 51.1: f(x) = 4 x 2
 51.2: f(x) = x 2 + 2x
 51.3: f(x) =  x 2 + 4x  1
 51.4: f(x) = x 2 + 8x + 3
 51.5: f(x) = 2 x 2  4x + 1
 51.6: f(x) = 3 x 2 + 10x
 51.7: f(x) =  x 2 + 7
 51.8: f(x) = x 2  x  6
 51.9: f(x) = 4 x 2 + 12x + 9
 51.10: f(x) = x 2  4x + 1
 51.11: NEWSPAPERS Due to increased production costs, the Daily News must i...
 51.12: f(x) = 2 x
 51.13: f(x) = 5 x 2
 51.14: f(x) = x 2 + 4
 51.15: f(x) = x 2  9
 51.16: f(x) = 2 x 2  4
 51.17: f(x) = 3 x 2 + 1
 51.18: f(x) = x 2  4x + 4
 51.19: f(x) = x 2  9x + 9
 51.20: f(x) = x 2  4x  5
 51.21: f(x) = x 2 + 12x + 36
 51.22: f(x) = 3 x 2
 51.23: f(x) =  x 2  9
 51.24: f(x) = x 2  8x + 2
 51.25: f(x) = x 2 + 6x  2
 51.26: f(x) = 4x  x 2 + 1
 51.27: f(x) = 3  x 2  6x
 51.28: f(x) = x 2  10x  1
 51.29: f(x) = x 2 + 8x + 15
 51.30: f(x) = x 2 + 12x  28
 51.31: f(x) = 14x  x 2  109
 51.32: Write the equation of the axis of symmetry and find the coordinates...
 51.33: According to this model, what is the maximum height of the arch?
 51.34: What are the domain and range of the function? What domain and rang...
 51.35: Find the maximum height reached by the object and the time that the...
 51.36: Interpret the meaning of the yintercept in the context of this pro...
 51.37: f(x) = 3 x 2 + 6x  1
 51.38: f(x) = 2 x 2 + 8x  3
 51.39: f(x) = 3 x 2  4x
 51.40: f(x) = 2 x 2 + 5x
 51.41: f(x) = 0.5 x 2  1
 51.42: f(x) = 0.25 x 2  3x
 51.43: f(x) = _1 2 x 2 + 3x + _9 2
 51.44: f(x) = x 2  _2 3 x  _8 9
 51.45: f(x) = 2x + 2 x 2 + 5
 51.46: f(x) = x  2 x 2  1
 51.47: f(x) = 7  3 x 2 + 12x
 51.48: f(x) = 20x + 5 x 2 + 9
 51.49: f(x) =  _1 2 x 2  2x + 3
 51.50: f(x) = _3 4 x 2  5x  2
 51.51: Write an algebraic expression for the kennels length.
 51.52: What are reasonable values for the domain of the area function?
 51.53: What dimensions produce a kennel with the greatest area?
 51.54: Find the maximum area of the kennel.
 51.55: GEOMETRY A rectangle is inscribed in an isosceles triangle as shown...
 51.56: What ticket price would give the most income for the Drama Club?
 51.57: If the Drama Club raised its tickets to this price, how much income...
 51.58: f(x) = 3 x 2  7x + 2
 51.59: f(x) = 5 x 2 + 8x
 51.60: f(x) = 2 x 2  3x + 2
 51.61: f(x) = 6 x 2 + 9x
 51.62: f(x) = 7 x 2 + 4x + 1
 51.63: f(x) = 4 x 2 + 5x
 51.64: OPEN ENDED Give an example of a quadratic function that has a domai...
 51.65: CHALLENGE Write an expression for the minimum value of a function o...
 51.66: Writing in Math Use the information on page 236 to explain how inco...
 51.67: ACT/SAT The graph of which of the following equations is symmetrica...
 51.68: REVIEW In which equation does every real number x correspond to a n...
 51.69: 2x + 3y = 8 70. x + 4y = 9 x  2y = 3
 51.70: x + 4y = 9 x  2y = 3 3x + 2y = 3
 51.71: 2 1 5 2
 51.72: 4 1 3 1
 51.73: 2 0 1 5 3 1 2 4
 51.74: [1 3] 4 3 2 2 1 0
 51.75: [4 1 3] + [6 5 8]
 51.76: [2 5 7]  [3 8 1]
 51.77: 4 7 2 5 4 11 9
 51.78: 2 3 7 0 _1 3 12 4
 51.79: CONCERTS The price of two lawn seats and a pavilion seat at an outd...
 51.80: 4a  3b = 4 81. 2r + s = 1 82. 3x  2y = 3 3a  2b = 4
 51.81: 2r + s = 1 82. 3x  2y = 3 3a  2b = 4 r  s = 8
 51.82: 3x  2y = 3 3a  2b = 4 r  s = 8 3x + y = 3
 51.83: Graph the system of equations y = 3x and y  x = 4. State the solu...
 51.84: (6, 7), (0, 5)
 51.85: (3, 2), (1, 4)
 51.86: (3, 2), (5, 6)
 51.87: (2, 8), (1, 7)
 51.88: (3, 8), (7, 22)
 51.89: (4, 21), (9, 12)
 51.90: x  3 = 7
 51.91: 4 d + 2 = 12
 51.92: 5 k  4 = k + 8
 51.93: GEOMETRY The formula for the surface area of a regular pyramid is S...
 51.94: f(x) = x 2 + 2x  3, x = 2
 51.95: f(x) =  x 2  4x + 5, x = 3
 51.96: f(x) = 3 x 2 + 7x, x = 2
 51.97: f(x) = _2 3 x 2 + 2x  1, x = 3
Solutions for Chapter 51: Graphing Quadratic Functions
Full solutions for Algebra 2, Student Edition (MERRILL ALGEBRA 2)  1st Edition
ISBN: 9780078738302
Solutions for Chapter 51: Graphing Quadratic Functions
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Algebra 2, Student Edition (MERRILL ALGEBRA 2) was written by and is associated to the ISBN: 9780078738302. Chapter 51: Graphing Quadratic Functions includes 97 full stepbystep solutions. This textbook survival guide was created for the textbook: Algebra 2, Student Edition (MERRILL ALGEBRA 2), edition: 1. Since 97 problems in chapter 51: Graphing Quadratic Functions have been answered, more than 53660 students have viewed full stepbystep solutions from this chapter.

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

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).

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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].

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.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

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.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

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.

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