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Potomac Falls High School
Precalculus Semester Exam Review
2012 – 2013
 This review covers the major topics in the material that will be tested on the
semester exam. It is not necessarily all inclusive and additional study and
problem solving practice may be required to fully prepare for the semester exam.
 Place answers in the blanks, when provided.
 Use additional paper, if necessary.
 Calculators may be used; however, the semester exam will have non-calculator
portions. Therefore, prudence suggests you prepare with and without a
calculator so you can handle any contingency.
Name:___________________
Date:_______________
Period:______
Teacher:____________
Precalculus Semester Exam Review
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CONICS
Questions 1 – 4: Write the equation of a circle, in standard form, with the given conditions.
1) r = 5, (h, k) = (2, –3)
2) Center at the point (–4, 5) and tangent to the y-axis.
3) Center at the point (4, –7) and containing the point (–2, 6).
4) With endpoints af a diameter at the points (5, 7) and (–2, –2).
Questions 5 – 10: Convert to standard form. Identify the features specified for each.
5) y 2  4 y  4 x  4  0
Vertex: ________
Focus: _________
Directrix: _______
Length of the latus rectum: ______
6) 2 x 2  2 y 2  12 x  8 y  24  0
Center: ________
Radius: ________
Precalculus Semester Exam Review
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7) y 2  4 x 2  16 x  2 y  19  0
Center: ________
Transverse axis:: _________ (equation)
Vertices: ________________
Foci: ___________________
Equations of asymptotes:
__________________________
8) 2 x 2  3 y 2  8x  6 y  5  0
Center: ________
Major axis: ____________ (equation)
Minor axis: ____________ (equation)
Vertices: ________________
Foci: ___________________
9) An arch in the form of a semi-ellipse is 60 feet wide and 20 feet high at the center. Find the heightof
the arch 10 feet from the center.
Precalculus Semester Exam Review
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10) The cables of a suspension bridge are in the shape of a parabola. The towers supporting the cables are
500 feet apart and 120 feet high. If the cables are at a height of 15 feet midway between the towers,
what is the height of the cables at a point 100 feet from the center of the bridge?
11) A sealed-beam headlight is in the shape of a paraboloid of revolution. The bulb, which is placed at
the focus, is 1 inch from the vertex. If the depth is to be 2 inches, what is the diameter of the headlight
at its opening?
12) A whispering gallery is 80 feet wide. The height of the ceiling in the center is 25 feet. Where are the
foci in relation to the walls of the gallery?
Precalculus Semester Exam Review
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CONICS
State the dimensions of the following matrices.
 3 2 7 9 
13)  1 0 3 5 


 8 2 10 6 
 3 2 7 9 
14)  1 0 3 5 


 8 2 10 6 
9 
15)  6 
 5 
8 17 
6
 7 5 15 

16) 
 1 143 2 


 11 13 3 
Solve for the variable(s):
 3 5 
 0 2  3 11 
17) 
 3



 25 2
 x 4  15 14
5 6 
0 1
 12.5
10 7 
 1 2 
 23





18) 5  8 x   4  2 3   2  16





 1 6 
 4 11 
 y
 7 8 
 5 3 
 27.5
13 
13.5
100 

37 
14 
Find the product. If the product is not defined, explain why.
5
19) 3 1  
7 
5 2 
 3 7
22)  0 4  
2 0 
1 6  
Precalculus Semester Exam Review
 1 0 
20) 
  4 6
 5 4
9 1
1 3 0  

23) 
  4 3 
2
12

4

  2 4 


5
9 3  0 1 
21) 


0 2   4 2 
Solve for the variables.
 2 1 2 1   6 
24)  3 2 4  x   19

   
 0 2 4  3   y 
 9 2
 4 1 3 
   y 4
25) 
2
1


  13 8 
 2 x 1 


 1 1 
Questions 26 – 27: Write an inventory matrix and a cost per item matrix. Then use multiplication
to write a total cost matrix.
26) A softball team needs to buy 12 bats at $21 each, 45 balls at $4 each, and 15 uniforms at $30 each.
27) A teacher is buying supplies for two art classes. For class 1, the teacher buys 24 tubes of paint, 12
brushes, and 17 canvasses. For class 2, the teacher buys 20 tubes of paint, 14 brushes, and 15
canvasses. Each tube of paint costs #3.35, each brush costs $1.75, and each canvass costs $4.50.
Questions 28 – 29: Find the inverse of each matrix, if it exists. No calculator.
 2 6 
28) 

 1 3 
14 8 
29) 

 6 4
Precalculus Semester Exam Review
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Questions 30 – 31: Solve for X. 2x2 by hand, 3x3 with calculator.
 5 4
 10 
30) 
X 


 3 2 
 16 
 2 1 0 
 5 


31) 1 4 2 X  15 


 
 3 2 1 
 7 
Questions 32 – 33: Solve the following systems using matrices. Check your answers. All 2x2
systems must be solved by hand. Systems 3x3 and larger may be solved with a
calculator. However, show the matrix equation for all systems.
3x  7 y  7
32) 
7 x  3 y  3
 x  5 y  10 z  13

33) 2 x  y  3 z  18
4 x  6 y  12 z  7

34) A flower farmer wants to plant three types of bulbs: gladiolas, irises, and tulips. The gladiolas cost
$75per acre to plant, the irises cost $100 per acre to plant, and the tulips cost $50 per acre to plant.
The farmer wants to plant 200 acres of bulbs and spend a total of $15,000. The farmer decides to plant
twice as many gladiolas as irises. Write a system of equations and use a matrix to find the total
number of acres of each type of flower.
Precalculus Semester Exam Review
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FUNCTIONS
35) The cost of having a carpet installed is $25.00 for delivery and $1.50 per square yard for the actual
installation.
a) Find a linear equation that models the cost of having a carpet delivered and installed.
b) Find the number of square yards of carpet installed if the bill for delivery and installation is $60.25
without the tax.
36) Graph: y  x 2  2
37) Given f ( x)  x 2  1 , evaluate each of the following and simplify:
a) f (2)
b) f ( a  b)
 2
c) f  
 x
38) Find the domain and range of each function:
a) g ( x)  x  1.5
D: ___________
b) h( x) 
R: ____________
3
x  x  20
2
D: ___________
R: ____________
Precalculus Semester Exam Review
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39) Find the inverse of each function. Is the function one-to-one?
a) f ( x) 
x
2
b) g ( x) 
1
2x  3
40) Graph each function and determine its domain and range.
 x  1, x  0
a) f ( x)  
 x  1, x  0
b) g ( x)  x  3
D: ____________
D: ____________
R: ____________
R: ____________
41) Determine whether each function is even, odd, or neither. Show your algebraic check.
a) f ( x)  2 x 4  3 x 2  4
Precalculus Semester Exam Review
b) g ( x)  3x5  2 x3  x 2  1
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42) Graph the reflection of each function in the given line.
a) x-axis
c) Line y  x
b) y-axis
43) Graph each equation by determining the basic function and using transformations. Write the “basic”
function from which each is derived.
a) y 
b) y  2 x  3
1
x2
y  ________
y  ________
44) The height, s, in feet, of a ball thrown into the air is given by s  16t 2  96t  112 , where t is the
time in seconds. Find the maximum height and the time the ball takes to reach this height.
maximum height = ____________
time to reach max height = ______
Precalculus Semester Exam Review
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TRIGONOMETRY
11
counterclockwise rotation, find the measure of the angle in degrees. Round answer to three
9
decimal places.
angle measure = _________
45) For an
46) Express 218 18' 46" in decimal form. Round answer to nearest thousandth.
47) Express
7
in degrees.
5
_____________
48) Express 318 in radians. Leave answer in terms of  .
49) If cos  
_____________
_____________
12
, and  is in Quadrant IV, determine the exact value of sin  .
13
sin   ________
50) The terminal side of an angle  in standard position passes through the point ( 7, 24) . Determine the
exact values of the six trigonometric functions.
sin   _______
csc   _______
cos   _______
sec   _______
tan   _______
cot   _______
Precalculus Semester Exam Review
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51) Determine the exact value of tan 690 .
_____________
52) Evaluate sin 62.2 to four decimal places.
_____________
53) If 0    360 and cot   1.4176 , determine  to the nearest tenth of a degree. _____________
For each function determine the amplitude, period, phase shift, vertical shift, and the equations for
two vertical asymptotes, if applicable.
Function
54) f ( x)  3cos
Amp
Period
Phase
Vertical
shift
Shift

4
55) g ( x)  sin(3   )  1


56) h( x)  tan  2    2
3

Precalculus Semester Exam Review
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Domain
Range
57) Write a sine function with an amplitude of 3, with a period of


, and with a phase shift of
2
6
y = _________________________
58) Write a tangent function with a vertical stretch of 3, a period of 2 , and a phase shift of

.
4
y = _________________________
59) Using your knowledge of the Unit Circle, find the exact values:
a) sin

3

 9
e) sec  
 2



7

4
 2
b) tan  
 3



c) csc
 7
f) cos  
 6



 15
g) sin  
 4
d) cot



13

6
 17
h) tan 
 6



60) Given a point on the terminal side of an angle, find the exact value of the requested trig function.
a)

P 8, 17

sin  
b)
1 1
P , 
c)  3 4 
tan  
P  3, 5 
cos  
61) You are riding a bicycle with wheels that have a radius of 25 inches. If you are traveling at 18 miles
per hour, what is the rotational speed of the tires in revolutions per second? What is the angular
speed of the wheels in radians per second?
Precalculus Semester Exam Review
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