 3.3.3.3.1: Graph each linear function. See Examples 1 and 2.f 1x2 = 2x
 3.3.3.3.2: Graph each linear function. See Examples 1 and 2.f 1x2 = 2x
 3.3.3.3.3: Graph each linear function. See Examples 1 and 2.f 1x2 = 2x + 3
 3.3.3.3.4: Graph each linear function. See Examples 1 and 2.f 1x2 = 2x + 6
 3.3.3.3.5: Graph each linear function. See Examples 1 and 2.f 1x2 = 12x
 3.3.3.3.6: Graph each linear function. See Examples 1 and 2.f 1x2 = 13x
 3.3.3.3.7: Graph each linear function. See Examples 1 and 2.f 1x2 = 12x  4
 3.3.3.3.8: Graph each linear function. See Examples 1 and 2.f 1x2 = 13x  2
 3.3.3.3.9: The graph of f1x2 = 5x follows. Use this graph to match each linear...
 3.3.3.3.10: The graph of f1x2 = 5x follows. Use this graph to match each linear...
 3.3.3.3.11: The graph of f1x2 = 5x follows. Use this graph to match each linear...
 3.3.3.3.12: The graph of f1x2 = 5x follows. Use this graph to match each linear...
 3.3.3.3.13: Graph each linear function by finding x and yintercepts. Then wri...
 3.3.3.3.14: Graph each linear function by finding x and yintercepts. Then wri...
 3.3.3.3.15: Graph each linear function by finding x and yintercepts. Then wri...
 3.3.3.3.16: Graph each linear function by finding x and yintercepts. Then wri...
 3.3.3.3.17: Graph each linear function by finding x and yintercepts. Then wri...
 3.3.3.3.18: Graph each linear function by finding x and yintercepts. Then wri...
 3.3.3.3.19: Graph each linear function by finding x and yintercepts. Then wri...
 3.3.3.3.20: Graph each linear function by finding x and yintercepts. Then wri...
 3.3.3.3.21: Graph each linear equation. See Examples 6 and 7.x = 1
 3.3.3.3.22: Graph each linear equation. See Examples 6 and 7.y = 5
 3.3.3.3.23: Graph each linear equation. See Examples 6 and 7.y = 0
 3.3.3.3.24: Graph each linear equation. See Examples 6 and 7.x = 0
 3.3.3.3.25: Graph each linear equation. See Examples 6 and 7.y + 7 = 0
 3.3.3.3.26: Graph each linear equation. See Examples 6 and 7.x  3 = 0
 3.3.3.3.27: Match each equation below with its graph.y = 2
 3.3.3.3.28: Match each equation below with its graph.x = 3
 3.3.3.3.29: Match each equation below with its graph.x  2 = 0
 3.3.3.3.30: Match each equation below with its graph.y + 1 = 0
 3.3.3.3.31: Graph each linear equation. See Examples 1 through 7.x + 2y = 8
 3.3.3.3.32: Graph each linear equation. See Examples 1 through 7.x  3y = 3
 3.3.3.3.33: Graph each linear equation. See Examples 1 through 7.3x + 5y = 7
 3.3.3.3.34: Graph each linear equation. See Examples 1 through 7.3x  2y = 5
 3.3.3.3.35: Graph each linear equation. See Examples 1 through 7.x + 8y = 8
 3.3.3.3.36: Graph each linear equation. See Examples 1 through 7.x  3y = 9
 3.3.3.3.37: Graph each linear equation. See Examples 1 through 7.5 = 6x  y
 3.3.3.3.38: Graph each linear equation. See Examples 1 through 7. 4 = x  3y
 3.3.3.3.39: Graph each linear equation. See Examples 1 through 7.x + 10y = 11
 3.3.3.3.40: Graph each linear equation. See Examples 1 through 7.x + 9 = y
 3.3.3.3.41: Graph each linear equation. See Examples 1 through 7.y = 32
 3.3.3.3.42: Graph each linear equation. See Examples 1 through 7.x = 32
 3.3.3.3.43: Graph each linear equation. See Examples 1 through 7.2x + 3y = 6
 3.3.3.3.44: Graph each linear equation. See Examples 1 through 7.4x + y = 5
 3.3.3.3.45: Graph each linear equation. See Examples 1 through 7.x + 3 = 0
 3.3.3.3.46: Graph each linear equation. See Examples 1 through 7.y  6 = 0
 3.3.3.3.47: Graph each linear equation. See Examples 1 through 7.f1x2 = 34x + 2
 3.3.3.3.48: Graph each linear equation. See Examples 1 through 7.f1x2 = 43x + 2
 3.3.3.3.49: Graph each linear equation. See Examples 1 through 7.f1x2 = x
 3.3.3.3.50: Graph each linear equation. See Examples 1 through 7.f1x2 = x
 3.3.3.3.51: Graph each linear equation. See Examples 1 through 7.f1x = x
 3.3.3.3.52: Graph each linear equation. See Examples 1 through 7.f1x2 = 2x
 3.3.3.3.53: Graph each linear equation. See Examples 1 through 7.f1x2 = 4x  13
 3.3.3.3.54: Graph each linear equation. See Examples 1 through 7.f1x2 = 3x +34
 3.3.3.3.55: Graph each linear equation. See Examples 1 through 7.x = 3
 3.3.3.3.56: Graph each linear equation. See Examples 1 through 7.f1x2 = 3
 3.3.3.3.57: Solve the following. See Sections 2.6 and 2.7.x  3 0 = 6
 3.3.3.3.58: Solve the following. See Sections 2.6 and 2.7.0 x + 2 0 6 4
 3.3.3.3.59: Solve the following. See Sections 2.6 and 2.7.2x + 5 0 7 3
 3.3.3.3.60: Solve the following. See Sections 2.6 and 2.7.0 5x 0 = 10
 3.3.3.3.61: Solve the following. See Sections 2.6 and 2.7.3x  4 0 2
 3.3.3.3.62: Solve the following. See Sections 2.6 and 2.7. 7x  2 0 5
 3.3.3.3.63: Simplify. See Section 1.3.6  32  8
 3.3.3.3.64: Simplify. See Section 1.3.4  51  0
 3.3.3.3.65: Simplify. See Section 1.3.8  1 223  1 22
 3.3.3.3.66: Simplify. See Section 1.3.12  1 3210  1 92
 3.3.3.3.67: Simplify. See Section 1.3.0  65  0
 3.3.3.3.68: Simplify. See Section 1.3.2  23  5
 3.3.3.3.69: Think about the appearance of each graph. Without graphing, determi...
 3.3.3.3.70: Think about the appearance of each graph. Without graphing, determi...
 3.3.3.3.71: Think about the appearance of each graph. Without graphing, determi...
 3.3.3.3.72: Think about the appearance of each graph. Without graphing, determi...
 3.3.3.3.73: Broyhill Furniture found that it takes 2 hours to manufactureeach t...
 3.3.3.3.74: While manufacturing two different digital camera models,Kodak found...
 3.3.3.3.75: The cost of renting a car for a day is given by the linear function...
 3.3.3.3.76: The cost of renting a piece of machinery is given by the linearfunc...
 3.3.3.3.77: The yearly cost of tuition (instate) and required fees forattendin...
 3.3.3.3.78: The yearly cost of tuition (instate) and required fees for attendi...
 3.3.3.3.79: In your own words, explain how to find x and yintercepts.
 3.3.3.3.80: Explain why it is a good idea to use three points to graph a linear...
 3.3.3.3.81: Discuss whether a vertical line ever has a yintercept.
 3.3.3.3.82: Discuss whether a horizontal line ever has an xintercept
 3.3.3.3.83: The graph of f 1x2 or y = 4x is given below. Without actually grap...
 3.3.3.3.84: The graph of f 1x2 or y = 4x is given below. Without actually grap...
 3.3.3.3.85: It is true that for any function f 1x2, the graph of f 1x2 + k is t...
 3.3.3.3.86: It is true that for any function f 1x2, the graph of f 1x2 + k is t...
 3.3.3.3.87: It is true that for any function f 1x2, the graph of f 1x2 + k is t...
 3.3.3.3.88: It is true that for any function f 1x2, the graph of f 1x2 + k is t...
 3.3.3.3.89: Use a graphing calculator to verify the results of each exerciseExe...
 3.3.3.3.90: Use a graphing calculator to verify the results of each exerciseExe...
 3.3.3.3.91: Use a graphing calculator to verify the results of each exerciseExe...
 3.3.3.3.92: Use a graphing calculator to verify the results of each exerciseExe...
Solutions for Chapter 3.3: Graphing Linear Functions
Full solutions for Intermediate Algebra  6th Edition
ISBN: 9780321785046
Solutions for Chapter 3.3: Graphing Linear Functions
Get Full SolutionsSince 92 problems in chapter 3.3: Graphing Linear Functions have been answered, more than 58813 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Intermediate Algebra, edition: 6. Intermediate Algebra was written by and is associated to the ISBN: 9780321785046. Chapter 3.3: Graphing Linear Functions includes 92 full stepbystep solutions.

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

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

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

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

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.

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

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.

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.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

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

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.

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.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

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

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

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.