 9.3.1: Is each algebraic expression a polynomial? If so, how many terms do...
 9.3.2: Expand each expression. On your calculator, enter the original expr...
 9.3.3: Copy each rectangle diagram and fill in the missing values. Then wr...
 9.3.4: Convert each equation from vertex form to general form. Check your ...
 9.3.5: Draw a rectangle diagram to represent each expression. Then write a...
 9.3.6: Consider the graph of the parabola y = x2 4x + 7.a. What are the co...
 9.3.7: Heather thinks she has found a shortcut to the rectangle diagram me...
 9.3.8: APPLICATION The quadratic equation y = 0.0056x2 + 0.14x relates a v...
 9.3.9: The function h(t) = 4.9(t 0.4)2 + 2.5 describes the height of a sof...
 9.3.10: Is the expression on the right equivalent to the expression on the ...
 9.3.11: APPLICATION The Yoyo Warehouse uses the equation y = 85x2 + 552.5x...
 9.3.12: Use a threebythree rectangle diagram to square each trinomial. a....
 9.3.13: What is the general form of y = (x + 4)2? Write a paragraph describ...
 9.3.14: Is this parabola a graph of a function?
 9.3.15: Use the graph of f (x) to evaluate each expression. Then think of t...
 9.3.16: The equation y = 0.0024x2 + 0.81x + 2.0 models the path of a golf b...
Solutions for Chapter 9.3: From Vertex to General Form
Full solutions for Discovering Algebra: An Investigative Approach  2nd Edition
ISBN: 9781559537636
Solutions for Chapter 9.3: From Vertex to General Form
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Discovering Algebra: An Investigative Approach, edition: 2. Since 16 problems in chapter 9.3: From Vertex to General Form have been answered, more than 6007 students have viewed full stepbystep solutions from this chapter. Chapter 9.3: From Vertex to General Form includes 16 full stepbystep solutions. Discovering Algebra: An Investigative Approach was written by and is associated to the ISBN: 9781559537636.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Elimination.
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.

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.

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

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

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.

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.

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

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).

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

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

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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

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

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.