 2.2.1: Solve the equation . 21x + 32  1 =
 2.2.2: Solve the equation x 2  9 = 0.
 2.2.3: The points, if any, at which a graph crosses or touches the coordin...
 2.2.4: The xintercepts of the graph of an equation are those xvalues for...
 2.2.5: If for every point on the graph of an equation the point is also on...
 2.2.6: If the graph of an equation is symmetric with respect to the yaxis...
 2.2.7: If the graph of an equation is symmetric with respect to the origin...
 2.2.8: True or False To find the yintercepts of the graph of an equation,...
 2.2.9: True or False The ycoordinate of a point at which the graph crosse...
 2.2.10: True or False If a graph is symmetric with respect to the xaxis, t...
 2.2.11: In 1116, determine which of the given points are on the graph of th...
 2.2.12: In 1116, determine which of the given points are on the graph of th...
 2.2.13: In 1116, determine which of the given points are on the graph of th...
 2.2.14: In 1116, determine which of the given points are on the graph of th...
 2.2.15: In 1116, determine which of the given points are on the graph of th...
 2.2.16: In 1116, determine which of the given points are on the graph of th...
 2.2.17: In 1728, find the intercepts and graph each equation by plotting po...
 2.2.18: In 1728, find the intercepts and graph each equation by plotting po...
 2.2.19: In 1728, find the intercepts and graph each equation by plotting po...
 2.2.20: In 1728, find the intercepts and graph each equation by plotting po...
 2.2.21: In 1728, find the intercepts and graph each equation by plotting po...
 2.2.22: In 1728, find the intercepts and graph each equation by plotting po...
 2.2.23: In 1728, find the intercepts and graph each equation by plotting po...
 2.2.24: In 1728, find the intercepts and graph each equation by plotting po...
 2.2.25: In 1728, find the intercepts and graph each equation by plotting po...
 2.2.26: In 1728, find the intercepts and graph each equation by plotting po...
 2.2.27: In 1728, find the intercepts and graph each equation by plotting po...
 2.2.28: In 1728, find the intercepts and graph each equation by plotting po...
 2.2.29: In 2938, plot each point. Then plot the point that is symmetric to ...
 2.2.30: In 2938, plot each point. Then plot the point that is symmetric to ...
 2.2.31: In 2938, plot each point. Then plot the point that is symmetric to ...
 2.2.32: In 2938, plot each point. Then plot the point that is symmetric to ...
 2.2.33: In 2938, plot each point. Then plot the point that is symmetric to ...
 2.2.34: In 2938, plot each point. Then plot the point that is symmetric to ...
 2.2.35: In 2938, plot each point. Then plot the point that is symmetric to ...
 2.2.36: In 2938, plot each point. Then plot the point that is symmetric to ...
 2.2.37: In 2938, plot each point. Then plot the point that is symmetric to ...
 2.2.38: In 2938, plot each point. Then plot the point that is symmetric to ...
 2.2.39: In 3950, the graph of an equation is given. (a) Find the intercepts...
 2.2.40: In 3950, the graph of an equation is given. (a) Find the intercepts...
 2.2.41: In 3950, the graph of an equation is given. (a) Find the intercepts...
 2.2.42: In 3950, the graph of an equation is given. (a) Find the intercepts...
 2.2.43: In 3950, the graph of an equation is given. (a) Find the intercepts...
 2.2.44: In 3950, the graph of an equation is given. (a) Find the intercepts...
 2.2.45: In 3950, the graph of an equation is given. (a) Find the intercepts...
 2.2.46: In 3950, the graph of an equation is given. (a) Find the intercepts...
 2.2.47: In 3950, the graph of an equation is given. (a) Find the intercepts...
 2.2.48: In 3950, the graph of an equation is given. (a) Find the intercepts...
 2.2.49: In 3950, the graph of an equation is given. (a) Find the intercepts...
 2.2.50: In 3950, the graph of an equation is given. (a) Find the intercepts...
 2.2.51: In 5154, draw a complete graph so that it has the type of symmetry ...
 2.2.52: In 5154, draw a complete graph so that it has the type of symmetry ...
 2.2.53: In 5154, draw a complete graph so that it has the type of symmetry ...
 2.2.54: In 5154, draw a complete graph so that it has the type of symmetry ...
 2.2.55: In 5570, list the intercepts and test for symmetry. y2 = x + 4
 2.2.56: In 5570, list the intercepts and test for symmetry. y2 = x + 9
 2.2.57: In 5570, list the intercepts and test for symmetry. y = 13 x
 2.2.58: In 5570, list the intercepts and test for symmetry. y = 15 x
 2.2.59: In 5570, list the intercepts and test for symmetry. x2 + y  9 = 0
 2.2.60: In 5570, list the intercepts and test for symmetry. x2  y  4 = 0
 2.2.61: In 5570, list the intercepts and test for symmetry. 9x2 + 4y2 = 36
 2.2.62: In 5570, list the intercepts and test for symmetry. 4x2 + y2 = 4
 2.2.63: In 5570, list the intercepts and test for symmetry. y = x3  27
 2.2.64: In 5570, list the intercepts and test for symmetry. y = x4  1
 2.2.65: In 5570, list the intercepts and test for symmetry. y = x2  3x  4
 2.2.66: In 5570, list the intercepts and test for symmetry. y = x2 + 4
 2.2.67: In 5570, list the intercepts and test for symmetry. y = 3xx2 + 9
 2.2.68: In 5570, list the intercepts and test for symmetry. y = x2  42x
 2.2.69: In 5570, list the intercepts and test for symmetry. y = x3x2  9
 2.2.70: In 5570, list the intercepts and test for symmetry. y = x4 + 12x5
 2.2.71: In 7174, draw a quick sketch of each equation. y = x3
 2.2.72: In 7174, draw a quick sketch of each equation. x = y2
 2.2.73: In 7174, draw a quick sketch of each equation. y = 1x
 2.2.74: In 7174, draw a quick sketch of each equation. y = 1x
 2.2.75: If is a point on the graph of , what is 13, b2 y = 4x + 1 b?
 2.2.76: If is a point on the graph of , what is 12, b2 2x + 3y = 2 b?
 2.2.77: If is a point on the graph of , what is y = x a?
 2.2.78: If is a point on the graph of , what is y = x a?
 2.2.79: Given that the point (1, 2) is on the graph of an equation that is ...
 2.2.80: If the graph of an equation is symmetric with respect to the yaxis...
 2.2.81: If the graph of an equation is symmetric with respect to the origin...
 2.2.82: If the graph of an equation is symmetric with respect to the xaxis...
 2.2.83: Microphones In studios and on stages, cardioid microphones are ofte...
 2.2.84: Solar Energy The solar electric generating systems at Kramer Juncti...
 2.2.85: (a) Graph , and , noting which graphs are the same. (b) Explain why...
 2.2.86: Explain what is meant by a complete graph.
 2.2.87: Draw a graph of an equation that contains two xintercepts; at one ...
 2.2.88: Make up an equation with the intercepts , and . Compare your equati...
 2.2.89: Draw a graph that contains the points , and . Compare your graph wi...
 2.2.90: An equation is being tested for symmetry with respect to the xaxis...
 2.2.91: Draw a graph that contains the points , , and (0, 2) that is symmet...
 2.2.92: yaxis Symmetry Open the yaxis symmetry applet. Move point A aroun...
 2.2.93: xaxis Symmetry Open the xaxis symmetry applet. Move point A aroun...
 2.2.94: Origin Symmetry Open the origin symmetry applet. Move point A aroun...
Solutions for Chapter 2.2: Graphs of Equations in Two Variables; Intercepts; Symmetry
Full solutions for College Algebra  9th Edition
ISBN: 9780321716811
Solutions for Chapter 2.2: Graphs of Equations in Two Variables; Intercepts; Symmetry
Get Full SolutionsChapter 2.2: Graphs of Equations in Two Variables; Intercepts; Symmetry includes 94 full stepbystep solutions. This textbook survival guide was created for the textbook: College Algebra, edition: 9. College Algebra was written by and is associated to the ISBN: 9780321716811. This expansive textbook survival guide covers the following chapters and their solutions. Since 94 problems in chapter 2.2: Graphs of Equations in Two Variables; Intercepts; Symmetry have been answered, more than 32211 students have viewed full stepbystep solutions from this chapter.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

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

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

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.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Iterative method.
A sequence of steps intended to approach the desired solution.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

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

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

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

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

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

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

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

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