 4.1.1: Find the slope of each line using a slope triangle or the slope for...
 4.1.2: Find the slope of the line through each pair of points. Then name a...
 4.1.3: Given the slope of a line and one point on the line, name two other...
 4.1.4: Run the LINES program five times. Start by playing the easy level o...
 4.1.5: Each table gives the coordinates of four points on a different line...
 4.1.6: Consider lines a and b shown in the graph at right. a. How are the ...
 4.1.7: APPLICATION Recall Hectors Internet use from the investigation. You...
 4.1.8: If a and c are the lengths of the vertical and horizontal segments ...
 4.1.9: This line has a slope of 1. Graph it on your own paper. a. Draw a s...
 4.1.10: APPLICATION A hotair balloonist gathered the data in this table. a...
 4.1.11: When you make a scatter plot of realworld data, you may see a line...
 4.1.12: The base of a triangle was recorded as 18.3 0.1 cm and the height w...
 4.1.13: Calista has five brothers. The mean of her brothers ages is 10 year...
 4.1.14: Enter {3, 1, 2, 8, 10} into list L1 on your calculator. a. Write a ...
 4.1.15: Convert each decimal number to a percent. a. 0.85 b. 1.50 c. 0.065 ...
 4.1.16: The equation 7x 10 = 2x + 3 is solved by balancing. Explain what ha...
Solutions for Chapter 4.1: A Formula for Slope
Full solutions for Discovering Algebra: An Investigative Approach  2nd Edition
ISBN: 9781559537636
Solutions for Chapter 4.1: A Formula for Slope
Get Full SolutionsThis textbook survival guide was created for the textbook: Discovering Algebra: An Investigative Approach, edition: 2. This expansive textbook survival guide covers the following chapters and their solutions. Since 16 problems in chapter 4.1: A Formula for Slope have been answered, more than 2870 students have viewed full stepbystep solutions from this chapter. Chapter 4.1: A Formula for Slope includes 16 full stepbystep solutions. Discovering Algebra: An Investigative Approach was written by Patricia and is associated to the ISBN: 9781559537636.

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).

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

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

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

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.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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

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.

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.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)ยท(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 .

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).

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

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

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

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