 8.2.1: Use f (x) = 2  x + 4  + 1 to find a. f (5) b. f (6) c. f (2) + 3 ...
 8.2.2: List L1 and list L2 contain coordinates for three points on the gra...
 8.2.3: Give the coordinates of the vertex for each graph.
 8.2.4: Use a calculator to graph each equation. Describe the graph as a tr...
 8.2.5: Write an equation for each of these transformations. a. Translate t...
 8.2.6: Describe each graph in Exercise 3 as a transformation of y =  x  ...
 8.2.7: This graph shows Beths distance from her teacher as she turns in he...
 8.2.8: Graph Y1(x) = abs(x) on your calculator. Predict what each graph wi...
 8.2.9: Describe how the graph of y = x2 will be transformed if you replace...
 8.2.10: APPLICATION The equation y = a bx models the decreasing voltage of ...
 8.2.11: MiniInvestigation Recall that an exponential equation in the form ...
 8.2.12: APPLICATION In 2004, the world population was estimated to be 6.4 b...
 8.2.13: The graph of a linear equation of the form y = bx passes through (0...
 8.2.14: Drews teacher gives skillbuilding quizzes at the start of each cla...
 8.2.15: Solve each system of equations.
Solutions for Chapter 8.2: Translating Graphs
Full solutions for Discovering Algebra: An Investigative Approach  2nd Edition
ISBN: 9781559537636
Solutions for Chapter 8.2: Translating Graphs
Get Full SolutionsDiscovering Algebra: An Investigative Approach was written by and is associated to the ISBN: 9781559537636. This textbook survival guide was created for the textbook: Discovering Algebra: An Investigative Approach, edition: 2. Chapter 8.2: Translating Graphs includes 15 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 15 problems in chapter 8.2: Translating Graphs have been answered, more than 7589 students have viewed full stepbystep solutions from this chapter.

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

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.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

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.

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.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

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].

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Outer product uv T
= column times row = rank one matrix.

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

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.

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

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

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).