 19.1: For Exercises 13, use the graph at the right. Name the ordered pair...
 19.2: For Exercises 13, use the graph at the right. Name the ordered pair...
 19.3: For Exercises 13, use the graph at the right. Identify the independ...
 19.4: The graph at the right represents Alexis speed as he rides his bike...
 19.5: Identify the graph that represents the altitude of a skydiver just ...
 19.6: PHYSICAL SCIENCE For Exercises 69, use the table and the informatio...
 19.7: PHYSICAL SCIENCE For Exercises 69, use the table and the informatio...
 19.8: PHYSICAL SCIENCE For Exercises 69, use the table and the informatio...
 19.9: PHYSICAL SCIENCE For Exercises 69, use the table and the informatio...
 19.10: For Exercises 1012, use the graph at the right. Name the ordered pa...
 19.11: For Exercises 1012, use the graph at the right. Name the ordered pa...
 19.12: For Exercises 1012, use the graph at the right. Identify the indepe...
 19.13: For Exercises 1315, use the graph at the right. Name the ordered pa...
 19.14: For Exercises 1315, use the graph at the right. Name the ordered pa...
 19.15: For Exercises 1315, use the graph at the right. Identify the indepe...
 19.16: The graph below represents Teresas temperature when she was sick. D...
 19.17: The graph below represents the altitude of a group of hikers. Descr...
 19.18: TOYS Identify the graph that displays the speed of a radiocontroll...
 19.19: INCOME In general, as people get older, their incomes increase unti...
 19.20: CARS Refer to the information at the left. A car was purchased new ...
 19.21: CHEMISTRY When ice is exposed to temperatures above 32F, it begins ...
 19.22: For Exercises 2225, use the table below. Identify the independent a...
 19.23: For Exercises 2225, use the table below. Identify the domain and ra...
 19.24: For Exercises 2225, use the table below. State whether the function...
 19.25: For Exercises 2225, use the table below. Predict the sum of the int...
 19.26: REASONING Compare and contrast dependent and independent variables.
 19.27: OPEN ENDED Give an example of a relation. Identify the domain and r...
 19.28: CHALLENGE Eva is 23 years older than Lisa. Draw a graph showing Eva...
 19.29: Writing in Math Use the data about concussions on page 53 to explai...
 19.30: What is the range for the function {(1, 3), (5, 7), (9, 11)}? A {1,...
 19.31: REVIEW If 3x  2y = 5 and x = 2, what value of y makes the equation...
 19.32: What is 121 ? (
 19.33: Identify the hypothesis and conclusion of each statement. You can s...
 19.34: Identify the hypothesis and conclusion of each statement. The expre...
 19.35: Identify the hypothesis and conclusion of each statement. Evaluate ...
 19.36: Write an algebraic expression for each verbal expression. the produ...
 19.37: Write an algebraic expression for each verbal expression. three tim...
Solutions for Chapter 19: Functions and Graphs
Full solutions for Algebra 1, Student Edition (MERRILL ALGEBRA 1)  1st Edition
ISBN: 9780078738227
Solutions for Chapter 19: Functions and Graphs
Get Full SolutionsAlgebra 1, Student Edition (MERRILL ALGEBRA 1) was written by and is associated to the ISBN: 9780078738227. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 19: Functions and Graphs includes 37 full stepbystep solutions. This textbook survival guide was created for the textbook: Algebra 1, Student Edition (MERRILL ALGEBRA 1) , edition: 1. Since 37 problems in chapter 19: Functions and Graphs have been answered, more than 34374 students have viewed full stepbystep solutions from this chapter.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

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

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

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

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

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.

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.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

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.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

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

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.

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 •

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

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

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.