- 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 Yo-yo Warehouse uses the equation y = 85x2 + 552.5x...
- 9.3.12: Use a three-by-three 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
Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).
Remove row i and column j; multiply the determinant by (-I)i + j •
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].
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 Fn-1c can be computed with ne/2 multiplications. Revolutionary.
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!).
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.
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
A sequence of steps intended to approach the desired solution.
A directed graph that has constants Cl, ... , Cm associated with the edges.
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.
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
Constant down each diagonal = time-invariant (shift-invariant) filter.
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