Math 1090
Practice Test 2
9 March, 2012
1. Be able do define/give an example of each of the following
(a) Matrix
(b) Scalar
(c) Zero matrix
(d) Square matrix
(e) Row vector
(f) Column vector
(g) Augmented matrix
(h) Elementary row operation
(i) Inverse Matrix
2. Use a matrix to solve the system of equations, if solutions exist.
(a)
1
y
3
= −x − 19
x + 31 y = 5
(b) 2x + y = 9
3x = y + 11
(c) y + 3z = −7
2x − y + 4z = −21
3y + 2z = 7
3. Describe the end behavior of the graphs of the following polynomials (e.g., up on the
left, down on the right)
(a) y = x2 + 2
(b) y = −5x5 − x3 + 6x2 − 3
(c) y = −4x12 − 4x11 + 6x7 − 8x4 + x2 − 13
(d) y = x3 − 1
4. Give the degree and leading coefficient of each polynomial.
(a) 2x2 − 3x3
(b) 2x4 − 6x2 + 12
5. Solve the following quadratic equations using the method indicated.
(a) (square root)
(x + 2)2 = 3
(b) (factoring)
x2 − 2x + 63 = 0
(c) (completing the square)
x2 − 4x − 9 = 0
(d) (quadratic formula)
3x2 + 4x = 3


1 3
4 7 

6. Let A = 
2 −1
4 6
(a) What is the size of A?
(b) What is a12 ?
(c) What is AT ?


4 −2
3 −4 6
4
−1
7. Let A =
, B = −6 1 , and C =
2 1 −8
3 0
0
3
Compute the following, if possible:
(a) A + B
(b) A + C T
(c) B + C
(d) AT + B
(e) A − B
(f) A − B T
(g) AC
(h) CA
(i) BC
(j) CB
8. Let g(x) = −2(x + 3)2 + 6
(a) Find the vertex.
(b) Find the axis of symmetry.
(c) Find the roots.
(d) Is the parabola conave up or concave down?
(e) Is the vertex a minimum or a maximum?
9. Are these functions polynomials?
(a) f (x) = 3x2 + 1
(b) g(x) = πx4 − 1
(c) h(x) = 3xπ − 2
1
(d) k(x) = x 2 + 1
(e) p(x) = x3 − x5 + 1
10. Solve with an augmented matrix.
(a)
5x + y = 4
15x + 3y = 21
(b)
9x − 4y = 47
4x − 8y = 52
(c)
x + 2y = 6
8x + 16y = 48
11. Let f (x) = x2 − 2x − 24.
(a) Find the vertex of this parabola.
(b) Find the axis of symmetry.
(c) Find the roots.
(d) Is the parabola concave up or concave down?
(e) Is the vertex a maximum or a minimum?
(f) Sketch the graph of the function.
12. Solve the system of equations using a inverse matrix.
2x + 4y = 8
5x − y = 31
13. Match the polynomials to their graphs. (Hint: Pay attention to degree, leading coefficient, and end behavior.)
(a) y = x4 − 2x3 + 1
(b) y = −x4 + x2
(c) y = 12 x3 − 2x2
(d) y = −x3
(i).
(ii).
(iii).
(iv).
14. If a company’s profit function is given by P (x) = −5(x − 35)2 + 10000, what is the
company’s maximum possible profit?
15. Let
f (x) =
3x2 + 6x + 3
x2 − 3x − 10
(a) Find the domain.
(b) Find the vertical asymptotes, if any.
(c) Find the horizontal asymptotes, if any.
(d) Find the y-intercept, if any.
(e) Find the x-intercepts, if any.
(f) Sketch the graph.
16. Find the number of units needed to break even with the following revenue and cost
functions. Remember there may be more than one break-even point!)
R(x) = −x2 + 1600x
C(x) = 1500x + 1600
17. Solve these matrix equations:
(a)
3
12 −9
A=
2
8 −6
(b)
1 −2
2 −7
0 −19
X+
=
3 5
10 1
15 20
18. Sketch each graph. Pay attention to x- and y-intercepts and end behavior.
(a) f (x) = x3 − x2 − 2x
(b) g(x) = −2x2 + 8
19. Jack’s basketball team scored 41 less than two times the number of points Dylan’s team
scored. The sum of both teams’ points was 106. How many points did each point score?
(Use an augmented matrix)
20. The party store sells fancy party hats for $1.50 and boring party hats for $0.80. Carmen
spent $104.50 for a total of 100 party hats. How many of each kind of party hats did
she buy?