OF A MIXTURE A 1000liter tank contains 50 liters of ...
 4.2.78: GEOMETRY A rectangular region of length and width has an area of 50...
 4.2.79: PAGE DESIGN A page that is inches wide and inches high contains 30 ...
 4.2.80: PAGE DESIGN A rectangular page is designed to contain 64 square inc...
 4.2.81: f x 3 x 1 x 2 x 1
 4.2.82: C x x 32 x
 4.2.83: MINIMUM COST The ordering and transportation cost (in thousands of ...
 4.2.84: MINIMUM COST The cost of producing units of a product is given by a...
 4.2.85: AVERAGE SPEED A driver averaged 50 miles per hour on the round trip...
 4.2.86: MEDICINE The concentration of a chemical in the bloodstream hours a...
 4.2.87: If the graph of a rational function has a vertical asymptote at it ...
 4.2.88: The graph of a rational function can never cross one of its asympto...
 4.2.89: The graph of has a slant asymptote.
 4.2.90: Every rational function has a horizontal asymptote.
 4.2.91: h x 6 2x 3 x f
 4.2.92: g x x 2 x 2 x 1
 4.2.93: WRITING Given a rational function how can you determine whether has...
 4.2.94: CAPSTONE
Solutions for Chapter 4.2: Graphs of Rational Functions
Full solutions for Algebra and Trigonometry  8th Edition
ISBN: 9781439048474
Solutions for Chapter 4.2: Graphs of Rational Functions
Get Full SolutionsChapter 4.2: Graphs of Rational Functions includes 94 full stepbystep solutions. Since 94 problems in chapter 4.2: Graphs of Rational Functions have been answered, more than 49127 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Algebra and Trigonometry, edition: 8. This expansive textbook survival guide covers the following chapters and their solutions. Algebra and Trigonometry was written by and is associated to the ISBN: 9781439048474.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

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.

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.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

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.

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

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.

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

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

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.

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

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

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

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.