 3.5.1: Fill in each blank so that the resulting statement is true.All rati...
 3.5.2: Fill in each blank so that the resulting statement is true.True or ...
 3.5.3: Fill in each blank so that the resulting statement is true.True or ...
 3.5.4: Fill in each blank so that the resulting statement is true.If the g...
 3.5.5: Fill in each blank so that the resulting statement is true.If the g...
 3.5.6: Fill in each blank so that the resulting statement is true.True or ...
 3.5.7: Fill in each blank so that the resulting statement is true.Compared...
 3.5.8: Fill in each blank so that the resulting statement is true.The grap...
 3.5.9: Fill in each blank so that the resulting statement is true.Based on...
 3.5.10: Use the graph of the rational function in the figure shown tocomple...
 3.5.11: Use the graph of the rational function in the figure shown tocomple...
 3.5.12: Use the graph of the rational function in the figure shown tocomple...
 3.5.13: Use the graph of the rational function in the figure shown tocomple...
 3.5.14: Use the graph of the rational function in the figure shown tocomple...
 3.5.15: Use the graph of the rational function in the figure shown tocomple...
 3.5.16: Use the graph of the rational function in the figure shown tocomple...
 3.5.17: Use the graph of the rational function in the figure shown tocomple...
 3.5.18: Use the graph of the rational function in the figure shown tocomple...
 3.5.19: Use the graph of the rational function in the figure shown tocomple...
 3.5.20: Use the graph of the rational function in the figure shown tocomple...
 3.5.21: In Exercises 2136, find the vertical asymptotes, if any, and theval...
 3.5.22: In Exercises 2136, find the vertical asymptotes, if any, and theval...
 3.5.23: In Exercises 2136, find the vertical asymptotes, if any, and theval...
 3.5.24: In Exercises 2136, find the vertical asymptotes, if any, and theval...
 3.5.25: In Exercises 2136, find the vertical asymptotes, if any, and theval...
 3.5.26: In Exercises 2136, find the vertical asymptotes, if any, and theval...
 3.5.27: In Exercises 2136, find the vertical asymptotes, if any, and theval...
 3.5.28: In Exercises 2136, find the vertical asymptotes, if any, and theval...
 3.5.29: In Exercises 2136, find the vertical asymptotes, if any, and theval...
 3.5.30: In Exercises 2136, find the vertical asymptotes, if any, and theval...
 3.5.31: In Exercises 2136, find the vertical asymptotes, if any, and theval...
 3.5.32: In Exercises 2136, find the vertical asymptotes, if any, and theval...
 3.5.33: In Exercises 2136, find the vertical asymptotes, if any, and theval...
 3.5.34: In Exercises 2136, find the vertical asymptotes, if any, and theval...
 3.5.35: In Exercises 2136, find the vertical asymptotes, if any, and theval...
 3.5.36: In Exercises 2136, find the vertical asymptotes, if any, and theval...
 3.5.37: In Exercises 3744, find the horizontal asymptote, if there is one,o...
 3.5.38: In Exercises 3744, find the horizontal asymptote, if there is one,o...
 3.5.39: In Exercises 3744, find the horizontal asymptote, if there is one,o...
 3.5.40: In Exercises 3744, find the horizontal asymptote, if there is one,o...
 3.5.41: In Exercises 3744, find the horizontal asymptote, if there is one,o...
 3.5.42: In Exercises 3744, find the horizontal asymptote, if there is one,o...
 3.5.43: In Exercises 3744, find the horizontal asymptote, if there is one,o...
 3.5.44: In Exercises 3744, find the horizontal asymptote, if there is one,o...
 3.5.45: In Exercises 4556, use transformations of f(x) = 1x or f(x) = 1x2to...
 3.5.46: In Exercises 4556, use transformations of f(x) = 1x or f(x) = 1x2to...
 3.5.47: In Exercises 4556, use transformations of f(x) = 1x or f(x) = 1x2to...
 3.5.48: In Exercises 4556, use transformations of f(x) = 1x or f(x) = 1x2to...
 3.5.49: In Exercises 4556, use transformations of f(x) = 1x or f(x) = 1x2to...
 3.5.50: In Exercises 4556, use transformations of f(x) = 1x or f(x) = 1x2to...
 3.5.51: In Exercises 4556, use transformations of f(x) = 1x or f(x) = 1x2to...
 3.5.52: In Exercises 4556, use transformations of f(x) = 1x or f(x) = 1x2to...
 3.5.53: In Exercises 4556, use transformations of f(x) = 1x or f(x) = 1x2to...
 3.5.54: In Exercises 4556, use transformations of f(x) = 1x or f(x) = 1x2to...
 3.5.55: In Exercises 4556, use transformations of f(x) = 1x or f(x) = 1x2to...
 3.5.56: In Exercises 4556, use transformations of f(x) = 1x or f(x) = 1x2to...
 3.5.57: In Exercises 5780, follow the seven steps on page 419 to grapheach ...
 3.5.58: In Exercises 5780, follow the seven steps on page 419 to grapheach ...
 3.5.59: In Exercises 5780, follow the seven steps on page 419 to grapheach ...
 3.5.60: In Exercises 5780, follow the seven steps on page 419 to grapheach ...
 3.5.61: In Exercises 5780, follow the seven steps on page 419 to grapheach ...
 3.5.62: In Exercises 5780, follow the seven steps on page 419 to grapheach ...
 3.5.63: In Exercises 5780, follow the seven steps on page 419 to grapheach ...
 3.5.64: In Exercises 5780, follow the seven steps on page 419 to grapheach ...
 3.5.65: In Exercises 5780, follow the seven steps on page 419 to grapheach ...
 3.5.66: In Exercises 5780, follow the seven steps on page 419 to grapheach ...
 3.5.67: In Exercises 5780, follow the seven steps on page 419 to grapheach ...
 3.5.68: In Exercises 5780, follow the seven steps on page 419 to grapheach ...
 3.5.69: In Exercises 5780, follow the seven steps on page 419 to grapheach ...
 3.5.70: In Exercises 5780, follow the seven steps on page 419 to grapheach ...
 3.5.71: In Exercises 5780, follow the seven steps on page 419 to grapheach ...
 3.5.72: In Exercises 5780, follow the seven steps on page 419 to grapheach ...
 3.5.73: In Exercises 5780, follow the seven steps on page 419 to grapheach ...
 3.5.74: In Exercises 5780, follow the seven steps on page 419 to grapheach ...
 3.5.75: In Exercises 5780, follow the seven steps on page 419 to grapheach ...
 3.5.76: In Exercises 5780, follow the seven steps on page 419 to grapheach ...
 3.5.77: In Exercises 5780, follow the seven steps on page 419 to grapheach ...
 3.5.78: In Exercises 5780, follow the seven steps on page 419 to grapheach ...
 3.5.79: In Exercises 5780, follow the seven steps on page 419 to grapheach ...
 3.5.80: In Exercises 5780, follow the seven steps on page 419 to grapheach ...
 3.5.81: In Exercises 8188, a. Find the slant asymptote of the graph ofeach ...
 3.5.82: In Exercises 8188, a. Find the slant asymptote of the graph ofeach ...
 3.5.83: In Exercises 8188, a. Find the slant asymptote of the graph ofeach ...
 3.5.84: In Exercises 8188, a. Find the slant asymptote of the graph ofeach ...
 3.5.85: In Exercises 8188, a. Find the slant asymptote of the graph ofeach ...
 3.5.86: In Exercises 8188, a. Find the slant asymptote of the graph ofeach ...
 3.5.87: In Exercises 8188, a. Find the slant asymptote of the graph ofeach ...
 3.5.88: In Exercises 8188, a. Find the slant asymptote of the graph ofeach ...
 3.5.89: In Exercises 8994, the equation for f is given by the simplifiedexp...
 3.5.90: In Exercises 8994, the equation for f is given by the simplifiedexp...
 3.5.91: In Exercises 8994, the equation for f is given by the simplifiedexp...
 3.5.92: In Exercises 8994, the equation for f is given by the simplifiedexp...
 3.5.93: In Exercises 8994, the equation for f is given by the simplifiedexp...
 3.5.94: In Exercises 8994, the equation for f is given by the simplifiedexp...
 3.5.95: In Exercises 9598, use long division to rewrite the equation for gi...
 3.5.96: In Exercises 9598, use long division to rewrite the equation for gi...
 3.5.97: In Exercises 9598, use long division to rewrite the equation for gi...
 3.5.98: In Exercises 9598, use long division to rewrite the equation for gi...
 3.5.99: A company is planning to manufacture mountain bikes. Thefixed month...
 3.5.100: A company that manufactures running shoes has a fixedmonthly cost o...
 3.5.101: The functionf(x) = 6.5x2  20.4x + 234x2 + 36 models the pH level, ...
 3.5.102: A drug is injected into a patient and the concentrationof the drug ...
 3.5.103: Among all deaths from a particular disease, the percentage thatis s...
 3.5.104: Among all deaths from a particular disease, the percentage thatis s...
 3.5.105: Among all deaths from a particular disease, the percentage thatis s...
 3.5.106: Among all deaths from a particular disease, the percentage thatis s...
 3.5.107: The bar graph shows the amount, in billions of dollars, thatthe Uni...
 3.5.108: What is a rational function?
 3.5.109: Use everyday language to describe the graph of a rationalfunction f...
 3.5.110: Use everyday language to describe the behavior of agraph near its v...
 3.5.111: If you are given the equation of a rational function, explainhow to...
 3.5.112: If you are given the equation of a rational function,explainhow to ...
 3.5.113: Describe how to graph a rational function.
 3.5.114: If you are given the equation of a rational function, how canyou te...
 3.5.115: Is every rational function a polynomial function? Why orwhy not? Do...
 3.5.116: Although your friend has a family history of heart disease,he smoke...
 3.5.117: Use a graphing utility to verify any five of your handdrawngraphs ...
 3.5.118: Use a graphing utility to graph y = 1x, y = 1x3, and 1x5 inthe same...
 3.5.119: Use a graphing utility to graph y = 1x2, y = 1x4, and y = 1x6in the...
 3.5.120: Use a graphing utility to graphf(x) = x2  4x + 3x  2 and g(x) = x...
 3.5.121: The rational functionf(x) = 27,725(x  14)x2 + 9  5x models the nu...
 3.5.122: In Exercises 122125, determine whether eachstatement makes sense or...
 3.5.123: In Exercises 122125, determine whether eachstatement makes sense or...
 3.5.124: In Exercises 122125, determine whether eachstatement makes sense or...
 3.5.125: In Exercises 122125, determine whether eachstatement makes sense or...
 3.5.126: In Exercises 126129, determine whether each statement is trueor fal...
 3.5.127: In Exercises 126129, determine whether each statement is trueor fal...
 3.5.128: In Exercises 126129, determine whether each statement is trueor fal...
 3.5.129: In Exercises 126129, determine whether each statement is trueor fal...
 3.5.130: In Exercises 130133, write the equation of a rational functionf(x) ...
 3.5.131: In Exercises 130133, write the equation of a rational functionf(x) ...
 3.5.132: In Exercises 130133, write the equation of a rational functionf(x) ...
 3.5.133: In Exercises 130133, write the equation of a rational functionf(x) ...
 3.5.134: Basic Car Rental charges $20 a day plus $0.10 per mile,whereas Acme...
 3.5.135: Identify the graphs (a)(d) in which y is a function of x.
 3.5.136: Which of the following graphs (a)(d) represent functionsthat have a...
 3.5.137: Exercises 137139 will help you prepare for the material coveredin t...
 3.5.138: Exercises 137139 will help you prepare for the material coveredin t...
 3.5.139: Exercises 137139 will help you prepare for the material coveredin t...
Solutions for Chapter 3.5: Rational Functions and Their Graphs
Full solutions for College Algebra  7th Edition
ISBN: 9780134469164
Solutions for Chapter 3.5: Rational Functions and Their Graphs
Get Full SolutionsSince 139 problems in chapter 3.5: Rational Functions and Their Graphs have been answered, more than 29708 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: College Algebra , edition: 7. Chapter 3.5: Rational Functions and Their Graphs includes 139 full stepbystep solutions. College Algebra was written by and is associated to the ISBN: 9780134469164. This expansive textbook survival guide covers the following chapters and their solutions.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

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.

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.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

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.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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 •

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

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.

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.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

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.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

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

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

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