 2.1: In 16, find the following for each pair of points: (a) The distance...
 2.2: In 16, find the following for each pair of points: (a) The distance...
 2.3: In 16, find the following for each pair of points: (a) The distance...
 2.4: In 16, find the following for each pair of points: (a) The distance...
 2.5: In 16, find the following for each pair of points: (a) The distance...
 2.6: In 16, find the following for each pair of points: (a) The distance...
 2.7: Graph by plotting points.
 2.8: List the intercepts of the graph below.
 2.9: In 916, list the intercepts and test for symmetry with respect to t...
 2.10: In 916, list the intercepts and test for symmetry with respect to t...
 2.11: In 916, list the intercepts and test for symmetry with respect to t...
 2.12: In 916, list the intercepts and test for symmetry with respect to t...
 2.13: In 916, list the intercepts and test for symmetry with respect to t...
 2.14: In 916, list the intercepts and test for symmetry with respect to t...
 2.15: In 916, list the intercepts and test for symmetry with respect to t...
 2.16: In 916, list the intercepts and test for symmetry with respect to t...
 2.17: In 1720, find the standard form of the equation of the circle whose...
 2.18: In 1720, find the standard form of the equation of the circle whose...
 2.19: In 1720, find the standard form of the equation of the circle whose...
 2.20: In 1720, find the standard form of the equation of the circle whose...
 2.21: In 2126, find the center and radius of each circle. Graph each circ...
 2.22: In 2126, find the center and radius of each circle. Graph each circ...
 2.23: In 2126, find the center and radius of each circle. Graph each circ...
 2.24: In 2126, find the center and radius of each circle. Graph each circ...
 2.25: In 2126, find the center and radius of each circle. Graph each circ...
 2.26: In 2126, find the center and radius of each circle. Graph each circ...
 2.27: In 2736, find an equation of the line having the given characterist...
 2.28: In 2736, find an equation of the line having the given characterist...
 2.29: In 2736, find an equation of the line having the given characterist...
 2.30: In 2736, find an equation of the line having the given characterist...
 2.31: In 2736, find an equation of the line having the given characterist...
 2.32: In 2736, find an equation of the line having the given characterist...
 2.33: In 2736, find an equation of the line having the given characterist...
 2.34: In 2736, find an equation of the line having the given characterist...
 2.35: In 2736, find an equation of the line having the given characterist...
 2.36: In 2736, find an equation of the line having the given characterist...
 2.37: In 3740, find the slope and yintercept of each line. Graph the lin...
 2.38: In 3740, find the slope and yintercept of each line. Graph the lin...
 2.39: In 3740, find the slope and yintercept of each line. Graph the lin...
 2.40: In 3740, find the slope and yintercept of each line. Graph the lin...
 2.41: In 4144, find the intercepts and graph each line. 2x  3y = 12
 2.42: In 4144, find the intercepts and graph each line. x  2y = 8
 2.43: In 4144, find the intercepts and graph each line. 12x +13y = 2
 2.44: In 4144, find the intercepts and graph each line. 13x  1y = 1
 2.45: Sketch a graph of .
 2.46: Sketch a graph of .
 2.47: Graph the line with slope containing the point .
 2.48: Show that the points , and are the vertices of an isosceles triangle.
 2.49: Show that the points , and are the vertices of a right triangle in ...
 2.50: The endpoints of the diameter of a circle are and. Find the center ...
 2.51: Show that the points , and lie on a line by using slopes.
 2.52: Mortgage Payments The monthly payment p on a mortgage varies direct...
 2.53: Revenue Function At the corner Esso station, the revenue R varies d...
 2.54: Weight of a Body The weight of a body varies inversely with the squ...
 2.55: Keplers Third Law of Planetary Motion Keplers Third Law of Planetar...
 2.56: Create four problems that you might be asked to do given the two po...
 2.57: Describe each of the following graphs in the plane. Give justifica...
 2.58: Suppose that you have a rectangular field that requires watering. Y...
Solutions for Chapter 2: Graphs
Full solutions for College Algebra  9th Edition
ISBN: 9780321716811
Solutions for Chapter 2: Graphs
Get Full SolutionsChapter 2: Graphs includes 58 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 58 problems in chapter 2: Graphs have been answered, more than 34214 students have viewed full stepbystep solutions from this chapter. College Algebra was written by and is associated to the ISBN: 9780321716811. This textbook survival guide was created for the textbook: College Algebra, edition: 9.

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.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

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.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

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

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.

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.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

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

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)ยท(b  Ax) = o.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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.

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.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.