 3.1: Evaluate each expression if a = 1, b = 4, and c = 3.
 3.2: Evaluate each expression if a = 1, b = 4, and c = 3.
 3.3: Evaluate each expression if a = 1, b = 4, and c = 3.
 3.4: Evaluate each expression if a = 1, b = 4, and c = 3.
 3.5: FOOD Noah is buying a sandwich with 1 type of meat, 2 types of chee...
 3.6: Solve each equation for y.
 3.7: Solve each equation for y.
 3.8: Solve each equation for y.
 3.9: Solve each equation for y.
 3.10: Solve each equation for y.
 3.11: Solve each equation for y.
 3.12: Graph each ordered pair on a coordinate grid.
 3.13: Graph each ordered pair on a coordinate grid.
 3.14: Graph each ordered pair on a coordinate grid.
 3.15: Graph each ordered pair on a coordinate grid.
 3.16: Graph each ordered pair on a coordinate grid.
 3.17: Graph each ordered pair on a coordinate grid.
 3.18: MAPS Taylor is looking at a map and needs to go 3 blocks east and 2...
 3.19: ENTERTAINMENT For Exercises 1820, use the following information. Th...
 3.20: ENTERTAINMENT For Exercises 1820, use the following information. Th...
 3.21: Determine the xintercept, yintercept, and zero of each linear fun...
 3.22: Determine the xintercept, yintercept, and zero of each linear fun...
 3.23: Graph each equation.
 3.24: Graph each equation.
 3.25: Graph each equation.
 3.26: Graph each equation.
 3.27: SOUND The distance d in kilometers that sound waves travel through ...
 3.28: Find the next three terms of each arithmetic sequence.
 3.29: Find the next three terms of each arithmetic sequence.
 3.30: Find the next three terms of each arithmetic sequence.
 3.31: Find the nth term of each arithmetic sequence described.
 3.32: Find the nth term of each arithmetic sequence described.
 3.33: MONEY The table represents Tiffanys income. Write an equation for t...
 3.34: Write an equation in function notation for each relation.
 3.35: Write an equation in function notation for each relation.
 3.36: ANALYZE TABLES For Exercises 3638, use the table below that shows t...
 3.37: ANALYZE TABLES For Exercises 3638, use the table below that shows t...
 3.38: ANALYZE TABLES For Exercises 3638, use the table below that shows t...
Solutions for Chapter 3: Functions and Patterns
Full solutions for Algebra 1, Student Edition (MERRILL ALGEBRA 1)  1st Edition
ISBN: 9780078738227
Solutions for Chapter 3: Functions and Patterns
Get Full SolutionsThis textbook survival guide was created for the textbook: Algebra 1, Student Edition (MERRILL ALGEBRA 1) , edition: 1. Chapter 3: Functions and Patterns includes 38 full stepbystep solutions. Since 38 problems in chapter 3: Functions and Patterns have been answered, more than 37067 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Algebra 1, Student Edition (MERRILL ALGEBRA 1) was written by and is associated to the ISBN: 9780078738227.

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

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

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.

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.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

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 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.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

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.

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