SESSION: 21 PAGE: 1 5/11/04 11:26 LOGIN IS-glenn PATH: @sunultra1/raid/CLS_tpc/GRP_va_sprg04/JOB_04-ribsg11/DIV_g11alg2-1
VIRGINIA
STANDARDS OF LEARNING ASSESSMENTS
Spring 2004 Released Test
END OF COURSE
ALGEBRA II
CORE 1
Property of the Virginia Department of Education
䉷 2004 by the Commonwealth of Virginia Department of Education, James Monroe Building, 101 N. 14th Street,
Richmond, Virginia, 23219. All rights reserved. Except as permitted by law, this material may not be reproduced or
used in any form or by any means, electronic or mechanical, including photocopying or recording, or by any
information storage or retrieval system, without written permission from the copyright owner. Commonwealth of
Virginia public school educators may photocopy or print any portion of these Released Tests for educational purposes
without requesting permission. All others should direct their requests to the Commonwealth of Virginia Department
of Education at (804) 225-2102, Division of Assessment and Reporting.
SESSION: 21 PAGE: 2 5/11/04 11:26 LOGIN IS-glenn PATH: @sunultra1/raid/CLS_tpc/GRP_va_sprg04/JOB_04-ribsg11/DIV_g11alg2-1
Algebra II
3
DIRECTIONS
Read and solve each question. Then mark the
space on the answer sheet for the best answer.
For this test you may assume that the value
of the denominator of a rational expression is
not zero.
2
(a ⴐ b)3
ⴢ
?
18
(a ⴐ b)2
SAMPLE
a3
6(a ⴐ 2)
ⴢ
ⴔ
a
aⴐ2
A
6
a2
B
6(a ⫹ 2)
a
C
6a2 墍
D
6a ⫹ 24a ⫹ 24
a4
Which is equivalent to
4
A
a⫹b
墍
9
B
(a ⫹ b)2
9
C
(a ⫹ b)5
36
D
18a ⫹ 9b
JY020606
墍
C
Which is a factored form of
2
1 ⴑ y3 ?
(1
G (1
H (1
J (1
F
1
JY010515
墌
C
L
Which indicates a correct application
of the distributive property?
a(b ⫹ c) ⫽ (ab) ⫹ (ac) 墍
B (a ⫹ b) ⫹ c ⫽ a ⫹ (b ⫹ c)
C (a ⫹ b) ⫹ c ⫽ c ⫹ (a ⫹ b)
D a ⫺ (b ⫹ c) ⫽ (a ⫺ b) ⫹ c
⫺
⫺
⫹
⫹
y)(1
y)(1
y)(1
y)(1
⫺
⫹
⫺
⫹
y
y
y
y
⫺
⫹
⫺
⫹
JY050107
墍
y2)
y2) 墍
y2)
y2)
C
A
5
3
Which is equivalent to 252 ?
125 墍
B 75
C 15
D 5
A
2
Which of the following may not be
true?
JY010622
墌
F
If a ⫽ b and b ⫽ c, then a ⫽ c.
C
G
If a ⫽
/ b and b ⫽
/ c, then a ⫽
/ c. 墍
H
If a ⫽ b, then b ⫽ a.
J
If a ⫽
/ b, then b ⫽
/ a.
2004 Commonwealth of Virginia Department of Education
2
JY03B401
墍
C
SESSION: 21 PAGE: 3 5/11/04 11:26 LOGIN IS-glenn PATH: @sunultra1/raid/CLS_tpc/GRP_va_sprg04/JOB_04-ribsg11/DIV_g11alg2-1
6
9
Which is equivalent to
Which is equivalent to
ⴑ
(1 ⴑ 2i)3 ?
JY170615
墌
F
C
7
(6) 2(30)(23)?
JY03A414
⫺11 ⫹ 2i 墍
G
13 ⫺ 10i
H
13 ⫹ 2i
J
1 ⫹ 8i
A
⫺288
B
⫺216
C
1
6
D
2
墍
9
墍
C
Which is a factored form of
x2 ⴑ 18x ⴐ 81?
JY050532
墌
C
L
8
(x
B (x
C (x
D (x
A
⫹
⫺
⫺
⫹
9)2
3) (x ⫹ 27)
9)2 墍
9) (x ⫺ 9)
10
(3 ⴑ i) ⴑ (5 ⴐ 2i)?
JY170301
Which expression is equivalent to
1
JY03A708
C
F
20
3
G
4墍
H
62
J
2412
F
2 ⫺ 3i
G
2 ⫺ 7i
H
⫺2 ⫹ i
J
⫺2 ⫺ 3i 墍
1
83 ⴐ 164?
墌
Which is equivalent to
1
1
2004 Commonwealth of Virginia Department of Education
3
墍
C
L
SESSION: 21 PAGE: 4 5/11/04 11:34 LOGIN IS-glenn PATH: @sunultra1/raid/CLS_tpc/GRP_va_sprg04/JOB_04-ribsg11/DIV_g11alg2-1
11
12
9
Which of the following sketches could
represent the graph of y ⴔ ⱍ x ⴑ a ⱍ ?
8
JY080404
墌
7
C
5
JY080505
墍
6
C
L
4
ArtCodes
3
JY080404.AR1
2
–9 –8 –7 –6 –5 –4 –3 –2 –1 O
–1
ArtCodes
F
1
1
2
3
4
5
6
7
8
JY080505.AR1
JY080505.AR2
JY080505.AR3
JY080505.AR4
9
–2
–3
–4
–5
–6
–7
–8
–9
G
Which of the following is most likely
the equation graphed above?
y
B y
C y
D y
A
⫽
⫽
⫽
⫽
(x ⫹ 2)2 ⫹ 1
5(x ⫺ 1)2 ⫺ 2
(x ⫺ 2)2 ⫹ 2
(x ⫺ 2)2 ⫺ 1 墍
墍
H
y
J
2004 Commonwealth of Virginia Department of Education
4
x
SESSION: 21 PAGE: 5 5/11/04 11:34 LOGIN IS-glenn PATH: @sunultra1/raid/CLS_tpc/GRP_va_sprg04/JOB_04-ribsg11/DIV_g11alg2-1
13
16
9
8
7
6
5
4
3
2
1
JY080102
墌
C
ArtCodes
JY080102.AR1
–9 –8 –7 –6 –5 –4 –3 –2 –1 O
–1
–2
–3
–4
–5
–6
–7
–8
–9
Which could be a graph of
y ⴔ ax3 ⴐ bx 2 ⴐ cx ⴐ d if a, b, c, and d
are real numbers and a ⬍ 0?
JY150105
墍
C
L
F
ArtCodes
JY150105.AR1
JY150105.AR2
JY150105.AR3
JY150105.AR4
1 2 3 4 5 6 7 8 9
y
x
G
Which type of function is shown?
Absolute value
B Exponential
C Linear
D Quadratic 墍
A
墍
H
14
Which is a zero of the function
f (x) ⴔ x2 ⴑ 2x ⴑ 24?
JY090528
墌
C
15
JY090612
墌
C
y
⫺6
G ⫺3
H 4
J 6 墍
F
J
Given: f (x) ⴔ x3 ⴑ 3x and g(x) ⴔ x2 ⴑ 8.
What is f (10) ⴑ g(10)?
608
B 783
C 862
D 878 墍
A
2004 Commonwealth of Virginia Department of Education
5
x
SESSION: 21 PAGE: 6 5/11/04 11:34 LOGIN IS-glenn PATH: @sunultra1/raid/CLS_tpc/GRP_va_sprg04/JOB_04-ribsg11/DIV_g11alg2-1
17
19
What is the value of
3
30
兺2
JY150517
墌
nⴐ1
?
JY160305
墍
nⴔ0
24
C
C
L
12
B 14
C 28
D 30 墍
A
18
ArtCodes
JY150517.AR1
12
6
-4 -3 -2 -1
O
1
2
3
4
-6
20
-12
Which function most likely represents
the graph shown above?
f (x)
B f (x)
C f (x)
D f (x)
A
18
JY160104
墌
C
⫽
⫽
⫽
⫽
x4
x3
x4
x4
⫺
⫹
⫹
⫺
12x2
12x2
10x2
10x2
⫹
⫺
⫺
⫹
27 墍
27
5
27
If the fifth and eighth terms of an
arithmetic sequence are ⴑ9 and ⴑ21,
respectively, what are the first four
terms of the sequence?
2,
G 3,
H 5,
J 7,
F
Wind chill factor (W) varies directly
with temperature (T) and inversely
with the wind velocity (v). If k is the
constant of proportionality, which
formula represents this relationship?
F
W ⫽ kTv
G
W⫽
kv
T
H
W⫽
kT
墍
v
J
W⫽
kT
v2
21
⫺1, ⫺4, ⫺7
⫺1, ⫺5, ⫺9
2, ⫺1, ⫺4
3, ⫺1, ⫺5 墍
2
0.5
–
3
0.75
3
–
8
y
B y
C y
D y
6
⫽
⫽
⫽
⫽
3⫺x ⫹ 1
x2 ⫺ 7
⫺ x2 ⫹ 1 墍
⫺3x ⫺ 3
墍
C
4
–
15
Which is a general rule for the
function containing the values shown
in the table?
A
2004 Commonwealth of Virginia Department of Education
–
JY200110
JY080525
墍
C
ArtCodes
JY080525.AR1
SESSION: 21 PAGE: 7 5/11/04 11:34 LOGIN IS-glenn PATH: @sunultra1/raid/CLS_tpc/GRP_va_sprg04/JOB_04-ribsg11/DIV_g11alg2-1
22
25
Which best represents the graph of
ⱍ 2x ⴑ 1 ⱍ
JY040502
⬎ 5?
Which graph represents a quadratic
equation with no real solutions?
JY060614
墌
C
F
G
ArtCodes
JY040502.AR1
JY040502.AR2
JY040502.AR3
JY040502.AR4
H
J
–7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7
A
–7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7
–7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7
墍
–7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7
B
23
JY040516
墌
C
What is the solution set for
墍
5
4
3
2
1
-5 -4 -3 -2 -1 O
-1
-2
-3
-4
-5
C
ArtCodes
JY060614.AR1
JY060614.AR2
JY060614.AR3
JY060614.AR4
1 2 3 4 5
5
4
3
2
1
-5 -4 -3 -2 -1 O
-1
-2
-3
-4
-5
1 2 3 4 5
ⱍ 2x ⴐ 5 ⱍ ⴔ 7?
{⫺6, 1} 墍
B {⫺1, 1}
C {⫺6}
D {1}
A
C
5
4
3
2
1
-5 -4 -3 -2 -1 O
-1
-2
-3
-4
-5
1 2 3 4 5
24
–9 –8 –7 –6 –5 –4 –3 –2 –1
0
1
2
3
4
5
JY040404
墌
C
L
ArtCodes
Which of the following inequalities
best represents the graph above?
F
JY040404.AR1
G
H
J
ⱍx ⫺ 1ⱍ
ⱍx ⫺ 7ⱍ
ⱍx ⫹ 3ⱍ
ⱍx ⫹ 3ⱍ
⬍
⬍
⬍
⬍
D
3
4
7
4墍
2004 Commonwealth of Virginia Department of Education
7
5
4
3
2
1
-5 -4 -3 -2 -1 O
-1
-2
-3
-4
-5
1 2 3 4 5
墍
SESSION: 21 PAGE: 8 5/11/04 11:34 LOGIN IS-glenn PATH: @sunultra1/raid/CLS_tpc/GRP_va_sprg04/JOB_04-ribsg11/DIV_g11alg2-1
26
29
What is the solution set for
What is the solution to
6x ⴑ 2
3x2 ⴑ 2
ⴔ
?
x
x
x2 ⴑ 16 ⴔ 0?
JY060521
墌
C
27
{⫺4}
G {⫺4, 4} 墍
H {0, 4}
J {4}
F
What is the solution set for
JY060412
C
L
墍
C
L
6
B 2 墍
C 公2
D 0
A
30
(3x ⴐ 1)2 ⴔ 0?
墌
JY070575
A
B
C
D
再 冎
再 冎
再 冎
公x ⴑ 2 ⴔ 4?
1
,1
3
⫺1,
JY070609
{2}
G {14}
H {16}
J {18} 墍
F
⫺1
3
⫺1
What is the solution set for
墍
C
墍
3
{⫺1}
31
What is the solution set for
公3y ⴐ 4 ⴔ 5?
JY070608
28
Which is a solution to
2x2 ⴑ 7x ⴐ 3 ⴔ 0?
JY060713
墌
C
F
A
{3}
B
{1}
x⫽3墍
C
G
x⫽2
H
x⫽
J
x ⫽ ⫺3
⫺1
D
2
2004 Commonwealth of Virginia Department of Education
8
再冎
再冎
1
3
1
9
墍
C
墍
SESSION: 21 PAGE: 9 5/11/04 11:34 LOGIN IS-glenn PATH: @sunultra1/raid/CLS_tpc/GRP_va_sprg04/JOB_04-ribsg11/DIV_g11alg2-1
32
35
Two factors of a polynomial, P(x), are
(x ⴐ 6) and (x ⴑ 3). Which is a zero of
P(x)?
What are the coordinates of
the vertex of
JY100609
墌
C
33
yⴔ
6
G 3 墍
H 1
J ⫺3
F
2
(x ⴑ 4)2 ⴐ 1?
3
JY180406
墍
C
L
(4, 1) 墍
B (⫺4, 1)
C (⫺4, ⫺1)
D (4, ⫺1)
A
What are the factors of a polynomial
function if its graph has x-intercepts at
ⴑ1, 0, and 4?
36
JY100510
墌
C
L
x, (x ⫺ 1), and (x ⫹ 4)
B x, (x ⫹ 1), and (x ⫺ 4) 墍
C (x ⫹ 1) and (x ⫺ 4)
D (x ⫺ 1) and (x ⫹ 4)
A
5
JY180506
墍
4
C
3
2
ArtCodes
JY180506.AR1
1
-5 -4 -3 -2 -1
O
1
2
3
4
5
-1
34
Which of the following is a zero of the
function
-2
f (x) ⴔ x3 ⴑ x2 ⴑ 14x ⴐ 24?
-4
JY100521
墌
C
-3
-5
⫺4 墍
G ⫺3
H ⫺2
F
J
Which equation most likely represents
the function shown by the graph?
0
y
G y
H y
J y
F
2004 Commonwealth of Virginia Department of Education
9
⫽
⫽
⫽
⫽
⫺(x
⫺(x
⫺(x
⫺(x
⫹
⫹
⫹
⫺
1)2
1)2
2)2
1)2
⫺
⫺
⫺
⫺
1
2墍
1
2
SESSION: 21 PAGE: 10 5/11/04 11:34 LOGIN IS-glenn PATH: @sunultra1/raid/CLS_tpc/GRP_va_sprg04/JOB_04-ribsg11/DIV_g11alg2-1
37
What is the graph of
39
(x ⴑ 2)2 ⴐ (y ⴐ 3)2 ⴔ 4?
The matrix shows the ways points can
be scored in a football game, and the
number of points given for each.
JY180602
墌
C
JY110510
An ellipse
B A circle 墍
C A parabola
D A hyperbola
A
Points
Touchdown
Field Goal
Safety
Extra Point Conversion
38
JY110604
墌
C
U4⫻2 ⴢ V2⫻3
G
H2⫻2 ⴢ I2⫻2
H
C2⫻4 ⴢ D4⫻3
J
P1⫻3 ⴢ Q1⫻3 墍
3
2
1
Field
Extra
Touchdowns Goals Safeties Points
Alton
3
0
1
2
Bakey
2
2
0
1
[
]
Which product could be used to give the
final score of the game?
冤 冥冋
6
A
3
3
0
1
2
2
2
2
0
1
册
1
B
C
[6
冋
3
2
1]
冋
3
0
1
2
2
2
0
1
3
0
1
2
2
2
0
1
册
冤冥
墍
[6
3
册
6
3
2
1
D
2004 Commonwealth of Virginia Department of Education
C
L
This matrix shows the ways points
were scored by each team in a game
between Alton and Bakey.
Which matrix product is not defined?
F
冤冥
6
墍
10
冋
3
0
1
2
2
2
0
1
册
2
1]
SESSION: 21 PAGE: 11 5/11/04 11:35 LOGIN IS-glenn PATH: @sunultra1/raid/CLS_tpc/GRP_va_sprg04/JOB_04-ribsg11/DIV_g11alg2-1
40
JY120511
墌
C
41
Mel and Ann were having breakfast at
a restaurant. Mel paid $3.28 for 3 eggs
and 1 sausage patty. Ann paid $3.56 for
2 eggs and 3 sausage patties. Which of
the following equations could be
solved to determine the cost of each
item?
F
G
冋 册冋 册 冋 册
3
1
x
2
3
y
⫽
3.28
3.56
Which matrix equation represents the
given system of linear equations?
A
墍
B
冋 册冋 册 冋 册
3
3
x
1
2
y
⫽
3.28
C
3.56
D
H
[x
y]
冋 册
⫽ [3.28
3.56]
J
[x
y]
冋 册
⫽ [3.28
3.56]
3
1
2
3
3
2
1
3
再3x8x ⴑⴐ 2y9y ⴔⴔ ⴑ52
2004 Commonwealth of Virginia Department of Education
11
冋 册冋 册 冋 册
冋 册冋 册 冋 册
冋 册冋 册 冋 册
冋 册冋 册 冋 册
⫺2
x
3
y
8
5
⫽
⫺2
9
3
⫺2
8
x
9
y
3
⫺2
xy
8
9
xy
3
⫺2
x
8
9
y
⫽
⫽
⫽
5
⫺2
5
⫺2
5
墍
⫺2
JY120613
墍
C
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42
43
JY130404
冦
9
墌
8
C
L
Which graph shows a solution to the
following system?
7
ArtCodes
2y ⴑ x ⱖ 0
JY130204
墍
x ⴐ 4y ⱕ 12
C
y ⬍ xⴐ2
6
JY130404.AR1
5
4
3
A
2
ArtCodes
JY130204.AR1
JY130204.AR2
JY130204.AR3
JY130204.AR4
O
1
A
1
2
3
4
5
6
7
8
9
The graph of the linear programming
model consists of polygon ABCDE and
its interior. Under these constraints,
which is the point where the maximum
value of 2x ⴐ 4y occurs?
B
O
A
G C
H D 墍
J E
F
2004 Commonwealth of Virginia Department of Education
12
C
O
D
O
墍
SESSION: 21 PAGE: 13 5/11/04 11:35 LOGIN IS-glenn PATH: @sunultra1/raid/CLS_tpc/GRP_va_sprg04/JOB_04-ribsg11/DIV_g11alg2-1
再 yy ⴔⴔ ⴑxxⴑⴐ2x4xⴑⴑ11
2
44
JY140303
墌
C
46
2
Which set of ordered pairs is the
solution to the system of equations
shown?
{(0,
G {(0,
H {(0,
J {(3,
F
3), (⫺1, 2)}
⫺1), (3, 2)} 墍
⫺1), (6, 23)}
2), (6, ⫺11)}
Which of the following scatterplots
would most likely be a representation
of the correlation between the heights
of a group of high school seniors and
their grade point averages, if y
represents the height and x represents
the grade point average?
JY190112
墍
C
L
F
ArtCodes
JY190112.AR1
JY190112.AR2
JY190112.AR3
JY190112.AR4
y
45
5
JY140407
墌
4
C
L
3
G
2
ArtCodes
1
JY140407.AR1
–5
–4
–3
–2
–1
O
1
2
3
4
5
x
–1
–2
–3
–4
–5
H
This is a portion of the graph of a
system of equations. Which is most
likely the solution set for the system?
{(0, ⫺3),
B {(⫺3, 0),
C {(0, ⫺3),
D {(0, ⫺3),
A
(2.4, 1.8)} 墍
(3, 0)}
(2.4, ⫺1.8)}
(1.8, 2.4)}
2004 Commonwealth of Virginia Department of Education
J
13
墍
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47
JY190708
墌
C
For a psychology report, Cathy
compared the length in minutes of
her little brother’s afternoon nap with
the hours of sleep he received the
previous night.
Hours of Sleep
x
4.5
5.5
6.0
7.0
7.5
8.0
8.0
8.5
8.5
9.0
9.5
10.0
10.5
11.0
48
y
B y
C y
D y
⫽
⫽
⫽
⫽
Minutes of Naptime
y
160
160
170
130
120
90
180
80
120
100
60
90
0
30
墍
Year ( x)
Number of
Students ( n)
1
66
2
3
4
72
82
90
5
6
100
106
ArtCodes
JY190455.AR1
Assuming the trend continues, which is
the best prediction for the number of
students that will be enrolled during
the 12th year of the program?
212
G 157 墍
H 142
J 126
F
⫺23.39x ⫹ 296.04 墍
⫺0.03x ⫹ 11.26
11.26x ⫺ 0.03
296.04x ⫺ 23.39
2004 Commonwealth of Virginia Department of Education
JY190455
C
Which is most likely the line of best fit
for the data?
A
The table shows the number of
students enrolled in the advanced
algebra program at Fair Oaks High
School during its first 6 years.
14
SESSION: 21 PAGE: 15 5/11/04 11:35 LOGIN IS-glenn PATH: @sunultra1/raid/CLS_tpc/GRP_va_sprg04/JOB_04-ribsg11/DIV_g11alg2-1
49
JY190556
墌
A city began recording its air pollution
index in 1980. The scatterplot shows
the index related to various numbers
of years since 1980.
C
L
50
Larry made a scatterplot showing the
apparent height of a football at
one-second intervals during the time
period the ball was in the air.
C
14
Height (feet)
Air Pollution Index, I
JY190556.AR1
墍
y
600
500
ArtCodes
JY190570
400
300
200
ArtCodes
12
JY190570.AR1
10
8
6
4
2
0
100
0
1
2
3
4
5
Time (seconds)
0
0
5
10
15
20
Years since 1980, t
Which is most likely the equation for
the curve of best fit for the
relationship?
If no steps are taken to control city air
pollution, which is the best estimate
for the pollution index in 2005?
y
G y
H y
J y
F
425
B 475 墍
C 525
D 600
A
2004 Commonwealth of Virginia Department of Education
15
⫽
⫽
⫽
⫽
⫺0.4x ⫹ 9.0
9.0x ⫹ 0.4
5.3x2 ⫺ 0.9x ⫹ 4.9
⫺0.9x2 ⫹ 5.3x ⫹ 4.9 墍
x
SESSION: 21 PAGE: 16 5/11/04 12:14 LOGIN IS-glenn PATH: @sunultra1/raid/CLS_tpc/GRP_va_sprg04/JOB_04-ribsg11/DIV_g11alg2-1
Answer Key
Test
Sequence
Correct
Answer
Reporting
Category
Reporting Category Description
1
A
001
Expressions and Operations
2
G
001
Expressions and Operations
3
A
001
Expressions and Operations
4
G
001
Expressions and Operations
5
A
001
Expressions and Operations
6
F
001
Expressions and Operations
7
C
001
Expressions and Operations
8
G
001
Expressions and Operations
9
D
001
Expressions and Operations
10
J
001
Expressions and Operations
11
D
002
Relations and Functions
12
H
002
Relations and Functions
13
D
002
Relations and Functions
14
J
002
Relations and Functions
15
D
002
Relations and Functions
16
H
002
Relations and Functions
17
A
002
Relations and Functions
18
J
002
Relations and Functions
19
D
002
Relations and Functions
20
H
002
Relations and Functions
21
C
002
Relations and Functions
22
H
003
Equations and Inequalities
23
A
003
Equations and Inequalities
24
J
003
Equations and Inequalities
25
D
003
Equations and Inequalities
26
G
003
Equations and Inequalities
27
C
003
Equations and Inequalities
28
F
003
Equations and Inequalities
29
B
003
Equations and Inequalities
30
J
003
Equations and Inequalities
31
C
003
Equations and Inequalities
32
G
004
Analytical Geometry
33
B
004
Analytical Geometry
34
F
004
Analytical Geometry
35
A
004
Analytical Geometry
36
G
004
Analytical Geometry
37
B
004
Analytical Geometry
38
J
005
Systems of Equations/Inequalities
39
C
005
Systems of Equations/Inequalities
40
F
005
Systems of Equations/Inequalities
41
D
005
Systems of Equations/Inequalities
42
H
005
Systems of Equations/Inequalities
43
C
005
Systems of Equations/Inequalities
44
G
005
Systems of Equations/Inequalities
45
A
005
Systems of Equations/Inequalities
46
G
002
Relations and Functions
47
A
002
Relations and Functions
48
G
002
Relations and Functions
49
B
002
Relations and Functions
50
J
002
Relations and Functions