Math 1090-002
23 October 2011
Exam II Review
The following problems are representative of those you will encounter on the exam.
Sections 2.1 and 2.2
You should be able to perform the following operations: matrix addition, multiplication of a matrix by a
scalar, transposition of a matrix, and matrix multiplication.
For the matrices given below, perform the indicated operation(s) or indicate that this is impossible.

1
−3

 0
A=
 −2

−1

4 

5 

1
"
F =
1.

"
B=
−5
3
2
1
−1
4
−1
0
2
1
3
−4
−5
−1
#
"
C=

#
2

G= 4
−3
A+G
1
−1
0
0
−1
−1
#


0 
3
"
I=
D=
h
1
0
#
0
1
0
−6
10
i
2.
2G − B T
3.
D+B
4.
FT + A
5.
IF
2
6.
AB
7.
BA
8.
DB T C
9.
100 C 2
10.
(I + C)F
3
Sections 2.3 and 2.4
You should be able to use Gauss-Jordan elimination to solve systems of linear equations and compute matrix
inverses. You should also be able to use an inverse matrix to solve a linear system.
Use Gauss-Jordan elimination to solve the following systems of equations.
11.
3x − 2y = 5
2x + y = 1
4
12.
x + 2y = −3
4x + 3y = 8
5
13.
2x − y = 1
−7x + 4y = −1
6
14.
x − 2y + z = −1
3x − 5y + 2z = 2
−x + 4y − 2z = 3
7
15.
−6x + z = −1
4x − y − z = 4
−x + 2y + z = −6
8
For each of the square matrices given below, compute the inverse matrix, or state that no inverse exists.
16.
"
A=
3
4
4
5
9
#
17.
"
B=
−3
−2
6
4
10
#
18.
"
C=
3
−4
2
−3
11
#
19.


−6
3
−2

D =  −7
−2
5

−3 
−1
12
2
20.

−1

F = 1
2
1
−2
0
13
2


−5 
1
Solve each system of linear equations using an inverse matrix. That is, write each system as a matrix
equation of the form AX = B, and solve for X by computing A−1 and then X = A−1 B.
21.
−5x − 4y = 6
7x + 6y = −2
14
22.
2x + z = −1
−3x + y − z = 5
x+z =7
15
23.
x−z =4
−2x + y + 3 = 1
−x + 2z = −5
16
Sections 3.1 and 3.2
You should be able to solve quadratic equations in one variable. Also, given a quadratic function, you should
be able to find its roots, and identify the important features of its graph, which is a parabola.
Solve the following equations. (These are problems 1.–8. of the Chapter 3 Review in the text, and their
answers can be found in the back of the book.) Hint: All of these equations can be put into the form
ax2 + bx + c = 0. We know how to solve this equation, because it’s the equation we must use to find the
roots or zeros of a quadratic function.
24.
4x2 − 25 = 0
For problems 25 and 26, complete the square to solve the equation.
25.
2x2 + 5x = 12
26.
x2 + 4x − 7 = 0
For the rest of the equations, use any method you like to solve.
27.
6x2 − 5 = 13x
17
28.
2(x − 1)2 − 8 = 0
29.
x+5=
30.
2
1
1
−
=
x−5 x+1
4
24
x
18
For the parabola described by each quadratic function below, find the following:
(a) the axis of symmetry;
(b) its concavity (up or down);
(c) its vertex, and whether this point is the maximum or the minimum;
(d) the x-intercept(s), if any. (These are the points corresponding to the roots or zeros of the function.)
Finally, sketch the graph of the function on the axes provided.
31.
y = 2(x − 1)2 + 5
19
32.
y = −x2 − 2
20
33.
y = (x + 4)2
21
34.
y = x2 − 6x + 5
22
35.
y = − 12 x2 − 6x − 12
23
Section 3.6
For each function, describe all transformations, and then draw the graph of the function on the axes provided.
(These are problems 39.–42. from the Chapter 3 Review.)
36. y = −|x + 1| − 3.
(a) Vertical and/or horizontal shift? By how much? In which direction(s) (i.e. up or down, left or right)?
(b) Vertical and/or horizontal reflection?
(c) Vertical deformation (stretch or shrink)? By what factor?
(d) Horizontal deformation (stretch or shrink)? By what factor?
(e) Plot the function.
24
37. y = 21 (x + 2)2 + 5.
(a) Vertical and/or horizontal shift? By how much? In which direction(s)?
(b) Vertical and/or horizontal reflection?
(c) Vertical deformation (stretch or shrink)? By what factor?
(d) Horizontal deformation (stretch or shrink)? By what factor?
(e) Plot the function.
25
38. y =
p
−4(x − 1) + 2.
(a) Vertical and/or horizontal shift? By how much? In which direction(s)?
(b) Vertical and/or horizontal reflection?
(c) Vertical deformation (stretch or shrink)? By what factor?
(d) Horizontal deformation (stretch or shrink)? By what factor?
(e) Plot the function.
26
39. y = −x2 − 6x + 1.
(a) Vertical and/or horizontal shift? By how much? In which direction(s)?
(b) Vertical and/or horizontal reflection?
(c) Vertical deformation (stretch or shrink)? By what factor?
(d) Horizontal deformation (stretch or shrink)? By what factor?
(e) Plot the function.
27