 2.1: In Exercises 1 and 2, find the distance between the points whose co...
 2.2: In Exercises 1 and 2, find the distance between the points whose co...
 2.3: In Exercises 3 and 4, find the midpoint of the line segment with th...
 2.4: In Exercises 3 and 4, find the midpoint of the line segment with th...
 2.5: In Exercises 5 to 8, graph each equation by plotting points
 2.6: In Exercises 5 to 8, graph each equation by plotting points
 2.7: In Exercises 5 to 8, graph each equation by plotting points
 2.8: In Exercises 5 to 8, graph each equation by plotting points
 2.9: In Exercises 9 to 12, find the x and yintercepts of the graph of ...
 2.10: In Exercises 9 to 12, find the x and yintercepts of the graph of ...
 2.11: In Exercises 9 to 12, find the x and yintercepts of the graph of ...
 2.12: In Exercises 9 to 12, find the x and yintercepts of the graph of ...
 2.13: In Exercises 13 and 14, determine the center and radius of the circ...
 2.14: In Exercises 13 and 14, determine the center and radius of the circ...
 2.15: In Exercises 15 and 16, find the equation in standard form of the c...
 2.16: In Exercises 15 and 16, find the equation in standard form of the c...
 2.17: In Exercises 17 to 20, determine whether the equation defines y as ...
 2.18: In Exercises 17 to 20, determine whether the equation defines y as ...
 2.19: In Exercises 17 to 20, determine whether the equation defines y as ...
 2.20: In Exercises 17 to 20, determine whether the equation defines y as ...
 2.23: Let f be a piecewisedefined function given by
 2.24: Let f be a piecewisedefined function given by
 2.25: In Exercises 25 to 28, determine the domain of the function represe...
 2.26: In Exercises 25 to 28, determine the domain of the function represe...
 2.27: In Exercises 25 to 28, determine the domain of the function represe...
 2.28: In Exercises 25 to 28, determine the domain of the function represe...
 2.29: Find the values of a in the domain of for which .
 2.30: Find the value of a in the domain of for which
 2.31: In Exercises 31 and 32, graph the given equation.
 2.32: In Exercises 31 and 32, graph the given equation.
 2.33: In Exercises 33 and 34, find the zero or zeros of the given function.
 2.34: In Exercises 33 and 34, find the zero or zeros of the given function.
 2.35: In Exercises 35 and 36, find each function value.
 2.36: In Exercises 35 and 36, find each function value.
 2.37: In Exercises 37 to 40, find the slope of the line between the point...
 2.38: In Exercises 37 to 40, find the slope of the line between the point...
 2.39: In Exercises 37 to 40, find the slope of the line between the point...
 2.40: In Exercises 37 to 40, find the slope of the line between the point...
 2.41: Graph using the slope and yintercept
 2.42: Graph using the slope and yintercept
 2.43: Graph 3x  4y = 8.
 2.44: Graph 2x + 3y = 9
 2.45: Find the equation of the line that passes through the point with co...
 2.46: Find the equation of the line that passes through the point with co...
 2.47: Find the equation of the line that passes through the points with c...
 2.48: Find the equation of the line that passes through the points with c...
 2.49: Find the slopeintercept form of the equation of the line that passe...
 2.50: Find the slopeintercept form of the equation of the line that passe...
 2.51: Find the slopeintercept form of the equation of the line that passe...
 2.52: Find the slopeintercept form of the equation of the line that passe...
 2.53: Sports The speed of a professional golfers swing and the speed of t...
 2.54: Food Science Newer heating elements allow an oven to reach a normal...
 2.55: In Exercises 55 to 60, use the method of completing the square to w...
 2.56: In Exercises 55 to 60, use the method of completing the square to w...
 2.57: In Exercises 55 to 60, use the method of completing the square to w...
 2.58: In Exercises 55 to 60, use the method of completing the square to w...
 2.59: In Exercises 55 to 60, use the method of completing the square to w...
 2.60: In Exercises 55 to 60, use the method of completing the square to w...
 2.61: In Exercises 61 to 64, find the vertex of the graph of the quadrati...
 2.62: In Exercises 61 to 64, find the vertex of the graph of the quadrati...
 2.63: In Exercises 61 to 64, find the vertex of the graph of the quadrati...
 2.64: In Exercises 61 to 64, find the vertex of the graph of the quadrati...
 2.65: In Exercises 65 and 66, find the requested value
 2.66: In Exercises 65 and 66, find the requested value
 2.67: . Height of a Ball A ball is thrown vertically upward with an initi...
 2.68: Delivery Cost A freight company has determined that its cost, in do...
 2.69: Agriculture A farmer wishes to enclose a rectangular region borderi...
 2.70: In Exercises 70 and 71, sketch a graph that is symmetric to the giv...
 2.71: In Exercises 70 and 71, sketch a graph that is symmetric to the giv...
 2.72: In Exercises 72 to 79, determine whether the graph of each equation...
 2.73: In Exercises 72 to 79, determine whether the graph of each equation...
 2.74: In Exercises 72 to 79, determine whether the graph of each equation...
 2.75: In Exercises 72 to 79, determine whether the graph of each equation...
 2.76: In Exercises 72 to 79, determine whether the graph of each equation...
 2.77: In Exercises 72 to 79, determine whether the graph of each equation...
 2.78: In Exercises 72 to 79, determine whether the graph of each equation...
 2.79: In Exercises 72 to 79, determine whether the graph of each equation...
 2.80: In Exercises 80 to 85, sketch the graph of g. a. Find the domain an...
 2.81: In Exercises 80 to 85, sketch the graph of g. a. Find the domain an...
 2.82: In Exercises 80 to 85, sketch the graph of g. a. Find the domain an...
 2.83: In Exercises 80 to 85, sketch the graph of g. a. Find the domain an...
 2.84: In Exercises 80 to 85, sketch the graph of g. a. Find the domain an...
 2.85: In Exercises 80 to 85, sketch the graph of g. a. Find the domain an...
 2.86: In Exercises 86 to 91, use the graph of f shown below to sketch a g...
 2.87: In Exercises 86 to 91, use the graph of f shown below to sketch a g...
 2.88: In Exercises 86 to 91, use the graph of f shown below to sketch a g...
 2.89: In Exercises 86 to 91, use the graph of f shown below to sketch a g...
 2.90: In Exercises 86 to 91, use the graph of f shown below to sketch a g...
 2.91: In Exercises 86 to 91, use the graph of f shown below to sketch a g...
 2.92: In Exercises 92 to 95, use the graph of f shown below to sketch a g...
 2.93: In Exercises 92 to 95, use the graph of f shown below to sketch a g...
 2.94: In Exercises 92 to 95, use the graph of f shown below to sketch a g...
 2.95: In Exercises 92 to 95, use the graph of f shown below to sketch a g...
 2.96: Let and . Find each of the following
 2.97: If , find the difference quotient
 2.98: If , find the difference quotient
 2.99: Ball Rolling on a Ramp The distance traveled by a ball rolling down...
 2.100: If and , find a. b. c. d. ( f g)(x) (g f )(x)
 2.101: If and , find
 2.102: Sports A soccer coach examined the relationship between the speed, ...
 2.103: Physics The rate at which water will escape from the bottom of a ru...
Solutions for Chapter 2: College Algebra and Trigonometry 7th Edition
Full solutions for College Algebra and Trigonometry  7th Edition
ISBN: 9781439048603
Solutions for Chapter 2
Get Full SolutionsCollege Algebra and Trigonometry was written by and is associated to the ISBN: 9781439048603. Chapter 2 includes 101 full stepbystep solutions. This textbook survival guide was created for the textbook: College Algebra and Trigonometry, edition: 7. Since 101 problems in chapter 2 have been answered, more than 12456 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

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

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

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!).

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

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 •

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

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.

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.

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.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.