 2.5.1: Describe how the graph of the function can be obtained from one of ...
 2.5.2: Describe how the graph of the function can be obtained from one of ...
 2.5.3: Describe how the graph of the function can be obtained from one of ...
 2.5.4: Describe how the graph of the function can be obtained from one of ...
 2.5.5: Describe how the graph of the function can be obtained from one of ...
 2.5.6: Describe how the graph of the function can be obtained from one of ...
 2.5.7: Describe how the graph of the function can be obtained from one of ...
 2.5.8: Describe how the graph of the function can be obtained from one of ...
 2.5.9: Describe how the graph of the function can be obtained from one of ...
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 2.5.12: Describe how the graph of the function can be obtained from one of ...
 2.5.13: Describe how the graph of the function can be obtained from one of ...
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 2.5.15: Describe how the graph of the function can be obtained from one of ...
 2.5.16: Describe how the graph of the function can be obtained from one of ...
 2.5.17: Describe how the graph of the function can be obtained from one of ...
 2.5.18: Describe how the graph of the function can be obtained from one of ...
 2.5.19: Describe how the graph of the function can be obtained from one of ...
 2.5.20: Describe how the graph of the function can be obtained from one of ...
 2.5.21: Describe how the graph of the function can be obtained from one of ...
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 2.5.31: Describe how the graph of the function can be obtained from one of ...
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 2.5.35: Describe how the graph of the function can be obtained from one of ...
 2.5.36: Describe how the graph of the function can be obtained from one of ...
 2.5.37: The point 112, 42 is on the graph of y = f 1x2. Find the correspon...
 2.5.38: The point 112, 42 is on the graph of y = f 1x2. Find the correspon...
 2.5.39: The point 112, 42 is on the graph of y = f 1x2. Find the correspon...
 2.5.40: The point 112, 42 is on the graph of y = f 1x2. Find the correspon...
 2.5.41: The point 112, 42 is on the graph of y = f 1x2. Find the correspon...
 2.5.42: The point 112, 42 is on the graph of y = f 1x2. Find the correspon...
 2.5.43: The point 112, 42 is on the graph of y = f 1x2. Find the correspon...
 2.5.44: The point 112, 42 is on the graph of y = f 1x2. Find the correspon...
 2.5.45: Given that f 1x2 = x2 + 3, match the function g with a transformati...
 2.5.46: Given that f 1x2 = x2 + 3, match the function g with a transformati...
 2.5.47: Given that f 1x2 = x2 + 3, match the function g with a transformati...
 2.5.48: Given that f 1x2 = x2 + 3, match the function g with a transformati...
 2.5.49: Write an equation for a function that has a graph with the given ch...
 2.5.50: Write an equation for a function that has a graph with the given ch...
 2.5.51: Write an equation for a function that has a graph with the given ch...
 2.5.52: Write an equation for a function that has a graph with the given ch...
 2.5.53: Write an equation for a function that has a graph with the given ch...
 2.5.54: Write an equation for a function that has a graph with the given ch...
 2.5.55: Write an equation for a function that has a graph with the given ch...
 2.5.56: Write an equation for a function that has a graph with the given ch...
 2.5.57: Write an equation for a function that has a graph with the given ch...
 2.5.58: Write an equation for a function that has a graph with the given ch...
 2.5.59: A graph of y = f 1x2 follows. No formula for f is given. In Exercis...
 2.5.60: A graph of y = f 1x2 follows. No formula for f is given. In Exercis...
 2.5.61: A graph of y = f 1x2 follows. No formula for f is given. In Exercis...
 2.5.62: A graph of y = f 1x2 follows. No formula for f is given. In Exercis...
 2.5.63: A graph of y = f 1x2 follows. No formula for f is given. In Exercis...
 2.5.64: A graph of y = f 1x2 follows. No formula for f is given. In Exercis...
 2.5.65: A graph of y = f 1x2 follows. No formula for f is given. In Exercis...
 2.5.66: A graph of y = f 1x2 follows. No formula for f is given. In Exercis...
 2.5.67: graph of y = g1x2 follows. No formula for g is given. In Exercises ...
 2.5.68: graph of y = g1x2 follows. No formula for g is given. In Exercises ...
 2.5.69: graph of y = g1x2 follows. No formula for g is given. In Exercises ...
 2.5.70: graph of y = g1x2 follows. No formula for g is given. In Exercises ...
 2.5.71: The graph of the function f is shown in figure (a). In Exercises 71...
 2.5.72: The graph of the function f is shown in figure (a). In Exercises 71...
 2.5.73: The graph of the function f is shown in figure (a). In Exercises 71...
 2.5.74: The graph of the function f is shown in figure (a). In Exercises 71...
 2.5.75: The graph of the function f is shown in figure (a). In Exercises 71...
 2.5.76: The graph of the function f is shown in figure (a). In Exercises 71...
 2.5.77: The graph of the function f is shown in figure (a). In Exercises 71...
 2.5.78: The graph of the function f is shown in figure (a). In Exercises 71...
 2.5.79: For each pair of functions, determine if g1x2 = f 1x2.f 1x2 = 2x4 ...
 2.5.80: For each pair of functions, determine if g1x2 = f 1x2.f 1x2 = 14 x...
 2.5.81: A graph of the function f 1x2 = x3  3x2 is shown below. Exercises ...
 2.5.82: A graph of the function f 1x2 = x3  3x2 is shown below. Exercises ...
 2.5.83: A graph of the function f 1x2 = x3  3x2 is shown below. Exercises ...
 2.5.84: A graph of the function f 1x2 = x3  3x2 is shown below. Exercises ...
 2.5.85: Determine algebraically whether the graph is symmetric with respect...
 2.5.86: Determine algebraically whether the graph is symmetric with respect...
 2.5.87: Determine algebraically whether the graph is symmetric with respect...
 2.5.88: Video Game Sales. Sales of the video game Wii Fit totaled 3.5 milli...
 2.5.89: Gift Cards. It is estimated that about $5 billion in gift cards giv...
 2.5.90: efiling Taxes. The number of tax returns filed electronically in 2...
 2.5.91: Use the graph of the function f shown below in Exercises 91 and 92....
 2.5.92: Use the graph of the function f shown below in Exercises 91 and 92....
 2.5.93: Use the graph of the function g shown below in Exercises 93 and 94....
 2.5.94: Use the graph of the function g shown below in Exercises 93 and 94....
 2.5.95: Graph each of the following using a graphing calculator. Before doi...
 2.5.96: Graph each of the following using a graphing calculator. Before doi...
 2.5.97: If 13, 42 is a point on the graph of y = f 1x2, what point do you k...
 2.5.98: Find the zeros of f 1x2 = 3x5  20x3 . Then, without using a graphi...
Solutions for Chapter 2.5: Transformations
Full solutions for College Algebra: Graphs and Models  5th Edition
ISBN: 9780321783950
Solutions for Chapter 2.5: Transformations
Get Full SolutionsSince 98 problems in chapter 2.5: Transformations have been answered, more than 27465 students have viewed full stepbystep solutions from this chapter. Chapter 2.5: Transformations includes 98 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: College Algebra: Graphs and Models, edition: 5. College Algebra: Graphs and Models was written by and is associated to the ISBN: 9780321783950.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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

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.

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 IIĀ· Condition numbers measure the sensitivity of the output to change in the input.

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].

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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].

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex 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!).

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

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.

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.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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

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.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

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}.