Precalculus A Semester Exam Review 2015-2016

PRECALCULUS A
Semester Exam Review
Precalculus A
Semester Exam Review
2015-2016
MCPS © 2015–2016
1
PRECALCULUS A
Semester Exam Review
The semester A examination for Precalculus consists of two parts. Part 1 is selected response on
which a calculator will NOT be allowed. Part 2 is short answer on which a calculator will be
allowed.
Pages with the
symbol indicate that a student should be prepared to complete items like
these with or without a calculator.
The formulas below are provided in the examination booklet.
Trigonometric Identities
sin 2   cos 2   1
sec2   1  tan 2 
sin      sin  cos   cos  sin 
sin  2   2sin  cos 
tan     
cos      cos  cos   sin  sin 
cos  2   cos 2   sin 2   2 cos 2   1  1  2sin 2 
tan   tan 
1  tan  tan 
1  cos 
 
sin    
2
2
csc2   1  cot 2 
tan  2  
2 tan 
1  tan 2 
sin 
   1  cos 
tan   

sin 
1  cos 
2
1  cos 
 
cos    
2
2
Triangle Formulas
Law of Sines:
sin A sin B sin C


a
b
c
Area of a Triangle:
Law of Cosines: c 2  a 2  b 2  2ab cos C
1
ab sin C
2
Arc Length
s  r (  in radians)
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s

360
 2 r  ( 
in degrees)
2
PRECALCULUS A
Semester Exam Review
PART 1 NO CALCULATOR SECTION
1.
 x 2 ,
if x  0
Sketch the graph of the piece-wise function f  x   
1  x , if x  0
2.
Look at the graph of the piece-wise function below.
Which of the following functions is represented by the graph?
A
 x 2 , if x  0
f  x  
 x , if x  0
B
 x 2 , if x  0
f  x  
 x , if x  0
C
 x , if x  0
f  x   2
 x , if x  0
D
 x , if x  0
f  x   2
 x , if x  0
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PRECALCULUS A
3.
Semester Exam Review
Look at the graph of the piece-wise function below.
y
O
x
Which type of discontinuity does the graph have at the following x-values?
4.
a.
x  3
b.
x 1
c.
x4
cx  7 , if x  5
Let f  x    2
, if x  5
x
What is the value of c that will make f continuous at x  5 ?
5.
4 x  c , if x  11
Let g  x   
5 x  2 , if x  11
What is the value of c that will make g continuous at x  11 ?
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PRECALCULUS A
6.
7.
Which of the following is true about the function f  x  
A
The function is continuous for all real numbers.
B
The function is discontinuous at x  3 only.
C
The function is discontinuous at x  4 only.
D
The function is discontinuous at x  3 and x  4 .
Semester Exam Review
x4
?
x 3
Look at the graph of a function below.
Does the graph represent an odd function, an even function, or a function that is neither
odd nor even?
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PRECALCULUS A
8.
Semester Exam Review
Look at the graph of a function below.
Does the graph represent an odd function, an even function, or a function that is neither
odd nor even?
9.
Determine whether each function below is odd, even, or neither odd nor even.
a.
f  x   sin x  x 3
b.
g  x   x2  4
c.
r  x   cos x  4 x
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PRECALCULUS A
10.
2
3
If f  x   x , which of the following statements is NOT true?
A
B
11.
Semester Exam Review
The graph of f is symmetric with respect to the y-axis.
f is an even function.
C
The range of f is all real numbers.
D
As x  , f  x    .
Look at the graph of the function below.
y
O
a.
What is the domain of this function?
b.
What is the range of this function?
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x
7
PRECALCULUS A
12.
Semester Exam Review
For each function below, find a formula for f 1  x  and state any restrictions on the
domain of f 1  x  .
13.
a.
f  x  x  2
b.
f  x   x3  4
True or False.
a.
The function g  x   5 f  x   2 represents a vertical stretch of the graph of f
by a factor of 5, followed by a vertical translation down 2 units.
b.
The function g  x  
1
2
x
f   represents a vertical and horizontal shrinking of the
4
graph of f .
14.
Match the transformations that would create the graph of g from the graph of f .
_______ g  x   3 f  x 
A
Stretch the graph of f horizontally
_______ g  x   f  3 x 
B
Stretch the graph of f vertically
1 
_______ g  x   f  x 
3 
C
Shrink the graph of f horizontally
1
f  x
3
D
Shrink the graph of f vertically
_______ g  x  
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PRECALCULUS A
Semester Exam Review
For items 15 and 16, use the graphs of f and g below.
f  x
g  x
y
y
x
15.
x
Which of the following represents the relationship between f  x  and g  x  ?
A
g  x   2 f  x  3
B
g  x 
C
g  x  f  2x   3
D
1 
g  x  f  x   3
2 
1
f  x  3
2
16.
Sketch the graph of y  f  x  .
17.
Write the definitions of the six circular functions of an angle  in standard position,
passing through the point  x, y  , with r  x 2  y 2 .
18.
If sin   
4
with cos   0 , what are the values of other five trigonometric functions?
5
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PRECALCULUS A
19.
20.
21.
Semester Exam Review
For each of the following, state the quadrant in which the terminal side of  lies.
a.
sin   0, tan   0
b.
cos   0, tan   0
c.
sec   0, csc   0
Convert to radian measure. Leave your answer in terms of  .
a.
40o
b.
165o
On the unit circle below the coordinates of point A are 1, 0  and the coordinates of point
B are  0.8, 0.6  . Find the value of the following.
a.
sin 
b.
cos 
c.
tan 
y
B

A
22.
Sketch the graphs of the six circular functions on the interval 2  x  2 .
23.
What are the periods of the six circular functions?
24.
What is the period of y  tan  8 x  ?
25.
What is the value of b such that y  cos  bx  has a period of
26.
What is the value of c such that y  csc  cx  has a period of 10?
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
3
x
?
10
PRECALCULUS A
27.
28.
Semester Exam Review
Determine the exact value of the following.
a.
 
sin  
6
b.
 5 
cos  
 4 
c.
 5 
tan  
 3 
d.
 3 
sin  
 2 
e.
cos  
f.
 
tan  
2
g.
 7 
tan  

 4 
h.
 4 
cos  

 3 
i.
 11 
sin  

 6 
j.
 
sin  
4
k.
 5 
cos  
 6 
l.
 7 
tan 

 6 
m.
 5 
sec  

 4 
n.
 5 
cot  
 6 
o.
 4 
csc 

 3 
p.
sec  
Complete the table for the inverse circular functions.
Sin 1 x
Cos 1 x
Tan 1 x
Domain
Range
29.
Identify the functions represented by the graphs below.
y
a.
–3
–2
–1
O
y
b.
–1
0
1
2
y
c.
1
x
–1
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x
3
0
1
x
11
PRECALCULUS A
30.
31.
Semester Exam Review
Determine the exact value of the following.
 3
a.
1
Sin 1  
2
b.
 1 
Cos 1  

2

c.
Tan 1
d.

3
Sin 1  

 2 
e.
1
Cos 1  
2
f.
Tan 1  1
g.
Sin 1  1
h.
Cos 1  0 
i.
Tan 1  0 
j.

 3 
cos  Sin 1 
 

2



k.
sin  Tan 1  1 
l.

 1 
tan  Cos 1  

2 


m.

 8 
sin  Csc1   
 5 

n.

 12  
tan  Sin 1   
 13  

o.

 11
Cos 1  cos 
 6




Write a sinusoidal equation for each of the following graphs.
y
a.
5
3
1
0
2
y
b.
2
c.
3
2
–4
2
4
x

6
–3
7
6
–6
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12
x
PRECALCULUS A
32.
Determine the equation that best describes a sine curve with amplitude 3, period of 6, and
a phase shift of
33.
Semester Exam Review

2
to the right.
For each equation below, state the amplitude, period, phase shift, vertical translation, and
any reflections of the sinusoid relative to the basic function f  x   sin x or g  x   cos x .
Sketch the graph, marking the x- and y-axes appropriately.
34.
a.
 
 
h  x   2sin  3  x     5
6 
 
b.
h  x   5cos   x  1 
c.
h  x   sin  4 x     2
Simplify each expression below as a single function of a single angle. Do not evaluate.
a.
35.
cos

7
cos
3

3
 sin sin
7
7
7
c.
sin 7 cos 3  cos 7 sin 3
d.
e.
 
1  2sin 2  
9
f.
cos8cos 5  sin 8sin 5
g.
1  cos 23o
2
h.
sin
If cot A 
7
5
7
5
cos
sin
 cos
13
13
13
13
12

and 0  A  , determine the following.
5
2
sin  2A 
b.
cos  2A 
c.
 A
sin  
2
d.
 A
cos  
2
Solve the following equations over the interval 0    360o .
a.
37.
b.
tan11o  tan 25o
1  tan11o tan 25o
a.
36.
2sin17o cos17o
2sin   2  0
b.
3cos   4  5cos   5
Solve the following equations on the interval 0  x  2 .
a.
tan x  1  0
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b.
2sin 2 x  3sin x  1  0
13
PRECALCULUS A
38.
Semester Exam Review
Complete the following chart.
Radius
6 inches
Angle (radians)
Arc length
4
5
6
15 feet

10 meters
39.
30 meters
Prove the following identities.
a.
sin  cot   cos 
b.
 sin x  cos x 
c.
csc x
 sin x
1  cot 2 x
d.
sin  cos 

 sec  csc 
cos  sin 
e.
sin  x  y   sin  x  y   2sin x cos y
f.
sin 2   sin 2  tan 2   tan 2 
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2
 1  sin  2 x 
14
PRECALCULUS A
Semester Exam Review
PART 2 CALCULATOR SECTION
A calculator may be used on items 40 through 54. Make sure that your calculator is in the
appropriate mode (radian or degree) for each item. Unless otherwise specified, answers should
be correct to three places after the decimal point.
40.
A ball on a string is swinging back and forth from a ceiling, as shown in the figure below.
d
Let d represent the distance that the center of the ball is from the wall at time t. Assume
that the distance varies sinusoidally with time.
When t  0 seconds, the ball is farthest from the wall, d  160 cm.
When t  3 seconds, the ball is closest to the wall, d  20 cm.
When t  6 seconds, the ball is farthest from the wall, d  160 cm.
a.
Sketch a graph of the distance as a function of time.
b.
Write a trigonometric equation for the distance as a function of time.
c.
What is the distance of the ball from the wall at t  5 seconds?
d.
What is the value of t the first time the ball is 40 cm from the wall? Your answer
should be correct to three places after the decimal point.
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PRECALCULUS A
41.
42.
43.
Semester Exam Review
Sara is riding a Ferris wheel. Her sister Kari starts a stopwatch and records some data.
Let h represent Sara’s height above the ground at time t. Kari notices that Sara is at the
highest point, 80 feet above the ground, when t  3 seconds. When t  7 seconds, Sara
is at the lowest point, 20 feet above the ground. Assume that the height varies
sinusoidally with time.
a.
Write a trigonometric equation for the height of Sara above the ground as a
function of time.
b.
What will Sara’s height be at t  11.5 seconds? Your answer should be correct to
three places after the decimal point.
c.
Determine the first two times, t  0 , when Sara’s height is 70 feet. Your answer
should be correct to three places after the decimal point.
At Ocean Tide Dock, the first low tide of the day occurs at midnight, when the depth of
the water is 2 meters, and the first high tide occurs at 6:30 a.m., with a depth of 8 meters.
Assume that the depth of the water varies sinusoidally with time.
a.
Sketch and label a graph showing the depth of the water as a function of the
number of hours after midnight.
b.
Determine a trigonometric function that models the graph.
c.
Suppose a ship requires at least three meters of water depth is planning to dock
after midnight. Determine the earliest possible time that the ship can dock.
Solve the following equations for  , where 0o    360o .
a.
3cos   9  7
b.
3sin 2   7 sin   2  0
44.
How many triangles ABC are possible if A  20o , b  40, and a  10 ?
45.
Given ABC , where A  41o , B  58o , and c  19.7 cm , determine the measure of
side b. Your answer should be correct to three places after the decimal point.
46.
In ABC , a  9, b  12, c  16 . What is the measure of B ? Your answer should be
correct to the nearest tenth of a degree.
47.
Determine the remaining measurements of a triangle with A  58o , a  11.4, b  12.8 .
Your answers (sides and angles) should be correct to the nearest tenth.
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PRECALCULUS A
Semester Exam Review
48.
From a point 200 feet from its base, the angle of elevation from the ground to the top of a
lighthouse is 55 degrees. How tall is the lighthouse? Your answer should be correct to
three places after the decimal point.
49.
A truck is travelling down a mountain. A sign says that the degree of incline is 7 degrees.
After the truck has travelled 1 mile (5280 feet), how many feet in elevation has the truck
fallen? Your answer should be correct to three places after the decimal point.
50.
The owner of a garage shown below plans to install a trim along the roof. The lengths
required are in bold. How many feet of trim should be purchased? Your answer should
be correct to three places after the decimal point.
50o
50o
20 feet
51.
An airplane needs to take a detour around a group of thunderstorms, as shown in the
figure below. How much farther does the plane have to travel due to the detour? Your
answer should be correct top three places after the decimal point.
34o
20o
50 miles
52.
Determine the area of triangle ABC if a  4, b  10, and mC  30o .
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PRECALCULUS A
53.
Semester Exam Review
A real estate appraiser wishes to find the value of the lot below.
160 feet
62o
250 feet
54.
a.
Determine the length of the third side of the lot.
b.
Find the area of the lot. Your answer should be correct to three places after the
decimal point.
c.
An acre is 43,560 square feet. If land is valued at $56,000 per acre, what is the
value of the lot? Your answer should be correct to the nearest cent.
Find the area of the quadrilateral below. Your answer should be correct to three places
after the decimal point.
35
50
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126o
72o
38
58
18