 7.3.1: In Exercises 14, find the focus and directrix of each parabola wit...
 7.3.2: In Exercises 14, find the focus and directrix of each parabola wit...
 7.3.3: In Exercises 14, find the focus and directrix of each parabola wit...
 7.3.4: In Exercises 14, find the focus and directrix of each parabola wit...
 7.3.5: In Exercises 516, find the focus and directrix of the parabola wit...
 7.3.6: In Exercises 516, find the focus and directrix of the parabola wit...
 7.3.7: In Exercises 516, find the focus and directrix of the parabola wit...
 7.3.8: In Exercises 516, find the focus and directrix of the parabola wit...
 7.3.9: In Exercises 516, find the focus and directrix of the parabola wit...
 7.3.10: In Exercises 516, find the focus and directrix of the parabola wit...
 7.3.11: In Exercises 516, find the focus and directrix of the parabola wit...
 7.3.12: In Exercises 516, find the focus and directrix of the parabola wit...
 7.3.13: In Exercises 516, find the focus and directrix of the parabola wit...
 7.3.14: In Exercises 516, find the focus and directrix of the parabola wit...
 7.3.15: In Exercises 516, find the focus and directrix of the parabola wit...
 7.3.16: In Exercises 516, find the focus and directrix of the parabola wit...
 7.3.17: In Exercises 1730, find the standard form of /he equation of each ...
 7.3.18: In Exercises 1730, find the standard form of /he equation of each ...
 7.3.19: In Exercises 1730, find the standard form of /he equation of each ...
 7.3.20: In Exercises 1730, find the standard form of /he equation of each ...
 7.3.21: In Exercises 1730, find the standard form of /he equation of each ...
 7.3.22: In Exercises 1730, find the standard form of /he equation of each ...
 7.3.23: In Exercises 1730, find the standard form of /he equation of each ...
 7.3.24: In Exercises 1730, find the standard form of /he equation of each ...
 7.3.25: In Exercises 1730, find the standard form of /he equation of each ...
 7.3.26: In Exercises 1730, find the standard form of /he equation of each ...
 7.3.27: In Exercises 1730, find the standard form of /he equation of each ...
 7.3.28: In Exercises 1730, find the standard form of /he equation of each ...
 7.3.29: In Exercises 1730, find the standard form of /he equation of each ...
 7.3.30: In Exercises 1730, find the standard form of /he equation of each ...
 7.3.31: In Exercises 3134, find the vertex, focus, and directrix of each p...
 7.3.32: In Exercises 3134, find the vertex, focus, and directrix of each p...
 7.3.33: In Exercises 3134, find the vertex, focus, and directrix of each p...
 7.3.34: In Exercises 3134, find the vertex, focus, and directrix of each p...
 7.3.35: In Exercises 3542, find the l!C.rtex,focus, and direc1rix of each ...
 7.3.36: In Exercises 3542, find the l!C.rtex,focus, and direc1rix of each ...
 7.3.37: In Exercises 3542, find the l!C.rtex,focus, and direc1rix of each ...
 7.3.38: In Exercises 3542, find the l!C.rtex,focus, and direc1rix of each ...
 7.3.39: In Exercises 3542, find the l!C.rtex,focus, and direc1rix of each ...
 7.3.40: In Exercises 3542, find the l!C.rtex,focus, and direc1rix of each ...
 7.3.41: In Exercises 3542, find the l!C.rtex,focus, and direc1rix of each ...
 7.3.42: In Exercises 3542, find the l!C.rtex,focus, and direc1rix of each ...
 7.3.43: In Exercises 4348, convert each equation to standard form by compl...
 7.3.44: In Exercises 4348, convert each equation to standard form by compl...
 7.3.45: In Exercises 4348, convert each equation to standard form by compl...
 7.3.46: In Exercises 4348, convert each equation to standard form by compl...
 7.3.47: In Exercises 4348, convert each equation to standard form by compl...
 7.3.48: In Exercises 4348, convert each equation to standard form by compl...
 7.3.49: In Exercises 4954, use the vertex and the direction in which the p...
 7.3.50: In Exercises 4954, use the vertex and the direction in which the p...
 7.3.51: In Exercises 4954, use the vertex and the direction in which the p...
 7.3.52: In Exercises 4954, use the vertex and the direction in which the p...
 7.3.53: In Exercises 4954, use the vertex and the direction in which the p...
 7.3.54: In Exercises 4954, use the vertex and the direction in which the p...
 7.3.55: In Exercises 5560, find the solution set for eadl system by graphi...
 7.3.56: In Exercises 5560, find the solution set for eadl system by graphi...
 7.3.57: In Exercises 5560, find the solution set for eadl system by graphi...
 7.3.58: In Exercises 5560, find the solution set for eadl system by graphi...
 7.3.59: In Exercises 5560, find the solution set for eadl system by graphi...
 7.3.60: In Exercises 5560, find the solution set for eadl system by graphi...
 7.3.61: The reflector of a flashlight is in the shape of a parabolic surfac...
 7.3.62: The reflector of a flashlight is in the shape of a parabolic surfac...
 7.3.63: A satellite dish, like the one shown below, is in the shape of a pa...
 7.3.64: In Exercise 63, if the diameter of the dish is halved and the depth...
 7.3.65: The towers of the Golden Gate Bridge connecting San Franc.isco to M...
 7.3.66: The towers of a suspension bridge a re 800 feet apart and rise 160 ...
 7.3.67: The parabolic arch shown in the figure is 50 feet a bove the water ...
 7.3.68: A satellite dish in the shape of a parabolic surface has a diameter...
 7.3.69: What is a parabola?
 7.3.70: Explain how to use y2  Sx to find the parabola's focus a nd direct...
 7.3.71: If you are given the standard form of the equation of a parabola wi...
 7.3.72: Describe one similarity and one d ifference between the graphs of y...
 7.3.73: How can you distinguish parabolas from other conic sections by look...
 7.3.74: Look at the satellite dish shown in Exercise 63. Why must the recei...
 7.3.75: Use a graphing utility to graph any five of the parabolas that you ...
 7.3.76: Use a graphing utility to graph any three of the parabolas that you...
 7.3.77: Use a gmphing utility to graph the parabolas in Exercises 7778. Wr...
 7.3.78: Use a gmphing utility to graph the parabolas in Exercises 7778. Wr...
 7.3.79: In Exercises 7980, write each equation as a quadraHc equation in y...
 7.3.80: In Exercises 7980, write each equation as a quadraHc equation in y...
 7.3.81: In Exercises 81~84, determine whethereadJ statemetll makes sense or...
 7.3.82: In Exercises 81~84, determine whethereadJ statemetll makes sense or...
 7.3.83: In Exercises 81~84, determine whethereadJ statemetll makes sense or...
 7.3.84: In Exercises 81~84, determine whethereadJ statemetll makes sense or...
 7.3.85: In Exercises 8588. determine whether each statement is true or fal...
 7.3.86: In Exercises 8588. determine whether each statement is true or fal...
 7.3.87: In Exercises 8588. determine whether each statement is true or fal...
 7.3.88: In Exercises 8588. determine whether each statement is true or fal...
 7.3.89: Find the focu.o.; and directTix of a parabola whose equation is of ...
 7.3.90: Write the standard form of the equation of a parabola whose points ...
 7.3.91: Consull Lhe research department o f your library or the Internet to...
 7.3.92: Exercises 9294 will help you prepare for the nwter;at covered in t...
 7.3.93: Exercises 9294 will help you prepare for the nwter;at covered in t...
 7.3.94: Exercises 9294 will help you prepare for the nwter;at covered in t...
Solutions for Chapter 7.3: The Parabola
Full solutions for College Algebra  6th Edition
ISBN: 9780321782281
Solutions for Chapter 7.3: The Parabola
Get Full SolutionsChapter 7.3: The Parabola includes 94 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 94 problems in chapter 7.3: The Parabola have been answered, more than 24907 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: College Algebra , edition: 6. College Algebra was written by and is associated to the ISBN: 9780321782281.

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).

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

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

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.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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.

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 .

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).

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.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

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.

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

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.

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