 11.1.1: Exer. 112: Find the vertex, focus, and directrix of the parabola. S...
 11.1.2: Exer. 112: Find the vertex, focus, and directrix of the parabola. S...
 11.1.3: Exer. 112: Find the vertex, focus, and directrix of the parabola. S...
 11.1.4: Exer. 112: Find the vertex, focus, and directrix of the parabola. S...
 11.1.5: Exer. 112: Find the vertex, focus, and directrix of the parabola. S...
 11.1.6: Exer. 112: Find the vertex, focus, and directrix of the parabola. S...
 11.1.7: Exer. 112: Find the vertex, focus, and directrix of the parabola. S...
 11.1.8: Exer. 112: Find the vertex, focus, and directrix of the parabola. S...
 11.1.9: Exer. 112: Find the vertex, focus, and directrix of the parabola. S...
 11.1.10: Exer. 112: Find the vertex, focus, and directrix of the parabola. S...
 11.1.11: Exer. 112: Find the vertex, focus, and directrix of the parabola. S...
 11.1.12: Exer. 112: Find the vertex, focus, and directrix of the parabola. S...
 11.1.13: Exer. 1318: Find an equation for the parabola shown in the figure.
 11.1.14: Exer. 1318: Find an equation for the parabola shown in the figure.
 11.1.15: Exer. 1318: Find an equation for the parabola shown in the figure.
 11.1.16: Exer. 1318: Find an equation for the parabola shown in the figure.
 11.1.17: Exer. 1318: Find an equation for the parabola shown in the figure.
 11.1.18: Exer. 1318: Find an equation for the parabola shown in the figure.
 11.1.19: Exer. 1930: Find an equation of the parabola that satisfies the giv...
 11.1.20: Exer. 1930: Find an equation of the parabola that satisfies the giv...
 11.1.21: Exer. 1930: Find an equation of the parabola that satisfies the giv...
 11.1.22: Exer. 1930: Find an equation of the parabola that satisfies the giv...
 11.1.23: Exer. 1930: Find an equation of the parabola that satisfies the giv...
 11.1.24: Exer. 1930: Find an equation of the parabola that satisfies the giv...
 11.1.25: Exer. 1930: Find an equation of the parabola that satisfies the giv...
 11.1.26: Exer. 1930: Find an equation of the parabola that satisfies the giv...
 11.1.27: Exer. 1930: Find an equation of the parabola that satisfies the giv...
 11.1.28: Exer. 1930: Find an equation of the parabola that satisfies the giv...
 11.1.29: Exer. 1930: Find an equation of the parabola that satisfies the giv...
 11.1.30: Exer. 1930: Find an equation of the parabola that satisfies the giv...
 11.1.31: Exer. 3134: Find an equation for the set of points in an xyplane t...
 11.1.32: Exer. 3134: Find an equation for the set of points in an xyplane t...
 11.1.33: Exer. 3134: Find an equation for the set of points in an xyplane t...
 11.1.34: Exer. 3134: Find an equation for the set of points in an xyplane t...
 11.1.35: Exer. 3538: Find an equation for the indicated half of the parabola.
 11.1.36: Exer. 3538: Find an equation for the indicated half of the parabola.
 11.1.37: Exer. 3538: Find an equation for the indicated half of the parabola.
 11.1.38: Exer. 3538: Find an equation for the indicated half of the parabola.
 11.1.39: Exer. 3940: Find an equation for the parabola that has a vertical a...
 11.1.40: Exer. 3940: Find an equation for the parabola that has a vertical a...
 11.1.41: Exer. 4142: Find an equation for the parabola that has a horizontal...
 11.1.42: Exer. 4142: Find an equation for the parabola that has a horizontal...
 11.1.43: A mirror for a reflecting telescope has the shape of a (finite) par...
 11.1.44: A satellite antenna dish has the shape of a paraboloid that is 10 f...
 11.1.45: A searchlight reflector has the shape of a paraboloid, with the lig...
 11.1.46: A flashlight mirror has the shape of a paraboloid of diameter 4 inc...
 11.1.47: A sound receiving dish used at outdoor sporting events is construct...
 11.1.48: Work Exercise 47 if the receiver is 9 inches from the vertex.
 11.1.49: a) The focal length of the (finite) paraboloid in the figure is the...
 11.1.50: The parabola has its focus at the origin and axis along the xaxis....
 11.1.51: A radio telescope has the shape of a paraboloid of revolution, with...
 11.1.52: A satellite will travel in a parabolic path near a planet if its ve...
Solutions for Chapter 11.1: Parabolas
Full solutions for Algebra and Trigonometry with Analytic Geometry  12th Edition
ISBN: 9780495559719
Solutions for Chapter 11.1: Parabolas
Get Full SolutionsSince 52 problems in chapter 11.1: Parabolas have been answered, more than 35114 students have viewed full stepbystep solutions from this chapter. Algebra and Trigonometry with Analytic Geometry was written by and is associated to the ISBN: 9780495559719. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 11.1: Parabolas includes 52 full stepbystep solutions. This textbook survival guide was created for the textbook: Algebra and Trigonometry with Analytic Geometry, edition: 12.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

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

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

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

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

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.

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

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.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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)ยท(b  Ax) = o.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

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

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

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.

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