 9.3.1: Is each algebraic expression a polynomial? If so, how many terms do...
 9.3.2: Expand each expression. On your calculator, enter the original expr...
 9.3.3: Copy each rectangle diagram and fill in the missing values. Then wr...
 9.3.4: Convert each equation from vertex form to general form. Check your ...
 9.3.5: Draw a rectangle diagram to represent each expression. Then write a...
 9.3.6: Consider the graph of the parabola y = x2 4x + 7.a. What are the co...
 9.3.7: Heather thinks she has found a shortcut to the rectangle diagram me...
 9.3.8: APPLICATION The quadratic equation y = 0.0056x2 + 0.14x relates a v...
 9.3.9: The function h(t) = 4.9(t 0.4)2 + 2.5 describes the height of a sof...
 9.3.10: Is the expression on the right equivalent to the expression on the ...
 9.3.11: APPLICATION The Yoyo Warehouse uses the equation y = 85x2 + 552.5x...
 9.3.12: Use a threebythree rectangle diagram to square each trinomial. a....
 9.3.13: What is the general form of y = (x + 4)2? Write a paragraph describ...
 9.3.14: Is this parabola a graph of a function?
 9.3.15: Use the graph of f (x) to evaluate each expression. Then think of t...
 9.3.16: The equation y = 0.0024x2 + 0.81x + 2.0 models the path of a golf b...
Solutions for Chapter 9.3: From Vertex to General Form
Full solutions for Discovering Algebra: An Investigative Approach  2nd Edition
ISBN: 9781559537636
Solutions for Chapter 9.3: From Vertex to General Form
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Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

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.

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

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.

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

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

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 columns of A.
Columns without pivots; these are combinations of earlier columns.

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.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

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.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

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.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

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