 8.3.1: The graph of any quadratic function, f(x) = ax2 + bx + c, a 0, is c...
 8.3.2: The vertex of a parabola is the ______________ point if a 7 0.
 8.3.3: The vertex of a parabola is the ______________ point if a 6 0.
 8.3.4: The vertex of the graph of f(x) = a(x  h) 2 + k, a 0, is the point...
 8.3.5: The xcoordinate of the vertex of the graph of f(x) = ax2 + bx + c,...
 8.3.6: If f(x) = a(x  h) 2 + k or f(x) = ax2 + bx + c, a 0, any xinterce...
 8.3.7: . If f(x) = a(x  h) 2 + k or f(x) = ax2 + bx + c, a 0, the yinter...
 8.3.8: In Exercises 58, the graph of a quadratic function is given. Write ...
 8.3.9: In Exercises 916, find the coordinates of the vertex for the parabo...
 8.3.10: In Exercises 916, find the coordinates of the vertex for the parabo...
 8.3.11: In Exercises 916, find the coordinates of the vertex for the parabo...
 8.3.12: In Exercises 916, find the coordinates of the vertex for the parabo...
 8.3.13: In Exercises 916, find the coordinates of the vertex for the parabo...
 8.3.14: In Exercises 916, find the coordinates of the vertex for the parabo...
 8.3.15: In Exercises 916, find the coordinates of the vertex for the parabo...
 8.3.16: In Exercises 916, find the coordinates of the vertex for the parabo...
 8.3.17: In Exercises 1738, use the vertex and intercepts to sketch the grap...
 8.3.18: In Exercises 1738, use the vertex and intercepts to sketch the grap...
 8.3.19: In Exercises 1738, use the vertex and intercepts to sketch the grap...
 8.3.20: In Exercises 1738, use the vertex and intercepts to sketch the grap...
 8.3.21: In Exercises 1738, use the vertex and intercepts to sketch the grap...
 8.3.22: In Exercises 1738, use the vertex and intercepts to sketch the grap...
 8.3.23: In Exercises 1738, use the vertex and intercepts to sketch the grap...
 8.3.24: In Exercises 1738, use the vertex and intercepts to sketch the grap...
 8.3.25: In Exercises 1738, use the vertex and intercepts to sketch the grap...
 8.3.26: In Exercises 1738, use the vertex and intercepts to sketch the grap...
 8.3.27: In Exercises 1738, use the vertex and intercepts to sketch the grap...
 8.3.28: In Exercises 1738, use the vertex and intercepts to sketch the grap...
 8.3.29: In Exercises 1738, use the vertex and intercepts to sketch the grap...
 8.3.30: In Exercises 1738, use the vertex and intercepts to sketch the grap...
 8.3.31: In Exercises 1738, use the vertex and intercepts to sketch the grap...
 8.3.32: In Exercises 1738, use the vertex and intercepts to sketch the grap...
 8.3.33: In Exercises 1738, use the vertex and intercepts to sketch the grap...
 8.3.34: In Exercises 1738, use the vertex and intercepts to sketch the grap...
 8.3.35: In Exercises 1738, use the vertex and intercepts to sketch the grap...
 8.3.36: In Exercises 1738, use the vertex and intercepts to sketch the grap...
 8.3.37: In Exercises 1738, use the vertex and intercepts to sketch the grap...
 8.3.38: In Exercises 1738, use the vertex and intercepts to sketch the grap...
 8.3.39: In Exercises 3944, an equation of a quadratic function is given. a....
 8.3.40: In Exercises 3944, an equation of a quadratic function is given. a....
 8.3.41: In Exercises 3944, an equation of a quadratic function is given. a....
 8.3.42: In Exercises 3944, an equation of a quadratic function is given. a....
 8.3.43: In Exercises 3944, an equation of a quadratic function is given. a....
 8.3.44: In Exercises 3944, an equation of a quadratic function is given. a....
 8.3.45: In Exercises 4548, give the domain and the range of each quadratic ...
 8.3.46: In Exercises 4548, give the domain and the range of each quadratic ...
 8.3.47: In Exercises 4548, give the domain and the range of each quadratic ...
 8.3.48: In Exercises 4548, give the domain and the range of each quadratic ...
 8.3.49: In Exercises 4952, write an equation of the parabola that has the s...
 8.3.50: In Exercises 4952, write an equation of the parabola that has the s...
 8.3.51: In Exercises 4952, write an equation of the parabola that has the s...
 8.3.52: In Exercises 4952, write an equation of the parabola that has the s...
 8.3.53: In Exercises 5356, write an equation of the parabola that has the s...
 8.3.54: In Exercises 5356, write an equation of the parabola that has the s...
 8.3.55: In Exercises 5356, write an equation of the parabola that has the s...
 8.3.56: In Exercises 5356, write an equation of the parabola that has the s...
 8.3.57: A person standing close to the edge on the top of a 160foot buildi...
 8.3.58: A person standing close to the edge on the top of a 200foot buildi...
 8.3.59: Among all pairs of numbers whose sum is 16, find a pair whose produ...
 8.3.60: Among all pairs of numbers whose sum is 20, find a pair whose produ...
 8.3.61: Among all pairs of numbers whose difference is 16, find a pair whos...
 8.3.62: Among all pairs of numbers whose difference is 24, find a pair whos...
 8.3.63: You have 600 feet of fencing to enclose a rectangular plot that bor...
 8.3.64: You have 200 feet of fencing to enclose a rectangular plot that bor...
 8.3.65: You have 50 yards of fencing to enclose a rectangular region. Find ...
 8.3.66: You have 80 yards of fencing to enclose a rectangular region. Find ...
 8.3.67: A rain gutter is made from sheets of aluminum that are 20 inches wi...
 8.3.68: A rain gutter is made from sheets of aluminum that are 12 inches wi...
 8.3.69: In Chapter 3, we saw that the profit, P(x), generated after produci...
 8.3.70: In Chapter 3, we saw that the profit, P(x), generated after produci...
 8.3.71: What is a parabola? Describe its shape.
 8.3.72: Explain how to decide whether a parabola opens upward or downward.
 8.3.73: Describe how to find a parabolas vertex if its equation is in the f...
 8.3.74: Describe how to find a parabolas vertex if its equation is in the f...
 8.3.75: A parabola that opens upward has its vertex at (1, 2). Describe as ...
 8.3.76: Use a graphing utility to verify any five of your handdrawn graphs ...
 8.3.77: a. Use a graphing utility to graph y = 2x2  82x + 720 in a standar...
 8.3.78: In Exercises 7881, find the vertex for each parabola. Then determin...
 8.3.79: In Exercises 7881, find the vertex for each parabola. Then determin...
 8.3.80: In Exercises 7881, find the vertex for each parabola. Then determin...
 8.3.81: In Exercises 7881, find the vertex for each parabola. Then determin...
 8.3.82: Make Sense? In Exercises 8285, determine whether each statement mak...
 8.3.83: Make Sense? In Exercises 8285, determine whether each statement mak...
 8.3.84: Make Sense? In Exercises 8285, determine whether each statement mak...
 8.3.85: Make Sense? In Exercises 8285, determine whether each statement mak...
 8.3.86: In Exercises 8689, determine whether each statement is true or fals...
 8.3.87: In Exercises 8689, determine whether each statement is true or fals...
 8.3.88: In Exercises 8689, determine whether each statement is true or fals...
 8.3.89: In Exercises 8689, determine whether each statement is true or fals...
 8.3.90: In Exercises 9091, find the axis of symmetry for each parabola whos...
 8.3.91: In Exercises 9091, find the axis of symmetry for each parabola whos...
 8.3.92: In Exercises 9293, write the equation of each parabola in f(x) = a(...
 8.3.93: In Exercises 9293, write the equation of each parabola in f(x) = a(...
 8.3.94: A rancher has 1000 feet of fencing to construct six corrals, as sho...
 8.3.95: The annual yield per lemon tree is fairly constant at 320 pounds wh...
 8.3.96: Solve: 2 x + 5 + 1 x  5 = 16 x2  25 . (Section 6.6, Example 5)
 8.3.97: Simplify: 1 + 2 x 1  4 x2 . (Section 6.3, Example 1)
 8.3.98: Solve using determinants (Cramers Rule): e 2x + 3y = 6 x  4y = 14....
 8.3.99: Exercises 99101 will help you prepare for the material covered in t...
 8.3.100: Exercises 99101 will help you prepare for the material covered in t...
 8.3.101: Exercises 99101 will help you prepare for the material covered in t...
Solutions for Chapter 8.3: Quadratic Functions and Their Graphs
Full solutions for Intermediate Algebra for College Students  6th Edition
ISBN: 9780321758934
Solutions for Chapter 8.3: Quadratic Functions and Their Graphs
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 8.3: Quadratic Functions and Their Graphs includes 101 full stepbystep solutions. This textbook survival guide was created for the textbook: Intermediate Algebra for College Students, edition: 6. Since 101 problems in chapter 8.3: Quadratic Functions and Their Graphs have been answered, more than 38558 students have viewed full stepbystep solutions from this chapter. Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758934.

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).

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Companion matrix.
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 nl  An).

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

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.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

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.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

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

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

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·

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).