 1.1.1: On a real number line the origin is assigned the number _________ ....
 1.1.2: If 3 and 5 are the coordinates of two points on the real number li...
 1.1.3: If 3 and 4 are the legs of a right triangle, the hypotenuse is ____...
 1.1.4: Use the converse of the Pythagorean Theorem to show that a triangle...
 1.1.5: The area of a triangle whose base is b and whose altitude is h is A...
 1.1.6: The area of a triangle whose base is b and whose altitude is h is A...
 1.1.7: If 1x, y2 are the coordinates of a point P in the xyplane, then x ...
 1.1.8: The coordinate axes divide the xyplane into four sections called _...
 1.1.9: If three distinct points P, Q, and R all lie on a line and if d1P, ...
 1.1.10: True or False The distance between two points is sometimes a negati...
 1.1.11: True or False The point 1 1, 42 lies in quadrant IV of the Cartesi...
 1.1.12: True or False The midpoint of a line segment is found by averaging ...
 1.1.13: In 13 and 14, plot each point in the xyplane. Tell in which quadra...
 1.1.14: In 13 and 14, plot each point in the xyplane. Tell in which quadra...
 1.1.15: Plot the points 12, 02, 12, 32, 12, 42, 12, 12, and 12, 12. Descr...
 1.1.16: Plot the points 10, 32, 11, 32, 1 2, 32, 15, 32, and 1 4, 32. Des...
 1.1.17: Plot the points 10, 32, 11, 32, 1 2, 32, 15, 32, and 1 4, 32. Des...
 1.1.18: Plot the points 10, 32, 11, 32, 1 2, 32, 15, 32, and 1 4, 32. Des...
 1.1.19: Plot the points 10, 32, 11, 32, 1 2, 32, 15, 32, and 1 4, 32. Des...
 1.1.20: Plot the points 10, 32, 11, 32, 1 2, 32, 15, 32, and 1 4, 32. Des...
 1.1.21: In 2126, select a setting so that each given point will lie within ...
 1.1.22: In 2126, select a setting so that each given point will lie within ...
 1.1.23: In 2126, select a setting so that each given point will lie within ...
 1.1.24: In 2126, select a setting so that each given point will lie within ...
 1.1.25: In 2126, select a setting so that each given point will lie within ...
 1.1.26: In 2126, select a setting so that each given point will lie within ...
 1.1.27: In 2732, determine the viewing window used.
 1.1.28: In 2732, determine the viewing window used.
 1.1.29: In 2732, determine the viewing window used.
 1.1.30: In 2732, determine the viewing window used.
 1.1.31: In 2732, determine the viewing window used.
 1.1.32: In 2732, determine the viewing window used.
 1.1.33: In 3344, find the distance d1P1 , P22 between the points P1 and P2 .
 1.1.34: In 3344, find the distance d1P1 , P22 between the points P1 and P2 .
 1.1.35: In 3344, find the distance d1P1 , P22 between the points P1 and P2 .
 1.1.36: In 3344, find the distance d1P1 , P22 between the points P1 and P2 .
 1.1.37: In 3344, find the distance d1P1 , P22 between the points P1 and P2 .
 1.1.38: In 3344, find the distance d1P1 , P22 between the points P1 and P2 .
 1.1.39: In 3344, find the distance d1P1 , P22 between the points P1 and P2 .
 1.1.40: In 3344, find the distance d1P1 , P22 between the points P1 and P2 .
 1.1.41: In 3344, find the distance d1P1 , P22 between the points P1 and P2 .
 1.1.42: In 3344, find the distance d1P1 , P22 between the points P1 and P2 .
 1.1.43: In 3344, find the distance d1P1 , P22 between the points P1 and P2 .
 1.1.44: In 3344, find the distance d1P1 , P22 between the points P1 and P2 .
 1.1.45: In 4548, find the length of the line segment. Assume that the endpo...
 1.1.46: In 4548, find the length of the line segment. Assume that the endpo...
 1.1.47: In 4548, find the length of the line segment. Assume that the endpo...
 1.1.48: In 4548, find the length of the line segment. Assume that the endpo...
 1.1.49: In 4954, plot each point and form the triangle ABC. Verify that the...
 1.1.50: In 4954, plot each point and form the triangle ABC. Verify that the...
 1.1.51: In 4954, plot each point and form the triangle ABC. Verify that the...
 1.1.52: In 4954, plot each point and form the triangle ABC. Verify that the...
 1.1.53: In 4954, plot each point and form the triangle ABC. Verify that the...
 1.1.54: In 4954, plot each point and form the triangle ABC. Verify that the...
 1.1.55: In 5562, find the midpoint of the line segment joining the points P...
 1.1.56: In 5562, find the midpoint of the line segment joining the points P...
 1.1.57: In 5562, find the midpoint of the line segment joining the points P...
 1.1.58: In 5562, find the midpoint of the line segment joining the points P...
 1.1.59: In 5562, find the midpoint of the line segment joining the points P...
 1.1.60: In 5562, find the midpoint of the line segment joining the points P...
 1.1.61: In 5562, find the midpoint of the line segment joining the points P...
 1.1.62: In 5562, find the midpoint of the line segment joining the points P...
 1.1.63: In 5562, find the midpoint of the line segment joining the points P...
 1.1.64: In 6368, tell whether the given points are on the graph of the equa...
 1.1.65: In 6368, tell whether the given points are on the graph of the equa...
 1.1.66: In 6368, tell whether the given points are on the graph of the equa...
 1.1.67: In 6368, tell whether the given points are on the graph of the equa...
 1.1.68: In 6368, tell whether the given points are on the graph of the equa...
 1.1.69: In 6976, the graph of an equation is given. List the intercepts of ...
 1.1.70: In 6976, the graph of an equation is given. List the intercepts of ...
 1.1.71: In 6976, the graph of an equation is given. List the intercepts of ...
 1.1.72: In 6976, the graph of an equation is given. List the intercepts of ...
 1.1.73: In 6976, the graph of an equation is given. List the intercepts of ...
 1.1.74: In 6976, the graph of an equation is given. List the intercepts of ...
 1.1.75: In 6976, the graph of an equation is given. List the intercepts of ...
 1.1.76: In 6976, the graph of an equation is given. List the intercepts of ...
 1.1.77: In 7788, graph each equation by hand by plotting points. Verify you...
 1.1.78: In 7788, graph each equation by hand by plotting points. Verify you...
 1.1.79: In 7788, graph each equation by hand by plotting points. Verify you...
 1.1.80: In 7788, graph each equation by hand by plotting points. Verify you...
 1.1.81: In 7788, graph each equation by hand by plotting points. Verify you...
 1.1.82: In 7788, graph each equation by hand by plotting points. Verify you...
 1.1.83: In 7788, graph each equation by hand by plotting points. Verify you...
 1.1.84: In 7788, graph each equation by hand by plotting points. Verify you...
 1.1.85: In 7788, graph each equation by hand by plotting points. Verify you...
 1.1.86: In 7788, graph each equation by hand by plotting points. Verify you...
 1.1.87: In 7788, graph each equation by hand by plotting points. Verify you...
 1.1.88: In 7788, graph each equation by hand by plotting points. Verify you...
 1.1.89: In 8996, graph each equation using a graphing utility. Use a graphi...
 1.1.90: In 8996, graph each equation using a graphing utility. Use a graphi...
 1.1.91: In 8996, graph each equation using a graphing utility. Use a graphi...
 1.1.92: In 8996, graph each equation using a graphing utility. Use a graphi...
 1.1.93: In 8996, graph each equation using a graphing utility. Use a graphi...
 1.1.94: In 8996, graph each equation using a graphing utility. Use a graphi...
 1.1.95: In 8996, graph each equation using a graphing utility. Use a graphi...
 1.1.96: In 8996, graph each equation using a graphing utility. Use a graphi...
 1.1.97: If the point (2, 5) is shifted 3 units right and 2 units down, what...
 1.1.98: If the point (1, 6) is shifted 2 units left and 4 units up, what a...
 1.1.99: The medians of a triangle are the line segments from each vertex to...
 1.1.100: An equilateral triangle is one in which all three sides are of equa...
 1.1.101: In 101104, find the length of each side of the triangle determined ...
 1.1.102: In 101104, find the length of each side of the triangle determined ...
 1.1.103: In 101104, find the length of each side of the triangle determined ...
 1.1.104: In 101104, find the length of each side of the triangle determined ...
 1.1.105: A major league baseball diamond is actually a square, 90 feet on a ...
 1.1.106: The layout of a Little League playing field is a square, 60 feet on...
 1.1.107: Refer to 105. Overlay a rectangular coordinate system on a major le...
 1.1.108: Refer to 106. Overlay a rectangular coordinate system on a Little L...
 1.1.109: A Ford Focus and a Mack truck leave an intersection at the same tim...
 1.1.110: A hotair balloon, headed due east at an average speed of 15 miles ...
 1.1.111: When a draftsman draws three lines that are to intersect at one poi...
 1.1.112: he figure illustrates how net sales of WalMart Stores, Inc., have ...
 1.1.113: Poverty thresholds are determined by the U.S. Census Bureau. A pove...
 1.1.114: Plot the points A = (1, 8) and M = (2, 3) in the xyplane. If M is...
 1.1.115: (a) Graph y = 2x2 , y = x, y = 0 x 0 , and y = 1 1x22 , noting whic...
 1.1.116: Make up an equation satisfied by the ordered pairs 12, 02, 14, 02, ...
 1.1.117: Draw a graph that contains the points 1 2, 12, 10, 12, 11, 32, an...
 1.1.118: Explain what is meant by a complete graph
 1.1.119: Write a paragraph that describes a Cartesian plane. Then write a se...
Solutions for Chapter 1.1: The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing Equations
Full solutions for Precalculus Enhanced with Graphing Utilities  6th Edition
ISBN: 9780132854351
Solutions for Chapter 1.1: The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing Equations
Get Full SolutionsSince 119 problems in chapter 1.1: The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing Equations have been answered, more than 77746 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Precalculus Enhanced with Graphing Utilities, edition: 6. This expansive textbook survival guide covers the following chapters and their solutions. Precalculus Enhanced with Graphing Utilities was written by and is associated to the ISBN: 9780132854351. Chapter 1.1: The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing Equations includes 119 full stepbystep solutions.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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).

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

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].

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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

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.

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.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

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.

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 •

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

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

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.