 91.1: y = 5
 91.2: y = 2(5 ) x
 91.3: y = (_1 5) x
 91.4: y = 3(4 ) x
 91.5: y = 2 (_1 3) x
 91.6: y = (0.5 )
 91.7: y = 0.3(5 ) x
 91.8: (0, 3) and (1, 6)
 91.9: (0, 18) and (2, 2)
 91.10: Write an exponential function that could be used to model the money...
 91.11: Assume that the amount of money continues to grow at the same rate....
 91.12: 2 n + 4 = _ 1 32
 91.13: 9 2y  1 = 27 y
 91.14: 4 3x + 2 = _1 256
 91.15: 5 2x + 3 125
 91.16: 3 3x  2 > 81
 91.17: 4 4a + 6 16
 91.18: y = 2(3 ) x
 91.19: y = 5(2 )
 91.20: y = 0.5(4 )
 91.21: y = 4 (_1 3) x
 91.22: y = 10(3.5 ) x
 91.23: y = 2(4 )
 91.24: y = 0.4 (_1 3) x
 91.25: y = 3 (_5 2)
 91.26: y = 3 0 x
 91.27: y = 0.2(5 ) x
 91.28: (0, 2) and (2, 32)
 91.29: (0, 3) and (1, 15)
 91.30: (0, 7) and (2, 63)
 91.31: (0, 5) and (3, 135)
 91.32: (0, 0.2) and (4, 51.2)
 91.33: (0, 0.3) and (5, 9.6)
 91.34: Write an exponential function to model the population y of bacteria...
 91.35: How many bacteria were there at 7 P.M. that day?
 91.36: If the principal, interest rate, and number of interest payments ar...
 91.37: Write an equation giving the amount of money you would have after t...
 91.38: Find the account balance after 20 years.
 91.39: 2 3x + 5 = 128
 91.40: 5 n  3 = _1 25
 91.41: (_1 9) m = 81 m + 4
 91.42: (_1 7) y  3 = 343
 91.43: 1 0 x  1 = 10 0 2x  3
 91.44: 3 6 2p = 2 16 p  1
 91.45: 3 n  2 > 27
 91.46: 2 2n _1 16
 91.47: 16 n < 8 n + 1
 91.48: 3 2 5p + 2 1 6 5p
 91.49: y =  (_1 5) x
 91.50: y = 2.5(5 ) x
 91.51: If a typical computer operates with a computational speed s today, ...
 91.52: Suppose your computer operates with a processor speed of 2.8 gigahe...
 91.53: Write an exponential function that could be used to model the U.S. ...
 91.54: Assume that the U.S. population continued to grow at least that fas...
 91.55: RESEARCH Estimate the population of the U.S. in the most recent cen...
 91.56: y = 2 x y = 2 x + 3
 91.57: y = 3 x y = 3 x + 1
 91.58: y = (_ 1 5 ) x y = (_ 1 5 ) x  2
 91.59: y = (_ 1 4 ) x y = (_ 1 4 ) x  1
 91.60: Describe the effect of changing the values of h and k in the equati...
 91.61: OPEN ENDED Give an example of a value of b for which y = b x repres...
 91.62: REASONING Identify each function as linear, quadratic, or exponenti...
 91.63: CHALLENGE Decide whether the following statement is sometimes, alwa...
 91.64: Writing in Math Use the information about womens basketball on page...
 91.65: ACT/SAT If 4 x + 2 = 48, then 4 x = A 3.0 B 6.4 C 6.9 D 12.0
 91.66: REVIEW If the equation y = 3x is graphed, which of the following va...
 91.67: 15 p + p = 16
 91.68: s  3 s + 4 = _6 s 2  16
 91.69: 2a  5 a  9 + _a a + 9 = _6 a 2  81
 91.70: y = x  2 71
 91.71: y = 2[[ x]] 7
 91.72: y = 8
 91.73: 1 0 0 1
 91.74: 2 5 4 10
 91.75: 5 11 6 3
 91.76: ENERGY A circular cell must deliver 18 watts of energy. If each squ...
 91.77: h(x) = 2x  1 7 g(x) = x  5
 91.78: h(x) = x + 3 g(x) = x 2 g
 91.79: h(x) = 2x + 5 g(x) = x + 3
Solutions for Chapter 91: Exponential Functions
Full solutions for Algebra 2, Student Edition (MERRILL ALGEBRA 2)  1st Edition
ISBN: 9780078738302
Solutions for Chapter 91: Exponential Functions
Get Full SolutionsAlgebra 2, Student Edition (MERRILL ALGEBRA 2) was written by and is associated to the ISBN: 9780078738302. Chapter 91: Exponential Functions includes 79 full stepbystep solutions. This textbook survival guide was created for the textbook: Algebra 2, Student Edition (MERRILL ALGEBRA 2), edition: 1. This expansive textbook survival guide covers the following chapters and their solutions. Since 79 problems in chapter 91: Exponential Functions have been answered, more than 53704 students have viewed full stepbystep solutions from this chapter.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

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.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

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.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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.

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.

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 •

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.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

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

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.