 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 8853 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 and is associated to the ISBN: 9781559537636.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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.

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.

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

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

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).

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.

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

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Outer product uv T
= column times row = rank one matrix.

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

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

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

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).