 7.6.1: . Complete the table by filling in the missing values for the side,...
 7.6.2: Solve each equation for x.a.  x  = 6 b. x2 = 36 c.  x  = 3.8 d....
 7.6.3: Solve each equation, if possible. a. 4.7 =  x  2.8 b. 41 = x2 2.8...
 7.6.4: Solve each equation for x. Use a calculator graph to check your ans...
 7.6.5: For what values of x is  x  x2? To check your answer, graph Y1 = ...
 7.6.6: For what values of y does the equation y = x2 have a. No real solut...
 7.6.7: Graph the function What other equation produces the same graph?
 7.6.8: Look at the table of squares in the Investigation Graphing a Parabo...
 7.6.9: MiniInvestigation Find the sum of each set of numbers in 9ac. a. t...
 7.6.10: Write an equation for the function represented in each table. Use y...
 7.6.11: This 4by4 grid contains squares of different sizes. a. How many o...
 7.6.12: Explain why the equation x2 = 4 has no solutions.
 7.6.13: The table shows exponential data. xy 0 4 126.5625 3 168.75 1 1000 a...
 7.6.14: Use properties of exponents to find an equivalent expression in the...
 7.6.15: Graph the functions f (x) = 3x 5 and g(x) =  x 3 . What do the tw...
Solutions for Chapter 7.6: Squares, Squaring, and Parabolas
Full solutions for Discovering Algebra: An Investigative Approach  2nd Edition
ISBN: 9781559537636
Solutions for Chapter 7.6: Squares, Squaring, and Parabolas
Get Full SolutionsThis textbook survival guide was created for the textbook: Discovering Algebra: An Investigative Approach, edition: 2. This expansive textbook survival guide covers the following chapters and their solutions. Discovering Algebra: An Investigative Approach was written by and is associated to the ISBN: 9781559537636. Since 15 problems in chapter 7.6: Squares, Squaring, and Parabolas have been answered, more than 4541 students have viewed full stepbystep solutions from this chapter. Chapter 7.6: Squares, Squaring, and Parabolas includes 15 full stepbystep solutions.

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.

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!

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.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Column space C (A) =
space of all combinations of the columns of A.

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

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

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

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).

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.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

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).

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

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}.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

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·

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.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).