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.
Teachers may provide feedback with the form available on the Assessment
website: assessment.dallasisd.org.
First 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 1
1
2
If f ( x )  3x  2 and g ( x )  2 x 2 , what is (g  f )(4) ?
A
–320
B
98
C
200
D
392
If f ( x) 
x 1
, what is f 1 (x) ?
3
A
3x
B
3x  1
C
3x  3
D
3
x 1
Dallas ISD - Example Items
Page 2 of 8
EXAMPLE ITEMS Pre-Calculus, Sem 1
3
Which coordinate grid represents the graph of the natural logarithmic function?
10 y
A
10 y
8
8
6
6
4
4
2
2
x
-10
-8
-6
-4
-2
2
4
6
8
10
C
x
-10
-8
-6
-4
-2
-2
-2
-4
-4
-6
-6
-8
-8
-10
-10
10 y
B
4
-8
-6
-4
8
8
6
6
4
4
2
2
If the function f ( x)
-2
4
6
8
10
2
4
6
8
10
10 y
x
-10
2
2
4
6
8
10
D
x
-10
-8
-6
-4
-2
-2
-2
-4
-4
-6
-6
-8
-8
-10
-10
 x 3 is changed to g(x)  (x  2)3  1, how is the graph of the function
transformed?
A
The graph of f(x) is translated 2 units right and 1 unit up to create the graph of g(x).
B
The graph of f(x) is translated 2 units right and 1 unit down to create the graph of g(x).
C
The graph of f(x) is translated 2 units left and 1 unit up to create the graph of g(x).
D
The graph of f(x) is translated 2 units left and 1 unit down to create the graph of g(x).
Dallas ISD - Example Items
Page 3 of 8
EXAMPLE ITEMS Pre-Calculus, Sem 1
5
A function is graphed on the coordinate grid as shown.
What are the domain and range for this function?
A
Domain: (– , 2)  (2, 5)  (5,
Range: (– , 0)  (0,
B
Domain: (– , 2)  (5,
Range: (– ,
C
Domain: (– ,
)
)
Domain: (– ,
Range: (– ,
D
)
)
)
)
Range: (– , 0)  (0,
)
Dallas ISD - Example Items
)
Page 4 of 8
EXAMPLE ITEMS Pre-Calculus, Sem 1
6
The function, f (x), is graphed as shown.
y
10
8
6
4
2
-10
-8
-6
-4
-2
2
4
6
8
x
10
-2
-4
-6
-8
-10
Based on this graph, f (x) has —
7
A
three real roots and one absolute maximum
B
four real roots and three local maxima
C
four real roots and three local extrema
D
five real roots and three local extrema
A function is shown.
f (x)

x2  9
x 3
Which statement best describes the discontinuities in the graph of f ( x ) ?
A
The graph of f (x ) has no discontinuities.
B
The graph of f (x ) has an infinite discontinuity at x
C
The graph of f ( x ) has a jump discontinuity at x
D
The graph of f ( x ) has a removable discontinuity at x
Dallas ISD - Example Items
 3.
 3.
 3.
Page 5 of 8
EXAMPLE ITEMS Pre-Calculus, Sem 1
8
A soccer player kicks a ball at an angle. The height of the ball, in feet, varies with time and is
given by the function
h(t )  16t 2  33t
where t is the time in seconds. What is the maximum height of the soccer ball to the nearest
foot?
Record the answer and fill in the bubbles on
the grid provided. Be sure to use the correct
place value.
9
A radioactive substance, Cesium-137, has a half-life of about 30 years. This means that, after
every 30 years, half of the amount of Cesium-137 present turns into something else. If the
initial amount of Cesium-137 was 70 grams, which exponential decay function represents the
amount of Cesium-137 remaining over time, t, in years?
t
A
f (t )  30 0.5  70
B
f (t )  70  0.530
C
f (t )  30 2  70
D
 t 
f (t )  70 ln 

 30 
t
t
Dallas ISD - Example Items
Page 6 of 8
EXAMPLE ITEMS Pre-Calculus, Sem 1
10
15
What is the sum of the arithmetic series

(3k  2)?
k 1
Record the answer and fill in the bubbles on
the grid provided. Be sure to use the correct
place value.
11
12
A theater has 38 rows of seats. The first row has 25 seats, the second row has 29 seats, the
third row has 33 seats, and so on. What is the total number of seats in the theater?
A
3,762
B
3,838
C
7,524
D
7,676
What is the sum of the infinite geometric series 
A

27
8
B

27
40
C

9
20
D
9
4
Dallas ISD - Example Items
9
3
1
1



 ...?
8
4
2
3
Page 7 of 8
EXAMPLE ITEMS Pre-Calculus, Sem 1
13
14
What is the fourth term in the expansion of (3x  2y )8 ?
A
108,864x 5y 3
B
90,720x 4y 4
C
90,720x 4 y 4
D
108, 864x 5y 3
The Richter scale is used to calculate the intensity of an earthquake based on the amount of
observed motion. An earthquake’s Richter scale measurement is calculated using the formula
R = 3.75 + log (a)
where a is the amplitude of the observed motion in micrometers (m). What is the
amplitude, a, of an earthquake that measures 5.75 on the Richter scale?
15
m
A
2
B
7.4
C
10
D
100
m
m
m
The population of a small town is modeled by the equation
P(n) = P0(1 + r)n
where n is the number of years. In 2004 the population was 45,000. If the annual rate of
increase was 7.5%, which equation determines the number of years it takes to double the
population?
A
P(n)  90000(1  0.075)n
B
45000  90000(1  r )7.5
C
90000  45000(1  0.075)n
D
90000  45000(1  7.5)n
Dallas ISD - Example Items
Page 8 of 8
EXAMPLE ITEMS Pre-Calculus, Sem 1
16
The Texas Department of Public Safety collected data on the stopping distances for a number of
different types of cars. They found that one brand’s stopping distance could be modeled by the
equation
y  0.05x 2  0.10 x  5.0
where y is the stopping distance in feet and x is the speed of the car in miles per hour. If a car
required 100 feet to stop, what was its approximate speed just before beginning to brake?
A
42.6 mph
B
44.6 mph
C
176.0 mph
D
515.0 mph
Dallas ISD - Example Items
EXAMPLE ITEMS Pre-Calculus, Sem 1
Answer
SE
Process Standards
1
C
P.2C
P.1B, P.1C, P.1F
2
B
P.2E
P.1B, P.1C, P.1E, P.1F
3
D
P.2F
P.1D, P.1E, P.1F
4
C
P.2G
P.1E, P.1F
5
A
P.2I
P.1E, P.1F
6
C
P.2I
P.1E, P.1F
7
D
P.2L
P.1C, P.1E, P.1F, P.1G
8
17
P.2N
P.1A, P.1B, P.1C, P.1F
9
B
P.2N
P.1A, P.1B, P.1E, P.1F
10
330
P.5A
P.1B, P.1C, P.1F
11
A
P.5C
P.1A, P.1B, P.1C, P.1F
12
B
P.5E
P.1B, P.1C
13
A
P.5F
P.1B, P.1C, P.1D, P.1F
14
D
P.5H
P.1A, P.1B, P.1C
15
C
P.5I
P.1A, P.1B, P.1C
16
A
P.5J
P.1B, P.1C, P.1D, P.1E, P.1F
Dallas ISD - Example Items