- 1.3.1: In the rectangular coordinate system, the horizontal number line is...
- 1.3.2: In the rectangular coordinate system, the vertical number line is c...
- 1.3.3: In the rectangular coordinate system, the point of intersection of ...
- 1.3.4: The axes of the rectangular coordinate system divide the plane into...
- 1.3.5: The first number in an ordered pair such as (8, 3) is called the __...
- 1.3.6: The ordered pair (4, 19) is a/an ______________ of the equation y =...
- 1.3.7: (4, -1)
- 1.3.8: (3, -2)
- 1.3.9: (-4, 0)
- 1.3.10: (0, -3)
- 1.3.11: y = x2 - 4
- 1.3.12: y = x2 - 9
- 1.3.13: y = x - 2
- 1.3.14: y = x + 2
- 1.3.15: y = 2x + 1
- 1.3.16: y = 2x - 4
- 1.3.17: y = - 1 2 x
- 1.3.18: y = - 1 2 x + 2
- 1.3.19: y = x + 1
- 1.3.20: y = x - 1
- 1.3.21: y = 2 x
- 1.3.22: y = -2 x
- 1.3.23: y = -x2
- 1.3.24: y = - 1 2 x2
- 1.3.25: y = x3
- 1.3.26: y = x3 - 1
- 1.3.27: [-5, 5, 1] by [-5, 5, 1]
- 1.3.28: [-10, 10, 2] by [-4, 4, 2]
- 1.3.29: [-20, 80, 10] by [-30, 70, 10]
- 1.3.30: [-40, 40, 20] by [-1000, 1000, 100]
- 1.3.31: Which equation corresponds to Y2 in the table? a. y2 = x + 8 b. y2 ...
- 1.3.32: Which equation corresponds to Y1 in the table? a. y1 = -3x b. y1 = ...
- 1.3.33: Does the graph of Y2 pass through the origin?
- 1.3.34: Does the graph of Y1 pass through the origin?
- 1.3.35: At which point does the graph of Y2 cross the x-axis?
- 1.3.36: At which point does the graph of Y2 cross the y-axis?
- 1.3.37: At which points do the graphs of Y1 and Y2 intersect?
- 1.3.38: For which values of x is Y1 = Y2?
- 1.3.39: The y-value is four more than twice the x-value.
- 1.3.40: The y-value is the difference between four and twice the x-value.
- 1.3.41: The y-value is three decreased by the square of the x-value.
- 1.3.42: The y-value is two more than the square of the x-value.
- 1.3.43: y = 5 (Let x = -3, -2, -1, 0, 1, 2, and 3.)
- 1.3.44: y = -1 (Let x = -3, -2, -1, 0, 1, 2, and 3.)
- 1.3.45: y = 1 x (Let x = -2, -1, - 1 2 , - 1 3 , 1 3 , 1 2 , 1, and 2.)
- 1.3.46: y = - 1 x (Let x = -2, -1, - 1 2 , - 1 3 , 1 3 , 1 2 , 1, and 2.)
- 1.3.47: Estimate the top marginal tax rate in 2010.
- 1.3.48: Estimate the top marginal tax rate in 1925.
- 1.3.49: For the period shown, during which year did the United States have ...
- 1.3.50: For the period from 1950 through 2010, during which year did the Un...
- 1.3.51: For the period shown, during which ten-year period did the top marg...
- 1.3.52: For the period shown, during which five-year period did the top mar...
- 1.3.53: At which age, estimated to the nearest year, do women have the leas...
- 1.3.54: At which age do men have the greatest number of awakenings during t...
- 1.3.55: Estimate, to the nearest tenth, the difference between the average ...
- 1.3.56: Estimate, to the nearest tenth, the difference between the average ...
- 1.3.57: As the blizzard got worse, the snow fell harder and harder.
- 1.3.58: The snow fell more and more softly.
- 1.3.59: It snowed hard, but then it stopped. After a short time, the snow s...
- 1.3.60: It snowed softly, and then it stopped. After a short time, the snow...
- 1.3.61: An airplane flew from Miami to San Francisco. a. Planes Height Seco...
- 1.3.62: At noon, you begin to breathe in. a. Volume of Air in Lungs Time af...
- 1.3.63: Measurements are taken of a persons height from birth to age 100. a...
- 1.3.64: You begin your bike ride by riding down a hill. Then you ride up an...
- 1.3.65: What is the rectangular coordinate system?
- 1.3.66: Explain how to plot a point in the rectangular coordinate system. G...
- 1.3.67: Explain why (5, -2) and (-2, 5) do not represent the same point.
- 1.3.68: Explain how to graph an equation in the rectangular coordinate system.
- 1.3.69: What does a [-20, 2, 1] by [-4, 5, 0.5] viewing rectangle mean?
- 1.3.70: Checking Account Balance
- 1.3.71: Hair Length
- 1.3.72: Use a graphing utility to verify each of your hand-drawn graphs in ...
- 1.3.73: The rectangular coordinate system provides a geometric picture of w...
- 1.3.74: There is something wrong with my graphing utility because it is not...
- 1.3.75: A horizontal line is not a graph that tells the story of the number...
- 1.3.76: I told my story with a graph, so I can be confident that there is a...
- 1.3.77: If the product of a points coordinates is positive, the point must ...
- 1.3.78: If a point is on the x-axis, it is neither up nor down, so x = 0.
- 1.3.79: If a point is on the y-axis, its x-coordinate must be 0.
- 1.3.80: The ordered pair (2, 5) satisfies 3y - 2x = -4.
- 1.3.81: You park your car at the garage for four hours on Tuesday and five ...
- 1.3.82: On Thursday, you paid $12 for parking at the garage. Describe how l...
- 1.3.83: Find the absolute value: -14.3. (Section 1.2, Example 1)
- 1.3.84: Simplify: [12 - (13 - 17)] - [9 - (6 - 10)] (Section 1.2, Examples ...
- 1.3.85: Simplify: 6x - 5(4x + 3) - 10. (Section 1.2, Example 13)
- 1.3.86: If -9 is substituted for x in the equation 4x - 3 = 5x + 6, is the ...
- 1.3.87: Simplify: 13 - 3(x + 2).
- 1.3.88: Simplify: 10a 3x + 1 2 b.
Solutions for Chapter 1.3: Graphing Equations
Full solutions for Intermediate Algebra for College Students | 6th Edition
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.
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
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
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.
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 Fn-1c can be computed with ne/2 multiplications. Revolutionary.
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
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.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = 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).
The diagonal entry (first nonzero) at the time when a row is used in elimination.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).
Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
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