 4.5.1: The Graph of a Rational Function
 4.5.2: If the numerator and the denominator of a rational function have no...
 4.5.3: The graph of a rational function never intersects a asymptote.
 4.5.4: The graph of a rational function sometimes intersects an oblique as...
 4.5.5: The graph of a rational function sometimes intersects an oblique as...
 4.5.6: R1x2 = x1x  222 x  2 (a) Find the domain of R. (b) Find the xint...
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 4.5.45: In 45 48, find a rational function that might have the given graph....
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 4.5.47: In 45 48, find a rational function that might have the given graph....
 4.5.48: In 45 48, find a rational function that might have the given graph....
 4.5.49: The concentration C of a certain drug in a patients bloodstream t h...
 4.5.50: The concentration C of a certain drug in a patients bloodstream t m...
 4.5.51: A rectangular area adjacent to a river is to be fenced in; no fence...
 4.5.52: The Doppler effect (named after Christian Doppler) is the change in...
 4.5.53: United Parcel Service has contracted you to design a closed box wit...
 4.5.54: A can in the shape of a right circular cylinder is required to have...
 4.5.55: A can in the shape of a right circular cylinder is required to have...
 4.5.56: A steel drum in the shape of a right circular cylinder is required ...
 4.5.57: Graph each of the following functions: y = x2  1 x  1 y = x3  1 ...
 4.5.58: Graph each of the following functions: y = x2 x  1 y = x4 x  1 y ...
 4.5.59: Write a few paragraphs that provide a general strategy for graphing...
 4.5.60: Create a rational function that has the following characteristics: ...
 4.5.61: Create a rational function that has the following characteristics: ...
 4.5.62: Create a rational function with the following characteristics: thre...
 4.5.63: Explain the circumstances under which the graph of a rational funct...
Solutions for Chapter 4.5: The Graph of a Rational Function
Full solutions for Precalculus Enhanced with Graphing Utilities  6th Edition
ISBN: 9780132854351
Solutions for Chapter 4.5: The Graph of a Rational Function
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Precalculus Enhanced with Graphing Utilities was written by and is associated to the ISBN: 9780132854351. Since 63 problems in chapter 4.5: The Graph of a Rational Function have been answered, more than 56420 students have viewed full stepbystep solutions from this chapter. Chapter 4.5: The Graph of a Rational Function includes 63 full stepbystep solutions. This textbook survival guide was created for the textbook: Precalculus Enhanced with Graphing Utilities, edition: 6.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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

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

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.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

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.

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

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

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

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

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.

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

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

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