 1.4.1.4.1: In Exercises 15, fill in the blanks.
 1.4.1.4.2: In Exercises 15, fill in the blanks.
 1.4.1.4.3: In Exercises 15, fill in the blanks.
 1.4.1.4.4: In Exercises 15, fill in the blanks.
 1.4.1.4.5: In Exercises 15, fill in the blanks.
 1.4.1.4.6: In Exercises 112, sketch the graphs of the three functions by hand ...
 1.4.1.4.7: In Exercises 112, sketch the graphs of the three functions by hand ...
 1.4.1.4.8: In Exercises 112, sketch the graphs of the three functions by hand ...
 1.4.1.4.9: In Exercises 112, sketch the graphs of the three functions by hand ...
 1.4.1.4.10: In Exercises 112, sketch the graphs of the three functions by hand ...
 1.4.1.4.11: In Exercises 112, sketch the graphs of the three functions by hand ...
 1.4.1.4.12: In Exercises 112, sketch the graphs of the three functions by hand ...
 1.4.1.4.13: Use the graph of to sketch each graph. To print an enlarged copy of...
 1.4.1.4.14: Use the graph of to sketch each graph. To print an enlarged copy of...
 1.4.1.4.15: In Exercises 1520, identify the parent function and describe the tr...
 1.4.1.4.16: In Exercises 1520, identify the parent function and describe the tr...
 1.4.1.4.17: In Exercises 1520, identify the parent function and describe the tr...
 1.4.1.4.18: In Exercises 1520, identify the parent function and describe the tr...
 1.4.1.4.19: In Exercises 1520, identify the parent function and describe the tr...
 1.4.1.4.20: In Exercises 1520, identify the parent function and describe the tr...
 1.4.1.4.21: In Exercises 2126, compare the graph of the function with the graph...
 1.4.1.4.22: In Exercises 2126, compare the graph of the function with the graph...
 1.4.1.4.23: In Exercises 2126, compare the graph of the function with the graph...
 1.4.1.4.24: In Exercises 2126, compare the graph of the function with the graph...
 1.4.1.4.25: In Exercises 2126, compare the graph of the function with the graph...
 1.4.1.4.26: In Exercises 2126, compare the graph of the function with the graph...
 1.4.1.4.27: In Exercises 2732, compare the graph of the function with the graph...
 1.4.1.4.28: In Exercises 2732, compare the graph of the function with the graph...
 1.4.1.4.29: In Exercises 2732, compare the graph of the function with the graph...
 1.4.1.4.30: In Exercises 2732, compare the graph of the function with the graph...
 1.4.1.4.31: In Exercises 2732, compare the graph of the function with the graph...
 1.4.1.4.32: In Exercises 2732, compare the graph of the function with the graph...
 1.4.1.4.33: In Exercises 3338, compare the graph of the function with the graph...
 1.4.1.4.34: In Exercises 3338, compare the graph of the function with the graph...
 1.4.1.4.35: In Exercises 3338, compare the graph of the function with the graph...
 1.4.1.4.36: In Exercises 3338, compare the graph of the function with the graph...
 1.4.1.4.37: In Exercises 3338, compare the graph of the function with the graph...
 1.4.1.4.38: In Exercises 3338, compare the graph of the function with the graph...
 1.4.1.4.39: In Exercises 3942, use a graphing utility to graph the three functi...
 1.4.1.4.40: In Exercises 3942, use a graphing utility to graph the three functi...
 1.4.1.4.41: In Exercises 3942, use a graphing utility to graph the three functi...
 1.4.1.4.42: In Exercises 3942, use a graphing utility to graph the three functi...
 1.4.1.4.43: In Exercises 4356, g is related to one of the six parent functions ...
 1.4.1.4.44: In Exercises 4356, g is related to one of the six parent functions ...
 1.4.1.4.45: In Exercises 4356, g is related to one of the six parent functions ...
 1.4.1.4.46: In Exercises 4356, g is related to one of the six parent functions ...
 1.4.1.4.47: In Exercises 4356, g is related to one of the six parent functions ...
 1.4.1.4.48: In Exercises 4356, g is related to one of the six parent functions ...
 1.4.1.4.49: In Exercises 4356, g is related to one of the six parent functions ...
 1.4.1.4.50: In Exercises 4356, g is related to one of the six parent functions ...
 1.4.1.4.51: In Exercises 4356, g is related to one of the six parent functions ...
 1.4.1.4.52: In Exercises 4356, g is related to one of the six parent functions ...
 1.4.1.4.53: In Exercises 4356, g is related to one of the six parent functions ...
 1.4.1.4.54: In Exercises 4356, g is related to one of the six parent functions ...
 1.4.1.4.55: In Exercises 4356, g is related to one of the six parent functions ...
 1.4.1.4.56: In Exercises 4356, g is related to one of the six parent functions ...
 1.4.1.4.57: Fuel Use The amounts of fuel (in billions of gallons) used by vans,...
 1.4.1.4.58: Finance The amounts (in billions of dollars) of home mortgage debt ...
 1.4.1.4.59: True or False? In Exercises 59 and 60, determine whether the statem...
 1.4.1.4.60: True or False? In Exercises 59 and 60, determine whether the statem...
 1.4.1.4.61: Exploration Use the fact that the graph of has xintercepts at and ...
 1.4.1.4.62: Exploration Use the fact that the graph of has xintercepts at and ...
 1.4.1.4.63: Exploration Use the fact that the graph of is increasing on the int...
 1.4.1.4.64: Exploration Use the fact that the graph of is increasing on the int...
 1.4.1.4.65: Library of Parent Functions In Exercises 6568, determine which equa...
 1.4.1.4.66: Library of Parent Functions In Exercises 6568, determine which equa...
 1.4.1.4.67: Library of Parent Functions In Exercises 6568, determine which equa...
 1.4.1.4.68: Library of Parent Functions In Exercises 6568, determine which equa...
 1.4.1.4.69: In Exercises 69 and 70, determine whether the lines and passing thr...
 1.4.1.4.70: In Exercises 69 and 70, determine whether the lines and passing thr...
 1.4.1.4.71: In Exercises 7174, find the domain of the function.
 1.4.1.4.72: In Exercises 7174, find the domain of the function.
 1.4.1.4.73: In Exercises 7174, find the domain of the function.
 1.4.1.4.74: In Exercises 7174, find the domain of the function.
Solutions for Chapter 1.4: Shifting, Reflecting, and Stretching Graphs
Full solutions for Precalculus With Limits A Graphing Approach  5th Edition
ISBN: 9780618851522
Solutions for Chapter 1.4: Shifting, Reflecting, and Stretching Graphs
Get Full SolutionsSince 74 problems in chapter 1.4: Shifting, Reflecting, and Stretching Graphs have been answered, more than 47940 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Precalculus With Limits A Graphing Approach, edition: 5. Precalculus With Limits A Graphing Approach was written by and is associated to the ISBN: 9780618851522. Chapter 1.4: Shifting, Reflecting, and Stretching Graphs includes 74 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

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

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

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

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.

Iterative method.
A sequence of steps intended to approach the desired solution.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

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

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!