Download Pre Calculus

Example Items
Pre-Calculus
Pre-Calculus Example Items are a representative set of
items for the ACP. Teachers may use this set of items along with the test
blueprint as guides to prepare students for the ACP. On the last page, the
correct answer and content SE is listed. The specific part of an SE that an
Example Item measures is NOT necessarily the only part of the SE that is
assessed on the ACP. None of these Example Items will appear on the ACP.
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Second Semester
2016–2017
Code #: 1121
ACP Formulas
Pre-Calculus/Pre-Calculus PAP
2016–2017
Trigonometric Functions and Identities
Pythagorean Theorem:
a2 + b2 = c2
Special Right Triangles:
30° - 60° - 90°
x, x 3, 2x
45° - 45° - 90°
x, x, x 2
Law of Sines:
sin A sin B sin C
=
=
a
b
c
Heron’s Formula:
A=
Law of Cosines:
a2 = b2 + c2 – 2bc cos A
b2 = a2 + c2 – 2ac cos B
c2 = a2 + b2 – 2ab cos C
Linear Speed:
v =
Angular Speed:
ω
s
t
sin θ =
Reciprocal Identities:
1
csc θ
1
csc θ =
Pythagorean
Identities:
Sum & Difference
Identities:
Double-Angle
Identities:
cos θ =
sec θ =
sin θ
sin2 θ + cos2 θ = 1
1
s ( s − a) ( s − b ) ( s − c )
θ
=
t
tan θ =
sec θ
1
cot θ =
cos θ
1 + tan2 θ = sec2 θ
1
cot θ
1
tan θ
1 + cot2 θ = csc2 θ
cos( α + β ) = cos α cos β − sin α sin β
sin(α + β ) = sin α cos β + cos α sin β
cos(α − β ) = cos α cos β + sin α sin β
sin(α − β ) = sin α cos β − cos α sin β
sin2x = 2 sin x cos x
cos 2 x = cos2 x − sin2 x
cos 2x = 2 cos2 x − 1
cos 2x = 1 − 2 sin2 x
Projectile Motion
1 2
gt + (v0 sin θ )t + y0
2
Vertical Position:
y =−
Vertical Free-Fall
Motion:
s(t ) = −
1 2
gt + v0t + s0
2
Horizontal Distance:
x = (v0 cos θ )t
v(t ) = − gt + v0
g ≈ 32
ft
m
≈ 9.8
sec2
sec2
Conic Sections
Parabola:
(x - h)2 = 4p(y - k)
(y - k)2 = 4p(x - h)
Circle:
x2 + y2 = r2
(x – h)2 + (y - k)2 = r2
Ellipse:
( x − h)
Hyperbola:
( x − h)
2
a2
+
2
a2
−
(y − k )
2
(y − k )
b2
2
=1
b2
( x − h)
2
+
b2
(y − k )
2
=1
a2
−
(y − k )
2
=1
a2
( x − h)
b2
2
=1
ACP Formulas
Pre-Calculus/Pre-Calculus PAP
2016–2017
Exponential Functions
Simple Interest:
I = prt
Compound Interest:
r

A = P 1 + 
n


Exponential Growth or
Decay:
N = N0 (1 + r )
nt
t
Continuous Compound
Interest:
A = Pert
Continuous
Exponential Growth or
Decay:
N = N0ekt
Sequences and Series
The nth Term of an
Arithmetic Sequence:
an = a1 + (n − 1)d
Sum of a Finite
Arithmetic Series:
a
Sum of a Finite
Geometric Series:
a
Sum of an Infinite
Geometric Series:
a
Binomial Theorem:
(a + b)
Permutations:
n
n
k =1
k
k =1
k
∞
n =1
n
Pr =
an = a1r n−1
n
(a + an )
2 1
=
n
The nth Term of a
Geometric Sequence:
=
a1(1 − r n )
, r ≠1
1−r
=
a1
, r ≠1
1−r
n
Sn =
a1 − an r
, r ≠1
1−r
= n C 0 an b0 + n C1 an −1 b1 + n C2 an − 2 b2 + ⋅ ⋅ ⋅ + n C n a0 b n
n!
(n − r )!
Combinations:
n
Cr =
n!
(n − r )! r !
Coordinate Geometry
Distance Formula:
d = ( x2 − x1 )2 + (y2 − y1 )2
Slope of a Line:
m=
Midpoint Formula:
 x + x2
M= 1
,
2

Quadratic Equation:
ax2 + bx + c = 0
y2 − y1
x2 − x1
y1 + y2 

2

Quadratic Formula:
Slope-Intercept Form of a Line:
y = mx + b
Point-Slope Form of a Line:
y − y1 = m(x − x1 )
Standard Form of a Line:
Ax + By = C
x =
−b ± b2 − 4ac
2a
HIGH SCHOOL
Page 1 of 8
EXAMPLE ITEMS Pre-Calculus, Sem 2
1
Which graph represents the curve given by the parametric equations x
y
 8 sin t over the interval 0  t 

2
?
A
C
B
D
Dallas ISD - Example Items
 3 cos t and
Page 2 of 8
EXAMPLE ITEMS Pre-Calculus, Sem 2
2
3
4
What is the rectangular form for the curve given by the parametric equations x
y  5t  3 ?
A
y  5x  33
B
y  5x  3
C
y  5x  27
D
y  5x  9
 t  6 and
Which pair of parametric equations represents a line that passes through points (2, 1)
and (0, –3)?
A
x
y
 2t
 4t  3
B
x
y
 2t
 8t  3
C
x
y
 2t
 4t  3
D
x  2t
y  8t  3
 5 
What are the rectangular coordinates of the point  5,
?
6 

A
 5 3 5
, 
 
2
2 

B
 5 5 3
  ,

2 
 2
C
5
5 3
 , 

2 
2
D
5 3
5
,  

2 
 2
Dallas ISD - Example Items
Page 3 of 8
EXAMPLE ITEMS Pre-Calculus, Sem 2
5
A polar equation is used to produce the graph of the rose shown.
Which equation is used to create the rose?
6
A
r
 2 sin(2 )
B
r
 2 sin(4 )
C
r
 3 sin(4 )
D
r
 3 sin(2 )
The intersection of a plane and a double-napped cone is shown in the diagram.
What type of conic section is formed by this intersection?
A
Circle
B
Ellipse
C
Hyperbola
D
Parabola
Dallas ISD - Example Items
Page 4 of 8
EXAMPLE ITEMS Pre-Calculus, Sem 2
7
The graph of an ellipse is shown.
Which equation represents this ellipse?
A
( x  1)2
(y  2)2

3
5
1
B
( x  1)2
(y  2)2

3
5
1
C
( x  1)2
(y  2)2

9
25
1
D
(x  1)2
(y  2)2

9
25
1
Dallas ISD - Example Items
Page 5 of 8
EXAMPLE ITEMS Pre-Calculus, Sem 2
8
A hyperbola has foci at (–1, –2) and (13, –2) and an eccentricity of
7
. What is the equation of
6
the hyperbola in standard form?
9
A
(x  6)2
(y  2)2

36
85
1
B
(x  6)2
(y  2)2

36
85
1
C
( x  6)2
(y  2)2

36
13
1
D
( x  6)2
(y  2)2

36
13
1
What is the exact value of the trigonometric function cos(–870°)?
A

B

C
1
2
D
10
3
2
1
2
3
2
What is the reference angle for an angle that measures 
A
7
4
B
5
4
C
3
4
D

4
Dallas ISD - Example Items
17
radians?
4
Page 6 of 8
EXAMPLE ITEMS Pre-Calculus, Sem 2
11
A cable holds an 80-foot pole straight upright, as shown.
Based on the given information, what is the approximate length of the cable, to the nearest tenth
of a foot?
Record the answer and fill in the bubbles on
the grid provided. Be sure to use the correct
place value.
12
A helicopter is flying from downtown Dallas to downtown Fort Worth. The distance between the
two cities is 32 miles.
45°
35°
?
Dallas
32 miles
Fort Worth
If the angle of depression from the helicopter to Dallas is 45° and the angle of depression to
Fort Worth is 35°, approximately how far is the helicopter from downtown Dallas?
A
18.6 miles
B
23.0 miles
C
26.0 miles
D
39.4 miles
Dallas ISD - Example Items
Page 7 of 8
EXAMPLE ITEMS Pre-Calculus, Sem 2
13
The diagram shows a boat that is anchored at point B in a river. There are two boat ramps on
the far side of the river, shown by points A and C. The boat is 120 meters from ramp A and
150 meters from ramp C.
If mABC
meter?
 110° , what is the approximate distance between the two boat ramps, to the nearest
Record the answer and fill in the bubbles on
the grid provided. Be sure to use the correct
place value.
14
Harrison walks to the library after school every day. When Harrison leaves school, he walks
16 blocks due West and then 12 blocks due North to get to the library. What is the magnitude
and direction of the resultant vector?
A
Magnitude: 20 blocks
Direction: W 41.4° N
B
Magnitude: 20 blocks
Direction: W 36.9° N
C
Magnitude: 28 blocks
Direction: W 41.4° N
D
Magnitude: 28 blocks
Direction: W 36.9° N
Dallas ISD - Example Items
Page 8 of 8
EXAMPLE ITEMS Pre-Calculus, Sem 2
15
16
If r = 4, –2, which graph represents –2r?
A
C
B
D
If u = 8, 12, –3, v = –4, 7, 14, and w = 2, –5, 6, what is 3u – 4v + 2w?
A
6, 14, 17
B
19, 24, 23
C
12, 54, 9
D
44, –2, –53
Dallas ISD - Example Items
EXAMPLE ITEMS Pre-Calculus, Sem 2
Answer
SE
Process Standards
1
B
P.3A
P.1B, P.1D, P.1E, P.1F
2
A
P.3B
P.1B, P.1D, P.1E, P.1F
3
C
P.3C
P.1B, P.1D, P.1E, P.1F
4
A
P.3D
P.1B, P.1C, P.1D, P.1E, P.1F
5
D
P.3E
P.1B, P.1D, P.1E, P.1F
6
B
P.3F
P.1F
7
C
P.3H
P.1B, P.1E, P.1F
8
D
P.3I
P.1B, P.1D, P.1E, P.1F
9
A
P.4A
P.1B, P.1C, P.1E, P.1F
10
D
P.4C
P.1B, P.1C, P.1E, P.1F
11
90.6
P.4E
P.1A, P.1B, P.1C, P.1F
12
A
P.4G
P.1A, P.1B, P.1C, P.1F
13
222
P.4H
P.1A, P.1B, P.1C, P.1F
14
B
P.4I
P.1A, P.1B, P.1C, P.1F
15
C
P.4J
P.1B, P.1C, P.1E, P.1F
16
D
P.4K
P.1B, P.1C, P.1E, P.1F
Dallas ISD - Example Items