Honors Precalculus A Semester Exam Review 2015-2016

HONORS PRECALCULUS A
Semester Exam Review
Honors Precalculus A
Semester Exam Review
2015-2016
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1
HONORS PRECALCULUS A
Semester Exam Review
The semester A examination for Honors 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)
s

360
 2 r  ( 
in degrees)
Linear Speed
v  r
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HONORS PRECALCULUS A
Semester A 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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HONORS PRECALCULUS A
3.
Semester A 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
Sketch the graph of the following piece-wise function. Be sure to indicate arrows, closed
circle endpoints and open circle endpoints.
4, if x  2

f  x    x 2  3, if  2  x  3
5  x, if 3  x  7

5.
Suppose that lim f  x   8 , and that f  4  is undefined.
x4
a.
Write a limit statement so that there is a removable discontinuity at x  4 .
b.
Write a limit statement so that there is a jump discontinuity at x  4 .
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HONORS 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 A Review
x4
?
x 3
cx  7 , if x  5
Let f  x    2
, if x  5
x
What is the value of c that will make f  x  continuous at x  5 ?
8.
 4 x  c , if x  11
Let g  x   
5 x  2 , if x  11
What is the value of c that will make f  x  continuous at x  11 ?
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HONORS PRECALCULUS A
9.
Semester A 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?
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HONORS PRECALCULUS A
10.
Semester A 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?
11.
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
d.
f  x   x 2 cos x  x sin x  4
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HONORS PRECALCULUS A
12.
2
3
If f  x   x , which of the following statements is NOT true?
A
B
13.
Semester A 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
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HONORS PRECALCULUS A
14.
Semester A Review
For each function below, find a formula for f 1  x  and state any restrictions on the
domain of f 1  x  .
15.
a.
f  x  x  2
b.
f  x   x3  4
c.
f  x 
x4
x2
Look at the graph of f below.
y
O
16.
x
a.
Sketch the graph of y  f  x  .
b.
Sketch the graph of y  f  2 x  .
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 .
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HONORS PRECALCULUS A
17.
Semester A Review
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  
For items 18 and 19, use the graphs of f and g below.
f  x
g  x
y
y
x
18.
19.
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
Sketch the graph of y  f  x  .
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HONORS PRECALCULUS A
20.
Semester A Review
Look at the graph of f below. The domain of f is 6  x  5 .
y
O
x
a.
Describe how to transform the graph of f  x  to g  x   2 f  x  1  3 .
b.
Sketch the graph of g .
c.
What is the domain of g ?
d.
What is the range of g ?
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HONORS PRECALCULUS A
21.
22.
23.
Semester A Review
Which of the following describes the right-end behavior of the function f  x  
A
lim f  x   
B
lim f  x   0
C
lim f  x   3
D
lim f  x   
3
?
x2
x 
x 
x 
x 
Let x and y be related by the equation x  y 2  17 .
a.
Determine the values of y if x = 0, 1, and 8.
b.
Is this relation a function? Justify your answer.
c.
Determine two functions defined implicitly by this relation.
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 .
4
with cos   0 , what are the values of other five trigonometric functions?
5
24.
If sin   
25.
For each of the following, state the quadrant in which the terminal side of  lies.
26.
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
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HONORS PRECALCULUS A
27.
Semester A Review
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.
28.
a.
sin 
b.
cos 
c.
tan 
y
B

x
A
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  
29.
Sketch the graphs of the six circular functions on the interval 2  x  2 .
30.
What are the periods of the six circular functions?
31.
What is the period of y  tan  8 x  ?
32.
What is the value of b such that y  cos  bx  has a period of
33.
What is the value of c such that y  csc  cx  has a period of 10?
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
3
?
13
HONORS PRECALCULUS A
34.
Semester A Review
Complete the table for the inverse circular functions.
Sin 1 x
Cos 1 x
Tan 1 x
Domain
Range
35.
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
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HONORS PRECALCULUS A
36.
37.
Semester A Review
Determine the exact (not a decimal) 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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x
HONORS PRECALCULUS A
38.
Determine the equation that best describes a sine curve with amplitude 3, period of 6, and
a phase shift of
39.
Semester A 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.
40.
41.
a.
 
 
h  x   2sin  3  x     5
6 
 
b.
h  x   5cos   x  1 
c.
h  x   sin  4 x     2
Assume that angle A and angle B are supplementary. Which of the following is equal to
cos A ?
A
sin B
B
 sin B
C
cos B
D
 cos B
Simplify each expression below as a single function of a single angle. Do not evaluate.
a.
2sin17 o cos17o
b.
cos

7
cos
3

3
 sin sin
7
7
7
c.
sin 7 cos 3  cos 7 sin 3
d.
tan11o  tan 25o
1  tan11o tan 25o
e.
 
1  2sin 2  
9
f.
cos8cos 5  sin 8sin 5
g.
1  cos 23o
2
h.
sin
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7
5
7
5
cos
sin
 cos
13
13
13
13
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HONORS PRECALCULUS A
42.
43.
If cot A 
45.
12

and 0  A  , determine the exact value of the following.
5
2
a.
sin  2A 
b.
cos  2A 
d.
 A
tan  
2
e.
 A
cos  
2
2sin   2  0
 A
sin  
2
b.
3cos   4  5cos   5
Solve the following equations on the interval 0  x  2 .
a.
tan 2 x  3  0
c.
2sin  2 x   1  0
b.
2sin 2 x  3sin x  1  0
Complete the following chart.
Radius
6 inches
10 meters
46.
c.
Solve the following equations over the interval 0    360o .
a.
44.
Semester A Review
Angle (radians)
Arc length
4
5
6
15 feet

30 meters
How many triangles are possible if a  40, b  80, and A  30o ?
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HONORS PRECALCULUS A
47.
Semester A Review
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 
g.
tan x  cot x  2 csc  2 x 
h.
cot  sec 

 sec2  csc 
cos  cot 
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2
 1  sin  2 x 
18
HONORS PRECALCULUS A
Semester Exam Review
PART 2 CALCULATOR SECTION
A calculator may be used on items 48 through 65. 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.
48.
A wheel of radius 12 cm turns at 7 revolutions per second. Determine the linear speed
of a point on the circumference of the wheel in meters per second.
49.
A moon makes a circular revolution around its planet in 80 hours. The radius of the
circular path is 20,000 miles.
50.
a.
What is the angular speed of the planet in radians per hour?
b.
What is the linear speed of the planet in feet per second?
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 from 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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HONORS PRECALCULUS A
51.
52.
53.
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
54.
How many triangles ABC are possible if A  20o , b  40, and a  10 ?
55.
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.
56.
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.
57.
Determine the remaining sides of a triangle with A  58o , a  11.4, b  12.8 . Your
answers (side and angles) should be correct to the nearest tenth.
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HONORS PRECALCULUS A
Semester Exam Review
58.
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.
59.
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.
60.
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
61.
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 to three places after the decimal point.
34o
20o
50 miles
62.
Determine the area of triangle ABC if a  4, b  10, and mC  30o .
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HONORS PRECALCULUS A
63.
Semester Exam Review
A real estate appraiser wishes to find the value of the lot below.
160 feet
62o
250 feet
64.
a.
How long is the third side of the lot? Your answer should be correct to three
decimal places.
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
65.
121.8o
18o
38
Figure NOT drawn to scale
58
Triangle ABC has an area of 2,400 with AB  80, AC  100 . Determine the two possible
measures of angle A. Your answers should be correct to the nearest tenth of a degree.
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