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,
content SE and SE justification are listed for each item.
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
2018–2019
Code #: 1121
ACP Formulas
Pre-Calculus/Pre-Calculus PAP
2018–2019
Trigonometric Functions and Identities
Pythagorean Theorem:
a2 + b2 = c2
Special Right Triangles:
30° - 60° - 90°
x, x 3, 2 x
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 Velocity:
v = r ω
Angular Velocity:
ω =
Reciprocal Identities:
Double-Angle
Identities:
t
cos θ =
1
csc θ =
Sum & Difference
Identities:
θ
1
csc θ
sin θ =
Pythagorean
Identities:
v = r
sec θ =
sin θ
sin2  + cos2  = 1
1
s ( s − a) ( s − b ) ( s − c )
θ
t
tan θ =
sec θ
1
1 + tan2  = sec2 
cot θ
1
cot θ =
cos θ
1
tan θ
1 + cot2  = csc2 
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
α − β ) sin α cos β − cos α sin β
sin(=
sin 2θ = 2 sin θ cos θ
cos
=
2θ cos2 θ − sin2 θ
cos 2θ= 1 − 2 sin2 θ
=
cos 2θ 2 cos2 θ − 1
Sequences and Series
The nth Term of an
Arithmetic Sequence:
an = a1 + (n − 1)d
Sum of a Finite
Arithmetic Series:
a
=
∑
n
k =1
k
n
(a + an )
2 1
n
a1 (1 − r n )
Sum of a Finite
=
ak
, r ≠1
∑
Geometric Series:
1−r
k =1
∞
a1
Sum of an Infinite
an
=
, r <1
∑
Geometric Series:
1−r
n =1
Binomial Theorem:
a + b)
(=
Permutations:
n
n
Pr =
The nth Term of a
Geometric Sequence:
n
S
=
n
=
Sn
an = a1r n−1
n
2a + (n − 1)d 
2 1
a1 − an r
, r ≠1
1−r
C0 an b0 + n C1 an −1b1 + n C2 an − 2 b2 + ⋅ ⋅ ⋅ + n Cr an − r br + ⋅ ⋅ ⋅ + n Cn a0 bn
n!
(n − r )!
Combinations:
n
Cr =
n!
(n − r )! r !
Projectile Motion
1 2
gt + h0
2
Vertical Position:
y= tv0 sin θ −
Vertical Free-Fall
Motion:
1
− gt 2 + v0t + s0
s(t ) =
2
Horizontal Distance:
x = tv0 cos θ
v(t ) =
−gt + v0
g ≈ 32
ft
m
≈ 9.8
sec2
sec2
ACP Formulas
Pre-Calculus/Pre-Calculus PAP
2018–2019
Conic Sections
Circle:
Standard Form:
Standard Form:
Parabola:
(x – h)2 + (y – k)2 = r2
(x – h)2 = 4p(y – k)
(y – k)2 = 4p(x – h)
Focus:
(h, k + p)
(h + p, k)
Directrix:
y=k–p
( x − h)
2
Standard Form:
Ellipse:
a2
Foci:
a, b, c Relationship:
+
Hyperbola:
b2
( x − h)
2
= 1
b2
−
(y − k )
(y − k
=
) ±
2
b2
(y − k )
2
=
1
b
( x − h)
a
e=
−
a2
( x − h)
(y − k
=
) ±
r

P 1 + 
n


Exponential Growth or
N
=
Decay:
N0 (1 + r )
Compound Interest:
nt
t
c2 = a2 + b2
c
a
e=
Continuous Compound
Interest:
A = Pert
Continuous
Exponential Growth or
Decay:
N = N0ekt
c
a
Coordinate Geometry
Distance Formula:
d=
( x2 − x1 )2 + (y2 − y1 )2
Slope of a Line:
m=
y2 − y1
x2 − x1
Midpoint Formula:
 x + x2
M = 1
,
2

Quadratic Equation:
ax2 + bx + c = 0
Slope-Intercept Form of a Line:
y1 + y2 

2

Quadratic Formula:
=
y mx + b
Point-Slope Form of a Line:
y − y1= m(x − x1 )
Standard Form of a Line:
Ax + By = C
=
1
a
( x − h)
b
I = prt
A
=
2
b2
Exponential Functions
Simple Interest:
=
1
a2
(h, k ± c)
c2 = a2 + b2
Eccentricity:
2
c2 = a2 – b2
a2
a, b, c Relationship:
(y − k )
c2 = a2 – b2
(h ± c, k)
Asymptotes:
+
(h, k ± c)
( x − h)
Foci:
(y − k )
(h ± c, k)
2
Standard Form:
x=h–p
2
x =
−b ± b2 − 4ac
2a
HIGH SCHOOL
EXAMPLE ITEMS Pre-Calculus, Sem 2
1
The graph of an ellipse is shown.
Which equation represents this ellipse?
2
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
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
Dallas ISD - Example Items
 t  6 and
EXAMPLE ITEMS Pre-Calculus, Sem 2
3
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.
4
 5 
What are the rectangular coordinates for 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
EXAMPLE ITEMS Pre-Calculus, Sem 2
5
6
 19 
What is the exact value of tan 
?
 6 
A
 3
B

3
3
C
3
3
D
3
Triangle ABC is shown.
Based on the information in the diagram, what is the approximate length of AC ?
7
A
6.59
B
8.11
C
12.33
D
15.18
If cos  
5
and sin   0 , what is cot  ?
13
A

12
5
B

5
12
C
5
12
D
12
5
Dallas ISD - Example Items
EXAMPLE ITEMS Pre-Calculus, Sem 2
8
A polar equation is used to produce the graph of the rose shown.
Which equation is used to create the rose?
9
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
EXAMPLE ITEMS Pre-Calculus, Sem 2
10
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?
11
12
A
18.6 miles
B
23.0 miles
C
26.0 miles
D
39.4 miles
Which equation represents the hyperbola with foci at (–8, 3) and (4, 3) and a transverse axis
that is 8 units long?
A
( x  2)2
(y  3)2

16
36
1
B
( x  2)2
(y  3)2

36
16
1
C
( x  2)2
(y  3)2

20
16
1
D
( x  2)2
(y  3)2

16
20
1
What is the exact value of the trigonometric function cos(–870°)?
A

B

C
1
2
D
3
2
1
2
3
2
Dallas ISD - Example Items
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
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
EXAMPLE ITEMS Pre-Calculus, Sem 2
15
16
17
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
y
 2t
 8t  3
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
A hyperbola has foci at (–1, –2) and (13, –2) and an eccentricity of
the hyperbola in standard form?
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
Dallas ISD - Example Items
7
. What is the equation of
6
EXAMPLE ITEMS Pre-Calculus, Sem 2
18
19
Which graph represents the curve given by the parametric equations x
over the interval 3
 t 2  3 and y  2t
 t  3?
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
20
21
If r = 4, –2, which graph represents –2r?
A
C
B
D
What is the reference angle for an angle that measures 300°?
A
30°
B
60°
C
120°
D
150°
Dallas ISD - Example Items
EXAMPLE ITEMS Pre-Calculus Key, Sem 2
Item#
Key
SE
SE Justification
1
C
P.3H
Use the characteristics of an ellipse to write the equation of an
ellipse with center (h, k).
2
A
P.3B
Convert parametric equations into rectangular relations.
3
90.6
P.4E
Solve problems involving trigonometric ratios in real-world
problems.
4
A
P.3D
Convert between rectangular coordinates and polar coordinates.
5
B
P.4A
Determine the relationship between the unit circle and the
definition of a periodic function to evaluate trigonometric functions
in mathematical problems.
6
C
P.4G
Use the Law of Sines in mathematical problems.
7
B
P.4E
Determine the value of trigonometric ratios of angles.
8
D
P.3E
Graph polar equations by plotting points.
9
B
P.3F
Determine the conic section formed when a plane intersects a
double-napped cone.
10
A
P.4G
Use the Law of Sines real-world problems.
11
D
P.3I
12
A
P.4A
13
222
P.4H
Use the Law of Cosines in real-world problems.
14
D
P.4C
Find the measure of reference angles and angles.
15
C
P.3C
Use parametric equations to model mathematical problems.
16
B
P.4I
Use vectors to model situations involving magnitude and direction.
17
D
P.3I
Use the characteristics of a hyperbola to write the equation of a
hyperbola with center (h, k).
18
A
P.3A
Graph a set of parametric equations.
19
D
P.4K
Apply vector addition and multiplication of a vector by a scalar in
mathematical problems.
20
C
P.4J
Represent the multiplication of a vector by a scalar geometrically.
21
B
P.4C
Find the measure of reference angles.
Use the characteristics of a hyperbola to write the equation of a
hyperbola with center (h, k).
Determine the relationship between the unit circle and the
definition of a periodic function to evaluate trigonometric functions
in mathematical problems.