 57.1: y = 3(x + 3 ) 2
 57.2: y = _1 3 (x  1 ) 2 + 3 3
 57.3: y = 2 x 2 + 16x  31
 57.4: y = 2 x 2 + 16x  31
 57.5: y = 5(x + 3 ) 2  1
 57.6: y = x 2 + 8x  3
 57.7: y = 3 x 2  18x + 11
 57.8: y O x (2, 0) (1, 4
 57.9: (3, 6) (5, 2)
 57.10: O x (2, 3) (4, 5)
 57.11: If the water lands 3 feet away from the jet, find a quadratic funct...
 57.12: Suppose a worker increases the water pressure so that the stream re...
 57.13: y = 4(x + 3) 2 + 1
 57.14: y = (x  5) 2  3
 57.15: y = _1 4 (x  2) 2 + 4
 57.16: y = _1 2 (x  3) 2  5
 57.17: y = x 2 + 6x + 2
 57.18: y = x 2  8x + 18
 57.19: What is the effect on the graph of the equation y = x 2 + 2 when th...
 57.20: What is the effect on the graph of the equation y = x 2 + 2 when th...
 57.21: y = 2(x + 3)
 57.22: y = _1 3 (x  1) 2 + 2
 57.23: y = x 2  4x + 8
 57.24: y = x 2  6x + 1
 57.25: y = 5x 2  6
 57.26: y = 8x 2 + 3
 57.27: 10 6 4 2 4 6 8642
 57.28: (3, 6) (4, 3)
 57.29: y O x (3, 0) (6, 6)
 57.30: (5, 4) (6, 1)
 57.31: (0, 5) (3, 8)
 57.32: (3, 2) (
 57.33: Which sprinkler angle will send water the highest? Explain your rea...
 57.34: Which sprinkler angle will send water the farthest? Explain your re...
 57.35: Which sprinkler angle will produce the widest path? The narrowest p...
 57.36: y = 4x 2 + 16x 11
 57.37: y = 5x 2  40x  80
 57.38: y =  _1 2 x 2 + 5x  _27 2
 57.39: y = _1 3 x 2  4x + 15
 57.40: y = 3x 2 + 12x
 57.41: y = 4x 2 + 24x
 57.42: y = 4x 2 + 8x  3
 57.43: y = 2x 2 + 20x 35
 57.44: y = 3x 2 + 3x 1
 57.45: y = 4x 2 12x 11
 57.46: Write an equation for a parabola with vertex at the origin and that...
 57.47: Write an equation for a parabola with vertex at (3, 4) and yinte...
 57.48: Write one sentence that compares the graphs of y = 0.2(x + 3) 2 + 1...
 57.49: Compare the graphs of y = 2(x  5) 2 + 4 and y = 2(x  4) 2  1.
 57.50: AEROSPACE NASAs KC135A aircraft flies in parabolic arcs to simulate...
 57.51: Find the time it will take for the diver to hit the water.
 57.52: Write an equation that models the divers distance above the water i...
 57.53: Find the time it would take for the diver to hit the water from thi...
 57.54: OPEN ENDED Write the equation of a parabola with a vertex of (2, 1...
 57.55: CHALLENGE Given y = ax 2 + bx + c with a 0, derive the equation for...
 57.56: FIND THE ERROR Jenny and Ruben are writing y = x 2 2x + 5 in vertex...
 57.57: CHALLENGE Explain how you can find an equation of a parabola using ...
 57.58: Writing in Math Use the information on page 286 to explain how the ...
 57.59: ACT/SAT If f(x) = x 2 5x and f(n) = 4, which of the following coul...
 57.60: REVIEW Which of the following most accurately describes the transla...
 57.61: 3x 2  6x + 2 = 0
 57.62: 4x 2 + 7x = 11
 57.63: 2x 2  5x + 6 = 0
 57.64: x 2 + 10x + 17 = 0
 57.65: x 2  6x + 18 = 0
 57.66: 4x 2 + 8x = 9
 57.67: 2x 2 + 3 < 0; x = 5
 57.68: 4x 2 + 2x  3 0; x = 1
 57.69: 4x 2  4x + 1 10; x = 2
 57.70: 6x 2 + 3x > 8; x = 0
Solutions for Chapter 57: Analyzing Graphs of Quadratic Functions
Full solutions for Algebra 2, Student Edition (MERRILL ALGEBRA 2)  1st Edition
ISBN: 9780078738302
Solutions for Chapter 57: Analyzing Graphs of Quadratic Functions
Get Full SolutionsAlgebra 2, Student Edition (MERRILL ALGEBRA 2) was written by and is associated to the ISBN: 9780078738302. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Algebra 2, Student Edition (MERRILL ALGEBRA 2), edition: 1. Chapter 57: Analyzing Graphs of Quadratic Functions includes 70 full stepbystep solutions. Since 70 problems in chapter 57: Analyzing Graphs of Quadratic Functions have been answered, more than 56137 students have viewed full stepbystep solutions from this chapter.

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).

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

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

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.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

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).

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Solvable system Ax = b.
The right side b is in the column space of A.

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.

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.

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).