- 5.1: For each of the functions given, use technology to answer the follo...
- 5.2: a Sketch the graphs of f(x) = x2 4 x and g(x) = x2 + 11 for 6 6 x 6...
- 5.3: Find the maximum value of: a y = x4 + 2x3 + 5x2 + x + 2 on the inte...
- 5.4: Sketch the graphs of f(x) = ex2 and g(x) = x2 on the same set of ax...
- 5.5: Part of the graph of y = f(x) is shown alongside. a Copy the graph,...
- 5.6: Answer the Opening page 136.
- 5.7: The graph of f(x) = x2 2x + 2 is translated 3 units right to g(x). ...
- 5.8: Suppose f(x) = x2 is transformed to g(x)=(x 3)2 + 2 . a Find the im...
Solutions for Chapter 5: GRAPHING FUNCTIONS
Full solutions for Mathematics for the International Student: Mathematics SL | 3rd Edition
Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).
Tv = Av + Vo = linear transformation plus shift.
Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)
Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.
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.
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.
lA-II = 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.
Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
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).
Every v in V is orthogonal to every w in W.
Outer product uv T
= column times row = rank one matrix.
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.