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Math III Review [578574]
Student
Class
Date
Read the following and answer the questions below:
The Ferris Wheel
The Ferris Wheel
“School’s out!” Ryan shouted gleefully to his older sister, Claire, as he
barged through the front door of his house and slung his backpack on the
floor. They attended the local high school, where Ryan had been a freshman
and Claire a junior. Summer vacation had finally begun.
Ryan and Claire had even more reason to celebrate because the family
vacation their parents had planned to Chicago, Illinois, was now only two
days away. Claire had wanted to visit Chicago ever since her friend Marco
returned from a visit there; she still remembers how he ranted and raved
about all there was to see and do in the “Windy City.”
Ryan and Claire had been doing research online about the numerous sights
to see and all the exciting things they should do on their visit. Claire was
very interested in seeing the downtown area with its prominent buildings
and skyscrapers, many having been designed by important architects. But
both Ryan and Claire were most excited about finally getting to go to Navy
Pier, an amusement park located on Lake Michigan that includes a musical
carousel, indoor mall, great food, and the famous Ferris wheel.
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“Hey, Ryan, did you know that the Navy Pier Ferris wheel is 150 feet high?”
Claire said in amazement. “According to this information, that’s as tall as a
13-story building.”
“You aren’t going to ride the Ferris wheel. You will be too frightened once
you see how high it really is,” Ryan teased.
“Oh, be quiet, Ryan! I won’t be too scared, especially since we will all be
together—that is, as long as you promise not to rock the carriage when we
get to the top. This website says that it has 40 carriages, or gondolas, that
you sit in, and each one will seat up to 6 people. Therefore, we will all
definitely be able to fit in one.”
Claire went on to read that during the 7-minute ride, she would be able to
see most of downtown. In fact, from the top of the Ferris wheel, on a clear
day, one could see 50 miles in all directions.
“Look! It says you can see the entire downtown area when you get to the
top.”
“That’s great, Claire. But the only sight I want to see is my eating a Chicagostyle hot dog with fries and a funnel cake.” Ryan laughed at his joke, but
Claire was too busy studying the website to even look away and roll her eyes
at his absurd comment, which would have been her normal reaction.
Claire loved to learn about how things were built, all the way from the
inception of the idea drafted on paper to the actual building of the structure,
which is why she planned to study architecture in college. She found the
subject fascinating and read as much as she could about the Ferris wheel,
wanting to know when, why, and how it came about.
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Claire discovered that the Ferris wheel was invented by George Washington
Ferris back in the 1890s. The first Ferris wheel was 25 stories high and was
made entirely of steel. It had a diameter of 250 feet and was supported by 2
towers, each 140 feet high. There were 36 enclosed carriages that could
hold slightly more than 1,400 passengers at any given time. The ride itself
lasted for 10 minutes, circling 2 full revolutions around; the first revolution
was slower than the second so that passengers could be loaded into the
carriages.
It debuted at the World’s Columbian Exposition, which is more commonly
known as the 1893 Chicago World’s Fair. George Ferris wanted the
attendees of the fair to marvel at his innovative invention and forget about
the Eiffel Tower, which had been revealed four years earlier at the Paris
International Exposition. Though the wheel was popular at first, it soon lost
its superstar appeal and was dismantled and eventually sold as scrap metal.
Claire looked up from her computer. She was more excited than ever,
knowing that soon she would experience all of the sites she had read about.
There was only one thing left to do: pack.
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1. Read “The Ferris Wheel” and answer the question.
Claire and Ryan’s carriage on the Navy Pier Ferris wheel travels 205 feet
counterclockwise from the loading point before they can get a clear view
of the city. How many radians, rounded to the nearest hundredth, does
the carriage rotate to reach this point?
2. Read “The Ferris Wheel” and answer the question.
Explain how the unit circle in the coordinate plane relates to the rotation
of the Navy Pier Ferris wheel. How does this comparison enable the
extension of trigonometric functions to all real numbers?
3. What is the simplified form of the expression
A.
B.
C.
D.
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4.
If
what is the value of
A.
B.
C.
D.
5. Which expression is equivalent to (3x2 – 5x + 4) + (2x2 – 7)?
A. 5x2 – 5x – 3
B. 5x2 – 5x – 11
C. 6x2 – 5x – 3
D. 5x4 – 5x – 3
6.
Which expression is equivalent to
A.
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B.
C.
D.
7.
What are the zeros of the polynomial function
A.
B.
C.
D.
8. Which function has a remainder of 3 when divided by
A.
B.
C.
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D.
9. Which polynomial has exactly 2 positive x-intercepts?
A.
B.
C.
D.
10.
Which graph best represents the function
A.
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B.
C.
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D.
11.
Let the function
What are all the x-intercepts
for the graph of
A.
B.
C.
D.
12.
A polynomial
that
can be expressed so
What is the value of a?
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A.
B.
C.
D.
13. Part A:
Prove the identity for the cube of a binomial:
Part B:
Explain the change in the formula when the binomial indicates addition
rather than subtraction.
14. For this task, assume that a and b are both positive and a is greater than b.
Part A. Use the diagram below to prove the polynomial identity
,
when a is the length of the largest, outer square, and
b is the length of the white square. Justify your reasoning.
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Part B. Draw a diagram that can be used to prove the polynomial identity
Use the diagram to prove the polynomial identity and
justify your reasoning.
Part C. Use polynomial division to prove the polynomial identity
Explain why polynomial division can be used to
prove this polynomial identity. Can the same method be used to prove the
polynomial identity
Explain why or why not.
Part D. There is also a geometric way of proving the difference of cubes
polynomial identity. Use the three-dimensional figures below to construct an
argument as to why
for
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and
Part E. Draw the figures that can help prove
and
Use the figures to construct an argument as to why
for the given values of a and b.
Part F. Could a similar geometric method be used to prove
Explain why or why not.
15. Which polynomial identity can be proved using the polynomial division
given below?
A.
B.
C.
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for
D.
16. The distributive property is used to prove which of the given identities
shown below are true?
A. I and II only
B. I and III only
C. II and III only
D. I, II, and III
17.
Which of these expressions is equivalent to
A.
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B.
C.
D.
18.
Given:
If
what is the value of
19. A computer repairman charges $50 to come to a home or office,
plus $30 per hour of work. During one week, he visits 12 homes
or offices earning $1,800. How many hours did the repairman
work?
A. 22 hours
B. 40 hours
C. 42 hours
D. 58 hours
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20. A high school is hosting a basketball tournament. Their goal is to
raise at least $1,500.00. Students can buy tickets for $3.00 and
non-students for $5.00. The seating capacity for the gym is 400
people. Which could represent the number of each type of ticket
sold to meet the high school’s goal and not exceed the capacity
of the gym?
A. 100 student, 200 non-student
B. 125 student, 175 non-student
C. 150 student, 350 non-student
D. 170 student, 229 non-student
21. Alma invests $300 in an account that compounds interest annually. After
2 years, the balance of the account is $329.49. To the nearest tenth of a
percent, what is the rate of interest on the account?
A. 6.9%
B. 5.4%
C. 4.8%
D. 4.4%
22. Trevor is making two types of bracelets.
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




Each Type P bracelet needs 12 inches of leather and 3
inches of string.
Each Type Q bracelet needs 4 inches of leather and 18
inches of string.
Trevor has 5 yards of leather and 6 yards of string.
x equals the number of Type P bracelets Trevor makes.
y equals the number of Type Q bracelets Trevor makes.
Which system of equations models the constraints on the
number of bracelets Trevor can make?
A. 12x + 4y ≤ 180
3x + 18y ≤ 216
x≥0
y≥0
B. 12x + 3y ≤ 180
4x + 18y ≤ 216
x≥0
y≥0
C. 12x + 4y ≤ 5
3x + 18y ≤ 6
x≥0
y≥0
D. 12x + 3y ≤ 5
4x + 18y ≤ 6
x≥0
y≥0
23. The escape velocity, v, with which a body should be projected so that it
overcomes the gravitational pull of the Earth is given as
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where g
is the acceleration due to gravity on the Earth and R is the radius of the
Earth.
Part A. Find the formula that can be used to calculate the acceleration
due to gravity on the Earth, given the escape velocity and the radius of
the Earth.
Part B. Find the formula that can be used to calculate the radius of the
Earth, given the acceleration due to gravity and the escape velocity.
Use words, numbers, and/or pictures to show your work.
24. The sum of three consecutive integers is 51. What is the value of
the largest integer?
A. 16
B. 17
C. 18
D. 19
25. Jacob stated that he solved the equation
using the addition and
multiplication property of equality. Which statement is true?
A.
Jacob added
to both sides and multiplied both sides by
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B. Jacob added
to both sides and multiplied both sides by
C.
Jacob added to both sides and multiplied both sides by
D. Jacob added to both sides and multiplied both sides by
26.
The equation
can be used to convert temperature from
degrees Fahrenheit
to degrees Celsius
first step in solving the equation for
A.
B.
C.
D.
27.
Which value is a solution to the equation
A.
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Which of these could be the
B.
C.
D.
28. Consider this equation.
Part A. Solve the equation for x, showing all steps and both resulting
values of x.
Part B. Do both values of x represent solutions to the equation? Explain
your answer.
Use words, numbers, and/or pictures to show your work.
29.
For what value of p is the expression
expression
A.
B.
C.
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equivalent to the
D.
30. Write and solve the equations for the situations listed below. Choose the
most efficient method to solve each equation and show your work.
Part A. A drawing room is in the shape of a square and has an area of
144 square feet. Write an equation to determine the side, s, of the room
length.
Part B. There is a table with a width of 3 feet less than its length and an
area of 10 square feet. Write and solve an equation to determine the
length and width of the table.
Part C. A sofa has a length that is 3 feet more than twice its width. If the
sofa occupies an area of 18 square feet, write and solve an equation to
determine the approximate dimensions of the sofa.
Part D. A rectangular plot of land is to be fenced in using 100 feet of
fencing. If you want the maximum area to be fenced in, what would be
the dimensions of the plot of land that is fenced?
Use words, numbers, and/or pictures to show your work.
31. Susan has a vegetable garden on a rectangular piece of property. Last
year, she divided her garden into three square sections:
In the south section, she planted beans.
In the middle section, she planted carrots.
In the north section, she planted cabbage.
This year, Susan makes two changes to her garden:
First, she uses some of the area of the cabbage section for a tool shed.
She builds the tool shed along the north end of the property. The tool
shed goes all the way across the north end of the property and measures
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2 yards from front to back. After subtracting the area taken up by the
tool shed, the area of Susan's three vegetable plots is 133 square yards.
Second, she decides to surround her garden with a fence to keep
the vegetables safe from animals. She will need to fence all four sides of
the garden. The tool shed will be outside of the fence.
Part A. Draw a picture to represent Susan's vegetable garden, the fence,
and the tool shed. Label each part of the picture.
Part B. Write an expression for the area that Susan will use this year to
plant vegetables, in terms of x, the length of the short edge of her
property. Set that expression equal to 133.
Part C. Solve the equation from Part B using two methods:
Use the method of completing the square to change the equation into the
form
work.
where p and q are real numbers. Solve for x. Show your
Use the quadratic formula. Solve for x. Show your work.
What is the length of the short side of the garden?
Part D. Determine how many yards of fencing Susan will need.
32.
If quadratic equation
what are the values of p and q?
is rewritten in the form of
A.
B.
C.
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D.
33. Which choice is an ordered pair that, for every real number k,
represents a point that lies on the graph of 30x – 5y = 10?
A. (k + 2, 6k + 10)
B. (k + 4, 6k + 20)
C. (3k, 18k + 2)
D. (5k, 30k + 2)
34.
If a point
the graph of
is on the graph of the equation
what is the value of b?
A.
B.
C.
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and also on
D.
Who Will Catch Up When?
35.
Three friends are packing gift boxes to be handed out at the high school
dance. Each box has ten sections to be filled. They each pack the boxes
at different rates.
Part A. The tables below show Kelsey’s and Andrew’s progress. The
variable t stands for the time that has passed since their starting time at
10:00 a.m.
Andrew had already packed 4 boxes the day before.
Fill in the tables, assuming that each person packs the boxes at a
constant rate.
Part B. Let the number of boxes Kelsey packs be represented by k(t) and
the number Andrew packs by a(t). Using the information from the tables,
write the functions for k(t) and a(t). Interpret each function in terms of
the context it represents.
Kelsey:
Andrew:
In terms of k(t) and a(t), what equation can be solved to find the time at
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which both Kelsey and Andrew have packed the same number of boxes?
Explain.
Part C. Graph and label the functions k(t) and a(t) on the same
coordinate grid below.
What is the solution of the equation that was written in part B that could
be used to find the time at which Kelsey and Andrew have packed the
same number of boxes? Explain how you can find the solution on
the graph and then verify your answer by solving the equation
algebraically.
Part D. The third friend, James, starts out packing the boxes very quickly
but then slows down. His approximate progress can be modeled by the
square root equation below, where t stands for the time that has passed
since their starting time at 10:00 a.m.
Fill in the table to show James’s progress. Round the values to the
nearest 0.1 box.
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Part E. Sketch a graph of the function j(t) on the coordinate grid above,
where k(t) and a(t) are already graphed.


Write the equation that could be used to find the time at which
Kelsey and James have packed the same number of
boxes. Approximate the solution or solutions to this equation
using the graph.
Write the equation that could be used to find the time at
which Andrew and James have packed the same number of boxes.
Approximate the solution or solutions to this equation
using the graph.
36. If f(x) = 5(2)x and g(x) = –2x + 46, for what positive value of x
does f(x) = g(x)?
A. 3
B. 5
C. 40
D. 46
37. Two functions are shown below.
f(x) = 2x + 2
g(x) = –2x + 6
For what value of x does f(x) = g(x)?
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A. 1
B. 2
C. 4
D. 6
38. What is the maximum number of intersections an exponential
function can have with a linear function?
A. 0
B. 1
C. 2
D. 3
39.
Let
Find all real values of x such that
40.
Which expression is equivalent to
where k is an even number?
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A.
B.
C.
D.
41. The expression for the amount of money earned on a savings account
compounded quarterly is given by the expression
where
represents the principal and is the time in years since the principal was
invested. Which expression is the equivalent form of the given expression
and shows the amount earned when the interest is compounded halfyearly?
A.
B.
C.
D.
42. Jesse and Shaun are comparing investment products to see who has the
better investment rate for their money.
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The interest on the money Jesse invested in Product X is compounded
annually. The value of the investment after n years can be found using
the formula
where
is the intial amount of money invested.
The interest on the money Shaun invested in Product Y is compounded
monthly. The value of the investment after m months can be found using
the formula
where
is the intial amount of money invested.
Part A. Rewrite Jesse's formula to find the approximate equivalent
monthly interest rate. Show your work.
Part B. Which product offers the best return on an investment? Use the
interest rates to justify your answer.
43. An athlete is training to run a marathon. She plans to run 2 miles the
first week. She increases the distance by 8% each week. Which function
models how far she will run in the nth week?
A.
B.
C.
D.
44. Cindy invested $2,800. The function V(t) = 2,800(1.025)t models
the value of Cindy’s investment after y months. The function
S(t) = 10t models the amount of money that Cindy has saved in
a safe at her house after t months. Which function C(t) models
the combined value of the investment and money in the safe?
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A. C(t) =2,810(1.025)t
B. C(t) = 2,800(1.025)11t
C. C(t) =(2,800 + 10t)(1.025)t
D. C(t) =2,800(1.025)t + 10t
45. A plane is at a height of 30,000 feet above the ground when it begins to
descend at a rate of 1,500 feet per minute. If
and
write
a recursive formula that can be used to determine the height of the plane
above the ground after n number of minutes.
46. The ingredients for a particular kind of European chocolates cost $12 per
box. The foil wrappers cost $0.05 per piece of chocolate. The box has x
pieces of chocolates in it. Which function represents the total cost per
piece of chocolate?
A.
B.
C.
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D.
47. Which recursive formula models the sequence shown below?
3, 1, 5, 9, . . .
–
A. NEXT = NOW + 4
B. NEXT = NOW – 4
C. NEXT = 4 • NOW
D. NEXT = 4 • NOW + 7
48. At the beginning of the school year, Jason’s dad gave him $50 to
put into his lunch account. Jason spends $2 each day on his
lunch. Which recursive formula models the amount of money
that Jason has in his account?
A. NEXT = NOW + 2
B. NEXT = NOW – 2
C. NEXT = 50 – 2 • NOW
D. NEXT = 52 – 2 • NOW
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49. If
and
the sequence?
which equation represents the explicit formula for
A.
B.
C.
D.
50. The first term of an arithmetic sequence is 3. The nth term of the sequence is found by using the
formula
Which other formula could be used to find the nth term?
A.
B.
C.
D.
51. Which transformation occurs to the graph of f(x) = x to produce
the graph of g(x) = x + 2?
A. down 2 units
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B. up 2 units
C. left 2 units
D. right 2 units
52.
If the graph of
is translated 2 units right and 4 units down,
which of these functions describes the transformed graph?
A.
B.
C.
D.
53. The function f(x) = 6x was replaced with f(x) + k resulting in the
function shown in the table below.
x
y
0
10
1
15
2
45
3
225
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What is the value of k?
A. 7
B. 8
C. 9
D. 10
54. The function f(x) = x – 2 was translated down 6 units, resulting
in the function g(x). Which function represents g(x)?
A. g(x) = 6x – 2
B. g(x) = 2x – 8
C. g(x) = x – 8
D. g(x) = x + 4
55.
A function
is defined as
Part A. Write a function
where
that represents the inverse of the function
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Part B. How do the domain and range of
compare with the domain
and range of the inverse function
Use words, numbers, and/or pictures to show your work.
56. The table below shows the attempts made by four students to find the
inverse of the function
Which student correctly found the inverse of the function?
A. Daniel
B. Jean
C. Scott
D. Sophia
57. The function h(t) = 200 – 16t represents the height of a ball
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dropped from 200 feet. How far had the ball traveled after falling
for 11 seconds?
A. 16 feet
B. 24 feet
C. 176 feet
D. 200 feet
58. The function f(t)=12,000(1.075)t models the value of an
investment t years from now. What is the meaning of the value
of f(5)?
A. the value of the investment 5 years ago
B. the value of the investment in 5 years
C. the initial value of the investment
D. the interest rate the investment earns
59. The table below shows the cost for a toy company to produce
different amounts of toys.
Toys Produced
1,000
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Cost
$122,000
3,000
5,000
7,000
$26,000
$10,000
$74,000
Assuming a quadratic relationship, about how many toys should
the company produce to minimize costs?
A. 1,000
B. 4,000
C. 5,000
D. 6,000
60. The table below shows the distance Chris is located from his
school at different times.
Time
(minutes)
0
3
6
9
12
15
Distance
(miles)
20
18
16
14
12
10
Assuming a linear relationship, how long will it take Chris to get
to school?
A. 20 minutes
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B. 24 minutes
C. 27 minutes
D. 30 minutes
61. Jimmy threw a baseball in the air from the roof of his house. The path
followed by the baseball can be modeled by the function
where t represents the time in seconds after the ball
was thrown and
represents its height, in feet, from the ground.
Part A. How high is the roof from the ground? How many seconds did it
take for the ball to hit the ground after it was thrown off the roof?
Part B. Jimmy wanted to throw the ball at a maximum height of 120 feet.
Did Jimmy's baseball reach this height after it was thrown? Explain your
answer.
Use words, numbers, and/or pictures to show your work.
62. Which function has the following features?



symmetry over the y-axis
increasing for all
y-intercept of 0
A.
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B.
C.
D.
63. Part A. Emma’s cell phone plan charges $0.20 for each text message.
The function
represents the cost of the total number of text
messages Emma sends and receives. If x represents the total number of
messages, what is the domain of the function
Part B. How will the domain of the function
limit of $25 on her monthly texting bill?
change if Emma puts a
Part C. The dollar amount of Emma’s prepaid call balance decreases by r
for each second of a call. The call balance left is modeled by the function
where b is the initial balance in dollars and s is the number of
seconds. What is the domain of the function
Part D. Emma changes her call plan from prepaid to pay-as-you-go. The
function
represents the total bill she pays after a month,
where A is the fixed monthly fees and
is the amount in dollars charged
for each second of calls she made. What is the reasonable domain of the
function
in terms of the given context?
64. The table below shows the population of a state during different
years.
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Year (x)
2004
2006
2007
2008
2010
Population (y)
8,500,000
8,900,000
9,000,000
9,200,000
9,500,000
What is the approximaterelative domain of the line of best fit
for the data?
A. x > 0
B. x > 1650
C. x > 1952
D. x > 2004
65. Function
has a minimum value of
and a maximum value of 8.
Which graph most likely represents function
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A.
B.
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C.
D.
66.
Which graph represents the function
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A.
B.
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C.
D.
67. Which is the graph of 3x – 2y = 4?
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A.
B.
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C.
D.
68.
Graphing
Part A. Consider the functions
the domain and range of each function?
and
What are
Part B. What are the x-intercepts and y-intercepts (if any) on
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On
Part C. Where are the functions increasing or decreasing?
Part D. What are the maximum points (if any) on
Part E. Sketch graphs of the functions
and
and
Part F. How are the graphs of the two functions in part A related? How do
you see these relationships in the equations?
Part G. Sketch graphs of the functions
and
Explain how you determined the coordinates of key points.
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Part H. How are the graphs of the three functions in part G related? How
do you see these relationships in the equations?
69. The height in feet,
a kangaroo reaches
seconds after it has jumped in
the air is modeled by the quadratic function
Which
equation shows the correctly factored version of the function and
the number of seconds it takes for the kangaroo to return to the ground?
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A.
8 seconds
B.
1.5 seconds
C.
8 seconds
D.
1.5 seconds
70. The function f(x) = 19,000(0.89)x models the value of a boat x
years after its purchase. Which statement correctly describes the
value of the boat?
A. The value is decreasing by 11% per year.
B. The value is decreasing by 89% per year.
C. The value is increasing by 11% per year.
D. The value is increasing by 89% per year.
71. Genevieve deposited $400 into her bank account. The
equation
can be used to calculate the value of her money
after t years. What is the annual interest rate she is earning on her
deposit?
A.
0.07%
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B.
1.07%
C.
7%
D. 107%
72. The function f(x) = 2,500(0.97)x models the value of an
investment after x months. Which statement is true about the
value of the investment?
A. The value of the investment increases by 3% each month.
B. The value of the investment decreases by 3% each month.
C. The value of the investment increases by 97% each month.
D. The value of the investment decreases by 97% each month.
73. Joseph compared the function f(x) = 3x2 + 2x – 1 to the
quadratic function that fits the values shown in the table below.
x
0
1
2
3
4
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g(x)
–
1
8
23
44
71
Which statement is true about the two functions?
A. The functions have the same y-intercepts.
B. The functions have the same x-intercepts.
C. The functions have the same vertex.
D. The functions have the same axis of symmetry.
74. Austin and Janda threw grappling hooks into the air. The function
gives the height, in feet, of Austin’s hook x seconds
after he threw it. The graph below shows the height, in feet, of Janda’s
hook x seconds after she threw it.
If both of them threw the grappling hooks at the same time, which of
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these statements is true?
A. Austin’s hook hit the ground first.
B. Austin’s hook reached its maximum height first.
C. Austin’s hook reached a greater maximum height.
D. Austin threw the hook from a greater initial height.
75. Two functions are shown below.
f(x) = 1.02x + 100
g(x) = 50(1.02)x
What is the smallest positive integer in which the value of g(x)
exceeds the value of f(x)?
A. 60
B. 59
C. 55
D. 50
76. Ronald invests $1000 at a simple interest rate of 10% for 4 years. His
best friend Rudy invests the same amount of money, but earns 10%
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interest compounded annually for 4 years.
Part A
Create a table to show the amount of Rudy's investment after each year.
Calculate the amount of Ronald's investment after 4 years.
Part B
Based on the amounts they made, which friend made the better
investment? Explain.
77. Clara’s and Michelle’s parents started saving for college in 1998.


Clara’s college fund can be modeled by the function f(x) =
500x + 2,500, where x is the number of years since 1998.
Michelle’s college fund can be modeled by the function g(x)
= 2,500(1.1)x, where x is the number of years since 1998.
About what year will Michelle’s college fund first exceed Clara’s
college fund?
A. 2013
B. 2015
C. 2017
D. 2019
78. Which table shows the function that increases at the fastest rate?
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A.
B.
C.
D.
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79.
For what value of x is it true that
A.
B.
C.
D.
80.
The bacteria in a certain Petri dish grow at a rate modeled by
where
represents the number of bacteria in the dish and t represents
the time in minutes since the introduction of the bacteria. Which
equation can be used to determine how many minutes will pass before
there are 68 bacteria in the dish, if the dish started with a single
bacterium?
A.
minutes
B.
minutes
C.
minutes
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D.
minutes
81.
What is the solution to the equation
A.
B.
C.
D.
82.
What is the solution to the equation
A.
B.
C.
D.
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83. Circle P, shown below, has a radius of 1 unit.
Which of these equations correctly identifies the relationship between
angle QPR, in radians, and the length, a, of arc QR?
A.
B.
C.
D.
84. Let
and
be two values such that
Express in terms of .
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but
85. In the right triangle pictured below, the measure of angle A, in radians,
is The side adjacent to angle A measures 6 cm and the side opposite
measures 5 cm.
Which of the following values is closest to
A.
B.
C.
D.
86.
Toni claims that the cosine of
is equal to the cosine of
Which equation could be used to justify Toni's claim?
A.
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.
B.
C.
D.
for any integer k
for any integer k
87. Which function best represents a sine curve that repeats every 12 units
and has a maximum of 42 and a minimum of 4?
A.
B.
C.
D.
88. George’s height above the ground as he rides a Ferris wheel ranges from
4 meters to 30 meters. If it takes 200 seconds to complete one
revolution, which sine function represents his height,
ground as a function of time,
A.
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from the
B.
C.
D.
89. A Ferris wheel with a diameter of 40 feet completes 2 revolutions in one
minute. The center of the wheel is 30 feet above the ground. If a person
taking a ride starts at the lowest point, which trigonometric function can
be used to model the rider’s height h(t) above the ground after t
seconds? (Consider the height of the rider negligible).
A.
B.
C.
D.
90. Nan draws the swinging end of a pendulum 10 centimeters to the left of
its rest position and releases it to swing. She wants to model the
horizontal displacement of the pendulum, d, as a function of time, t,
where
at the point of release. Which function family is best for Nan to
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use and why?
A.
, because
B.
, because
C.
, because
D.
, because
is an extremum
is an extremum
91.
What is the value of
simplest form.
if
92. Point P is a point on the unit circle.
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and
Write your answer in
Part A. Use the Pythagorean theorem and the diagram above to prove
the trigonometric identity
Part B. If
use the identity to find
Use words, numbers, and/or pictures to show your work.
93. Based on the diagram below, how can the Pythagorean identity
be shown?
A.
B.
C.
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D.
94. Let p represent a point on the unit circle, in the second quadrant. The
line including p and the origin has a slope of -2. What is the x-value of p?
A.
B.
C.
D.
95.
Similarity in Circles
Geometric similarity is an extremely useful concept. Similar figures are
alike except for their size; their corresponding angles are congruent, and
their corresponding parts are proportional. On the coordinate plane, one
figure can be mapped to the other by a series of transformations.
Part A. Consider these two equilateral triangles. Are they similar? How do
you know? Write a proportion showing the relationship of their sides.
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Part B. Are any two squares similar? Tell how you know.
Remember that the measure of each angle of a regular polygon is
where n is the number of sides. Can you make a general
statement about the similarity of two regular polygons (n-gons) with the
same number of sides? Explain your answer.
Part C. As the number of sides of a regular polygon increases, what
figure does it begin to look like?
What is a reasonable conclusion about the similarity of figures of this
kind of different sizes?
Part D. Consider these two circles on the coordinate plane. What is the
radius of circle A? Of circle B? Write the ratio. Write the ratios for the
diameters and circumferences of the two circles. Are the circles
proportional?
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Part E. You can also prove that two figures are similar by showing that a
series of transformations will map one figure to the other. What is the
equation for circle A?
Part F. What series of transformations will map circle A to circle B? Write
the equation for circle B. Are these two circles similar?
Part G. Any two circles can be centered at the origin through
translations. If both circles are centered at the origin, what one
transformation will map one to the other, proving their similarity? If the
equation of one circle is
and the radius of the other circle is f
times the radius of the first, what is the equation of the second circle?
What is the equation of the second circle if the center is NOT
In either case, no matter what the size or position of the
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circles, are all circles similar?
96. Which statement best explains why all circles are similar?
A. All circles have exactly one center point.
B. The diameter of all circles is twice the length of the radius.
C. All circles can be mapped onto any other circle using only
translations.
D. All circles can be mapped onto any other circle using a translation
and dilation.
97. Which property of quadrilaterals inscribed in a circle can be used to find
the value of x in the figure below?
A. The difference between an opposite pair of angles is 180°.
B. The difference between an opposite pair of angles is 0°.
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C. The sum of an opposite pair of angles is 180°.
D. The sum of an opposite pair of angles is 360°.
98.
In the figure given below,
If
is a diameter of the circle with center O.
what is
A. 60°
B. 70°
C. 80°
D. 110°
99.
In the figure below,
are radii of the circle with center O.
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Given that
what is
A. 20°
B. 35°
C. 40°
D. 80°
100.
In the figure shown below,
If
and
what is
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are radii of the circle with center
A. 56°
B. 62°
C.
D. 124°
101. Points A, B, C, and D lie on circle E as shown in the figure below.
Which statement must be true about the figure?
A.
B.
C.
D.
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102. The figure shown below is a circle.
Which statement must be true?
A.
B.
C.
D.
103. Amy is designing a piece of jewelry to sell in her craft store. She begins
with the triangular piece of silver, as shown below.
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Part A. Amy wants to add a circular piece of gold that will be inscribed
inside the triangular piece of silver. Use a compass and straightedge to
show how she can add the circular piece to the triangle above. Explain
the steps you used to perform the construction.
Part B. She needs to know the radius of the inscribed circle so that she
can calculate the circumference and area of the circular gold piece she
needs to make for the jewelry. Given that the silver triangle is a right
triangle with side lengths a, b, and c, find the equation Amy can use to
determine the radius of the circle, r. Explain your answer and draw a
diagram or use your construction in part A to support your reasoning.
Part C. Amy then decides to inscribe another similar silver triangle
inside a circular piece of copper so that each vertex of the triangle
touches the edge of the copper circle. Use a compass and straightedge
to construct her design below. Explain the steps you used to perform
the construction.
Part D. A couple of months ago, Amy designed a piece of jewelry with
a gold quadrilateral inscribed on a circular piece of silver. She found the
sketch of her design in her desk drawer, as shown below.
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Now Amy wants to produce an identical piece of jewelry but needs to
know the exact angle measures for the gold quadrilateral. What
geometric property about quadrilaterals can Amy use to find the
measures of the angles of her jewelry design? Use a paragraph proof to
justify your response.
Part E. What are the measures of the three missing angles in Amy’s
sketch of the piece of jewelry in part D? Explain how you know.
104. Quadrilateral
and angle
is inscribed in circle E. Angle
is an inscribed angle.
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is a central angle,
Which statement about the angles in this figure must be true?
A.
B.
C.
D.
105. In the given image,
and the angles are measured in radians.
Which of these must be true?
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A.
B.
C.
D.
106. Part A
Using the circle below, set up a proportion to determine the length, m,
of arc HI. Use for the central angle of a complete circle and
for the
circumference of the circle where r is the radius. Solve for m.
Part B
The circle below has a central angle which measures 2.1 radians and a
diameter of 3 inches. Find the length, in inches, of arc HJ.
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107. In circle C below, is measured in radians.
Which expression can be used to find the area of the shaded sector?
A.
B.
C.
D.
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108. Ashley is studying circle O, shown below.
She wrote the steps below.
What did Ashley derive?
A. that all circles are similar
B. the formula for arc length
C. the formula for the area of a sector of a circle
D. that a central angle has the same measure as the arc it subtends
109. Which of these correctly defines a ray?
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A. a part of a line with exactly two end points
B. a straight path that extends endlessly in both directions
C. a circular path such that every point on the path is equidistant from
its center
D. a part of a line that begins at a particular point and extends
endlessly in one direction
110. The distance between points A and P is the same as the distance
between points B and P. If P does not lie on a line joining the points A
and B, which of these conclusions are true?
I. All the points on
II. The angles
will be equidistant from point P.
and
III.
and
form
IV.
and
form arc APB.
are congruent.
A. I and III
B. II and IV
C. II and III
D. I and IV
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111. What is a definition of a line that is parallel to
A. a coplanar line that bisects
B. a coplanar line that does not intersect
C. a coplanar line that intersects
at a right angle
D. a coplanar line that intersects
but not at a right angle
112. Chris draws an image of two lines that lie in the same plane and are
equidistant at all points. Which of these describes the image drawn by
Chris?
A. an angle
B. parallel lines
C. intersecting lines
D. perpendicular lines
113. A proof of the Alternate Interior Angles Theorem, using parallel lines a
and b with transversal m, is shown below.
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Which property is used in step 4?
A. reflexive property
B. transitive property
C. associative property
D. commutative property
114. Two parallel lines are cut by a transversal x and a transversal y so that
x and y intersect at point Q as shown.
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Wong constructs the following argument:
Angle a is [missing reason 1] and therefore congruent to one angle in
the triangle formed by lines m, x, and y.
Angles b and c are [missing reason 2] and therefore congruent to two
other angles in the triangle.
The sum of the three angles in a triangle is 180 degrees. Therefore
.
What are the missing reasons in Wong’s argument?
A. Reason 1: an alternating exterior angle with one angle in the
triangle
Reason 2: vertical angles with the other two angles in the
triangle
B. Reason 1: an alternating exterior angle with one angle in the
triangle
Reason 2: complementary angles with the other two angles in
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the triangle
C. Reason 1: a corresponding angle with one angle in the
triangle
Reason 2: vertical angles with the other two angles in the
triangle
D. Reason 1: a corresponding angle with one angle in the
triangle
Reason 2: complementary angles with the other two angles in
the triangle
115.
John draws
and
with
as the perpendicular bisector of
are congruent to each other.
Which statement can John use this figure to prove?
A. Every isosceles triangle is a right triangle.
B. The base angles of any triangle are congruent.
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such that
C. The points on the perpendicular bisector of a side of a triangle are
equidistant from all of the vertices of the triangle.
D. The points on the perpendicular bisector of a side of a triangle are
equidistant from the vertices of the side it bisects.
116. Part A. In the figure below,
Based on facts about corresponding
angles and vertical angles, write a paragraph proof to show that the
measures of angles 1 and 8 are equal.
Part B. If it had not already been determined that
in this figure,
would the information that
be enough to verify
that p is in fact parallel to q? Justify your answer using
relevant theorems about lines and angles.
Use words, numbers, and/or pictures to show your work.
117. John writes the proof below to show that the sum of the angles in a
triangle is equal to 180º.
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Which of these reasons would John NOT need to use in his proof?
A. The sum of the angles on one side of a straight line is 180º.
B. If a statement about a is true and
replacing a with b is also true.
the statement formed by
C. When two parallel lines are cut by a transversal, the resulting
alternate interior angles are congruent.
D. When two parallel lines are cut by a transversal, the resulting
alternate exterior angles are congruent.
118. In triangle
and
the triangle sum theorem?
Which of these can be proved using
A.
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B.
C.
D.
119. In isosceles triangle
Roshni’s teacher asked different
students to draw different segments in triangle
Roshni drew a
median from vertex R to point T on side
as shown below.
The teacher asked the students to trade papers with a partner and
identify which type of segment their partner had
drawn. Roshni’s partner, Jose, looked at her drawing and told her he
thought that she had drawn an altitude.
Part A. Is
an altitude, median, neither, or both? Explain using a
paragraph proof.
Part B. Is
a perpendicular bisector of
Explain using the definition
of perpendicular bisector and your answer to part A.
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Part C. Does
below.
bisect
Explain using a two-column proof as shown
Part D. If Roshni draws a median from
to S, is the median also an
altitude? Why or why not? How does this compare to the result in part
A?
Use words, numbers, and/or pictures to show your work.
120. In
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Given the information above, which statement can be proved to be
true?
A. Triangle
B.
C.
is isosceles.
is perpendicular to
The length of
D. Triangle
is half the length of
is congruent to triangle
121. A proof of the base angle theorem is shown.
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Which statements correctly complete the proof?
A.
B.
C.
Math III Review Page 86/135
D.
122. The statements of a two-column proof are listed below.
What should the corresponding reasons be?
A. 1. Given; 2. Definition of congruency; 3. Definition of congruency;
4. SAS theorem; 5. CPCTC
B. 1. Given; 2. Definition of congruency; 3. Reflexive property; 4.
Hypotenuse-leg theorem; 5. CPCTC
C. 1. Given; 2. Definition of perpendicular lines; 3. Definition of
congruency; 4. SAS theorem; 5. CPCTC
D. 1. Given; 2. Definition of perpendicular lines; 3. Reflexive property;
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4. Hypotenuse-leg theorem; 5. CPCTC
123.
Properties of Parallelograms
In this task, you will be proving and comparing properties of certain
types of parallelograms.
Figure
below is a parallelogram. Extending the sides and drawing
other lines in the parallelogram can be helpful when proving the
properties of parallelograms.
Part A. Mark all the angles in the figure that are congruent to
Choose three pairs of these angles and explain how you know they are
congruent.
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Part B.
are both perpendicular to
Based on
properties of parallelograms, how do you know
How do you
know
Part C. Use the facts from parts A and B in a paragraph proof to prove
that the opposite sides of parallelogram
are congruent.
Part D. In the figure below, parallelogram
has diagonals that
intersect at point E. Use the properties of triangles and parallel lines in
the two-column proof template below to show that the diagonals of the
parallelogram bisect each other.
Part E. Figure
is a parallelogram with diagonals
intersecting at point Z. If
what can be determined about
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parallelogram
Use a paragraph proof to explain your answer. Add
labels and/or marks to the figure below to support your proof.
Part F. Use the two-column proof template and rhombus
shown
below to prove that the diagonals of a rhombus bisect its angles.
Part G. Compare the properties of parallelograms. Fill in the chart to
show whether a property is always, sometimes, or never true for each
type of parallelogram listed.
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124. The figure below shows the parallelogram ABCD with segments AC and
BD as diagonals.
Which of these correctly proves that the diagonals of a parallelogram
bisect each other?
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A.
B.
C.
Math III Review Page 92/135
D.
125. Consider the quadrilateral with vertices
Part A.
One way to prove that the quadrilateral is a parallelogram is to show
that the diagonals bisect each other. Explain two other different
methods that can be used to prove that the quadrilateral is a
parallelogram.
Part B.
Explain how Part A illustrates the claim that the diagonals of a
parallelogram bisect each other.
126. Quadrilateral MATH includes the points M(2,-4) and A(5,-2).
Part A: Find coordinates for T and H such that quadrilateral MATH is a
rectangle.
Part B: Prove that the resulting quadrilateral is a rectangle.
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127. An architect is designing a new city park. She draws a scale model that
includes plans for different park features.
Part A. The architect includes a triangular playground in her design of
the park. In her scale model, the playground is in the shape of an
isosceles triangle with a 112° angle and two legs that measure 1.5
inches. Use a protractor and a ruler to create the scale model of the
playground. Explain the steps you used to complete the construction.
Part B. A sprinkler will be placed in one area of the park. The region to
be watered by the sprinkler is shown below.
Use a compass and a straightedge to bisect the angle of the region the
sprinkler will water. Explain the steps you used to complete the
construction.
Part C. The architect’s plans for the park include a brick wall with a
flower garden in front of it, as shown by the diagram below.
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The flower garden will be divided into two equal halves with different
types of flowers planted in each half. What geometric construction can
be used to divide the garden into two equal halves? Perform the
construction on the diagram above and explain the steps you used to
complete the construction.
Part D. A walking trail is planned to form a straight path between the
parking lot and the athletic fields at the opposite end of the park, as
shown below.
The architect wants to design a bicycle route that goes through point X
and is parallel to the walking trail. Construct and label the bicycle route
on the diagram above. Explain the steps you used to complete the
construction.
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Part E. The architect wants to make another path for joggers. The new
path will intersect the walking trail and bicycle route and run
perpendicular to both. Use the diagram in part D to construct the
jogging path. Explain the steps you used to complete the construction.
128. Line m and point P are shown in the diagram.
Which method would not be used to construct a line through point P that is parallel to line m?
A. Draw a transversal through point P intersecting line m. Construct a pair of congruent
corresponding angles.
B. Draw a transversal through point P intersecting line m. Construct a pair of congruent
alternate interior angles.
C. Construct a line l through point P that is perpendicular to line m. Construct a line, k,
through point P that is parallel to line l.
D. Construct a line l through point P that is perpendicular to line m. Construct a line, k,
through point P that is perpendicular to line l.
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129.
The figure below shows
Part A. Construct
the same length as
using a compass and a
straightedge. Explain how you know the line segments are the same
length.
Part B. Use a compass and straightedge to construct the perpendicular
bisector of
Explain how you know the line segment you constructed
is a perpendicular bisector.
Use words, numbers, and/or pictures to show your work.
130.
Amanda is constructing a line parallel to the line segment
point P. So far, she has taken the steps shown below.
through
The first arc is the same distance from G as the second is from P.
Part A. What further steps does she need to take to complete the
construction?
Part B. Sketch an example of what the completed construction would
look like.
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Use words, numbers, and/or pictures to show your work.
131. April wants to construct quadrilateral
congruent to the
quadrilateral shown below using only a compass and a straightedge.
Part A. Her first step is to copy the base of the quadrilateral. Draw and
explain the steps April should use to construct
Part B. Next, April needs to copy
congruent to
Draw and explain the steps she
should use to construct an angle congruent to
vertex at point
along
with the
Part C. What are the next two steps that April needs to perform in order
to successfully continue her construction? Draw and explain the steps
she needs to take.
Part D. Complete April’s construction of quadrilateral
Draw and
explain the steps she needs to take to finish her copied figure.
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Use words, numbers, and/or pictures to show your work.
Correct Constructions
132.
Rajon is using a compass and a straightedge, along with his knowledge
of the characteristics of lines, angles, and triangles, to draw figures
according to certain specifications.
Part A. First, he has to draw the perpendicular bisector of segment
He starts by opening the compass so that the opening is greater than
half the length of the segment. He puts the point of the compass on one
end of the segment and draws an arc, as shown. What are the next
steps to drawing the bisector?
Part B. Use the tools to complete the drawing. Label the points where
the arcs intersect and
and label the midpoint of
Part C. Construct triangles
and
point
and use them to explain how you
know that the segment you have made is a bisector of
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Part D. Now, Rajon will draw a line parallel to line that passes through
point shown below. Explain how he can use a compass and a
straightedge to achieve this.
Part E. Draw the required line and explain how you know it is parallel to
the original line. Label points as necessary.
Part F. Finally, Rajon will bisect the angle below. Explain how he can use
a compass and a straightedge to bisect the angle.
Part G. Explain how you know that the two angles formed by the
bisector are congruent.
133. The equation shown below represents a circle. Which statement
describes the key features of the circle that can be determined from the
equation?
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134.
A. The circle has a center at
and radius of 2 units.
B. The circle has a center at
and radius of 2 units.
C. The circle has a center at
and radius of 4 units.
D. The circle has a center at
and radius of 4 units.
What is the equation of the circle that has a center at
radius of 4 units?
and a
A.
B.
C.
D.
135. What is the center and the radius of a circle given by the equation
A.
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B.
C.
D.
136. In the standard xy-coordinate plane, the horizontal line
the circle
at two points,
intersects
and
Part A: Complete the square to determine the standard form of the
equation of the circle:
Part B: Find a value for such that the distance from
Show your work or explain your reasoning.
to
is 10.
137. A circle with a radius of 3 units has its center at the origin. Which
equation gives any point
on the circle?
A.
B.
C.
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D.
138. A circle with its diameter is shown on the coordinate grid below.
Which equation represents the circle given above?
A.
B.
C.
D.
139. Given that a certain parabola has a focus at
answer the following questions.
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and a directrix at
Part A. What are the coordinates of the vertex of the parabola? Show
your work.
Part B. Does the parabola open up, down, left, or right? Explain your
answer using the locations of the focus and vertex.
Part C. Derive the equation that represents this parabola. Show your
work and explain your steps.
Part D. Sketch a graph of this parabola on the coordinate plane below.
Label the vertex of the parabola as well as the focus and directrix.
Use words, numbers, and/or pictures to show your work.
140. What is the vertex of a parabola with focus
A.
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and directrix
B.
C.
D.
141. What is the equation of a parabola with a focus of
and a directrix
that goes through a point that is ten units directly above the focus?
A.
B.
C.
D.
142. What is the equation of a parabola with a focus of
directrix of
and a
143. Katie is using graph paper to sketch a design for a flat wooden bench
that will have a rectangular seat. The front and side views that Katie
sketched are shown below.
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If Katie uses a scale on the graph paper in which each box represents
a 6-inch by 6-inch square, what are the dimensions of the seat of the
bench Katie is designing?
A. 2 feet long by 2.5 feet deep
B. 5 feet long by 2 feet deep
C. 5.5 feet long by 2.5 feet deep
D. 10 feet long by 5 feet deep
144.
Pet Fence
Dana is planning to build an enclosure in her yard so that her dogs can
play in a secure area. She is planning to use fencing that comes in rigid
6-foot-long sections. She cannot bend the individual sections, but she
can join them at any angle to form different polygons. Dana has enough
money to buy 24 sections of fencing, including one with a gate. Dana
plans to use all 24 sections of fencing when building the enclosure for
her dogs.
Part A. Dana first considers making a rectangular enclosure. In the table
below, list all possible ways Dana could use the fencing to make an
enclosure that has an area of at least 900 square feet. What is the
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greatest rectangular area Dana could enclose with the 24 sections of
fencing? Explain your answer.
Part B. Dana decides to sketch models of the rectangular enclosures.
She uses tick-marks to show each section of fencing on the models, and
she labels what will be the actual length and width of the enclosures. If
represents two pieces of fencing placed next to each other, use a
ruler or graph paper to sketch models of all of the possible enclosures
that have an area of at least 1,000 square feet. Label the models with
what will be the actual lengths and widths of the enclosures. How does
the area of each enclosure, in square feet, relate to the area of each
enclosure in fence section by fence section? Use the models you drew to
help explain your answer.
Part C. Dana is also considering making the enclosure in the shape of a
regular hexagon. Use a ruler or graph paper to sketch a model of a
regular hexagon with tick-marks to show how many fence sections
would be needed for each side. Include the length of each side, in feet.
Then, divide the hexagon into sections so that you can compute its area
in square feet. Show how you chose to divide the hexagon and show
your work for computing the area. When appropriate, leave side lengths
in radical form. For your final answer, round the area to the nearest
square foot.
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Part D. Dana’s sister suggested she make the enclosure in the shape of
a regular octagon. Use a ruler or graph paper to sketch a model of a
regular octagon with tick-marks to show how many fence sections
would be needed for each side. Include the length of each side, in feet.
Then, divide the octagon into sections so that you can compute its area
in square feet, and sketch your divisions on your model. Show your
work and label the lengths you used in your calculations. When
appropriate, leave side lengths in radical form. For your final answer,
round the area to the nearest square foot.
Part E. If Dana uses all 24 pieces of fencing as the sides of the
enclosure, how could Dana construct the enclosure in order to maximize
the area? Describe the configuration and explain your answer.
145. Build a Corner Cupboard
You are taking an interest in carpentry and want to design a corner
cupboard for books and knickknacks. A corner cupboard, or cabinet, is
shaped like a triangular prism and fits into the corner of a room. You
use geometric methods to work out the specifications for the cabinet.
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Part A. Given the space you have available, you are thinking that the
cabinet should come out 18 inches (in.) from the corner along each
wall. How wide will it be across the front? Show your work, and give
your answer to the nearest inch.
Part B. The back corner of the cabinet measures
measure of the two other angles of the cabinet?
What is the
How deep will the cabinet be, from the back corner to the center front
of the cabinet? Show your work, and give your answer to the nearest
inch.
Part C. You would like the cabinet to be 6 feet tall. At the bottom, there
will be 1 shelf enclosed by doors that are 2 feet high. There will be an
open shelf at the level of the top of the doors, 3 other evenly spaced
shelves, and then the top of the cabinet. You will have to buy extra
wood to allow for waste when you cut it, but what is the minimum
amount of wood you need for the sides, shelves, and top excluding the
doors? Show your work, and give your answer to the nearest square
foot.
Part D. What is the area of each shelf in square inches? What is the
total volume of the cabinet in cubic feet? Give your answer to the
nearest cubic foot. Show your work.
Part E. With triangular shelves, not all of the area is always usable
space. You have a set of books you want to put on one of the shelves,
with bookends at either end. Each book is 9 in. high, 6 in. wide, and
in. thick. If you put the books in a row across the shelf, with the spine
of each book at the edge of the shelf, what is the maximum number of
books you can put in the row? Draw a sketch, show your work, and
explain your answer.
Part F. You are thinking about making a small 3-in. rail for the top shelf,
as shown in darker gray below.
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Show how you could cut a piece of wood like the one below to make the
rail with the least waste possible. Label lengths and angles. What is the
minimum length of wood that you would need?
146. Sandy is designing cartons to hold cans of beans. The cartons are in the
shape of a rectangular prism, and the cans are cylinders measuring
6.858 centimeters (cm) in diameter and 10.16 cm tall. The carton must
be able to hold 30 cans on one layer and have a length no greater than
50 cm.
Due to restrictions on the machine that creates the cartons, Sandy first
completes her calculations and then rounds her results up to the next
0.25 cm.
Part A
Determine the dimensions for the base of the carton that would best
accommodate 30 cans with the least amount of space left over. Show
and explain your work.
Part B
Using the dimensions found for part A, determine the amount of area
NOT covered by the cans on the base level. Show your work. Round
your answer to the nearest hundredth.
Part C
The material for the cans weighs 0.0055 ounces per square centimeter.
How many ounces does each can weigh? Round your answer to the
nearest hundredth.
147.
is obtained by dilating
by a factor of 2, then translating it 5
units to the left. If
which statement must be correct?
A. Since translation preserves side lengths, the corresponding side
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B. Since translation preserves side lengths, the corresponding side
C. Since dilation results in proportional side lengths, the corresponding
side
D. Since dilation results in proportional side lengths, the corresponding
side
148. Triangles
and
shown below, are similar.
Which series of transformations best takes
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to
A. reflection over the y-axis, dilation by a factor of 0.75, and rotation
about point F by 90° counter-clockwise
B. reflection over the y-axis, dilation by a factor of 0.5, and rotation
about point F by 45° counter-clockwise
C. translation 16 units to the right, dilation by a factor of 0.75, and
rotation about point F by 90° counter-clockwise
D. translation 16 units to the right, dilation by a factor of 0.5, and
rotation about point F by 45° counter-clockwise
149. Given:
Which of the following statements cannot be proven true or false for all
cases?
A.
B.
C.
D.
150.
In the diagram below,
and the dimensions are in meters.
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What is the length of
A. 34.4 meters
B. 40 meters
C. 43.2 meters
D. 45 meters
151. Use the given image to answer part A and part B.
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Part A. What transformation(s) does
Be as specific as possible.
undergo to produce
Part B. What is true about the corresponding angles of these triangles?
Is this true for all triangles of this type? Explain.
Use words, numbers, and/or pictures to show your work.
152. In the figure below,
is the result of a dilation of
as the center of dilation.
with point B
Which statement best uses this information to explain why the AA
criterion can be used to prove
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A. A dilation takes a line not passing through the center of dilation to a
parallel line, so
parallel lines,
Because they are corresponding angles on
and
B. A dilation takes a line not passing through the center of dilation to a
parallel line, so
parallel lines,
Because they are corresponding angles on
and
C. By the reflexive property,
A dilation results in line
segments that are proportional in the scale factor of the dilation.
Therefore,
D. By the reflexive property,
A dilation results in line
segments that are proportional in the scale factor of the dilation.
Therefore,
153. Ms. Morales showed her geometry students the figure below and told
them that
divides
She asked them to use this information to prove that
and
proportionally.
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Hannah used the midpoint formula to show that
Jaden used the slope formula to prove that
and
and
have equal slopes.
Omar used the distance formula to show that
Which student or students used a formula that will help them prove the
relationship?
A. Jaden only
B. Omar only
C. Hannah and Omar only
D. Hannah, Jaden, and Omar
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154.
In the given
What is the value of x?
A. 12 in.
B. 16 in.
C. 20 in.
D. 30 in.
155.
Which expression is equivalent to the expression
A.
B.
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C.
D.
156.
Which number is equivalent to the complex number
A.
B.
C.
D.
157.
Use the expression
Part A. If
and
above expression?
Part B. If
and
complex form.
to answer the questions below.
what will be the real and imaginary part of the
write the above expression in simplest
Use words, numbers, and/or pictures to show your work.
158. Which of these correctly defines the complex number i?
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A.
B.
C.
D.
159.
Which expression is equivalent to
A.
B.
C.
D.
160.
Which expression is equivalent to
A.
B.
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C.
D.
161. Use
to answer the following questions.

What is

Define
Give answer in
in
form such that
form.
is a real number.
Use words, numbers, and/or pictures to show your work.
162. What is the value of
A.
B.
C.
D.
163.
What are the solutions of the equation
A.
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B.
C.
D.
164.
Which equation has
as two of its solutions?
A.
B.
C.
D.
165.
Which of these are the solutions of the equation
A.
B.
C.
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D.
166.
Which shows a solution of
A.
B.
C.
D.
167.
The equation
can be used to determine the kinetic energy of
an object, where m is the mass, in kilograms, and v is the velocity, in
meters per second. What are the units of kinetic energy?
A.
B.
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C.
D.
168. A recipe for cookies needs 6 tablespoons of butter per batch.
Logan is making 6 batches of cookies for a bake sale. How
much butter will Logan need? (Note: 4 tablespoons =
cup)
A. 2 cups plus 4 tablespoons
B. 2 cups plus 8 tablespoons
C. 9 tablespoons
D. 36 cups
169. John leaves his house for the local community center. He walks a
distance of 3 miles from his house in 45 minutes before stopping at a
store to pick up a bottle of water. From there, he walks to the
community center, which is 5 miles away from the store, in 1 hour.
What is John’s approximate average speed, in miles per minute, for
the entire time he is walking?
A. 0.067
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B. 0.076
C. 0.083
D. 0.150
170. Paul is a carpenter. He designs doors that are different sizes and
shapes. His prices are dependent on the size of the door he designs.
Part A. Paul wants to advertise his prices on a flyer. If he measures the
dimensions of the door in feet to calculate the surface area, what units
should he use to represent the cost?
Part B. A customer calls and asks Paul what his prices would be if he
measured the door in yards instead of feet. The equation
describes the cost, c, Paul charges per unit of area, a, when measuring
the dimensions of the door in feet. Write an equation to show the cost if
Paul wee to measure the dimensions of the door in yards.
Use words, numbers, and/or pictures to show your work.
171. Which expression results in an irrational number when simplified?
A.
B.
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C.
D.
172. Is the expression shown below rational?
A. Yes, because the sum of two irrational numbers is rational
B. No, because the sum of two irrational numbers is irrational
C. Yes, because the sum of a rational number and an irrational number
is rational
D. No, because the sum of a rational number and an irrational number
is irrational
173. The host of a television news program wants to predict the voters'
preferred candidate in the upcoming election. Which of the following
sampling processes would be the least subject to bias?
A. The host sets up a booth at the local shopping mall and asks
shoppers to participate in a survey.
B. The host asks viewers to call in, text, or visit the show's website to
participate in the survey.
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C. The host requires each of the show's employees to have four of their
neighbors participate in the survey.
D. The host acquires a list of all citizens who voted in the last election
and selects every 100th voter on the list to participate in the
survey.
174. As senior class president, Grace wants to help the cafeteria provide
more lunch items that the students like to eat. She designs a survey
that assesses the favorite foods of students. After school one day, on
her way to catch her bus, Grace administers the survey to 50 students
waiting at the bus loop where buses meet to take students home.
Part A
Will the results of this survey represent a random sample, a
convenience sample, or a self-selected sample? Explain why.
Part B
Which sampling method would be the best method in this situation?
Explain why.
175. Which characteristic in a statistical study is necessary in order for
conclusions to be drawn regarding the whole population, based on the
sample population?
A. The sample must be randomly selected.
B. The sample must not be randomly selected.
C. The population size must meet a minimum number requirement,
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based on the sample size.
D. The population size must meet a maximum number requirement,
based on the sample size.
176. A teacher assigned students to determine the number of red cars in the
student parking lot.

Student 1 looked at all the cars in the student parking lot and
recorded the number of red cars.

Student 2 looked at the first 10 cars entering the student parking
lot, counted the number of them that were red, and calculated the
percent. The student multiplied this percent times the number of
parking spaces in the parking lot to estimate the total number of red
cars.
Compare the methods used by the two students. Include the words
random, sample, and population in your comparison.
177. An excerpt from a voting report is shown below. Use the information in
the table to answer the questions below.
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Part A. The report lists a number and a percentage for each of the
categories. Why should we take into consideration the
voting percentages instead of simply looking at the number of people
who vote within each category?
Part B. Suppose a politician claims that more young citizens ages 18 to
24 vote than older citizens ages 75 and older. Do the data support this
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claim? Why or why not?
Part C. Overall, what are some trends that you see in the type of people
who vote? Who votes most frequently? Who votes less frequently?
Use words, numbers, and/or pictures to show your work.
178. A sports-and-exercise shop sampled its customer base one weekend
and asked 35 customers their age:
24, 24, 24, 24, 24, 24,
25, 25,
26, 26, 26, 26, 26, 26, 26, 26,
27, 27, 27,
28, 28, 28, 28, 28,
29, 29,
30, 30,
31,
33, 33, 33,
36,
45,
46
Which interval estimate of the population mean has a margin of error of
1.7 ?
A.
B.
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C.
D.
179. James wants to find out whether polyurethane swimsuits help swimmers
swim faster. To investigate, he chose seven volunteers from his swim
team to participate in an experiment. On two consecutive Sundays, he
had each volunteer swim 50 meters. On one Sunday, each swimmer
wore a polyurethane swimsuit, and on the other Sunday, each swimmer
wore an ordinary swimsuit. The times he recorded are listed below.
He then ran a simulation using a computer program to figure out what
differences in means could be expected to occur simply due to random
chance. Which statement best explains what James can conclude based
on the results of the simulation?
A. James can conclude that polyurethane swimsuits help swimmers
swim faster if the mean difference of the simulation is close to the
experimental mean difference.
B. James can conclude that polyurethane swimsuits help swimmers
swim faster if the mean difference of the simulation is less than the
experimental mean difference.
C. James can conclude that polyurethane swimsuits help swimmers
swim faster if the mean difference of the simulation is greater than
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the experimental mean difference.
D. James can conclude that polyurethane swimsuits do not help
swimmers swim faster if the mean difference is equal to the
experimental mean difference.
180. A survey was conducted where 150 high school students were asked the
average amount of time they spent doing household chores in one
week. The data collected resulted in a mean time of 180.5 minutes with
a standard deviation of 5.5 minutes. Which of these represents a 95%
confidence interval for the mean weekly hours spent doing household
chores of all high school students?
A. 171.5–189.5
B. 175–186
C. 178.25–182.75
D. 179.62–181.38
181. In a college math class, 500 students took a final exam. The final exam
results showed students had an average score of 65.3% with a standard
deviation of 5.2%. The scores on the final exam followed a normal
distribution curve with population percentages as shown below.
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How many students scored above 54.9% but below 70.5%?
A. 78
B. 82
C. 341
D. 409
182. A factory manufactures light bulbs and then packs them in boxes to be
shipped to its customers. Before each shipment, boxes are randomly
chosen and the bulbs inside are inspected. The number of bulbs found
to be defective in each box can be normally distributed. The mean
number of defective bulbs in each box is 12 with a standard deviation of
2.
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Use the normal curve shown above to answer the questions. Let
represent the mean and represent the standard deviation.
Part A. If any one box among the sample of inspected boxes chosen has
a total of 9 defective bulbs, what percentage of the sample boxes will
have more defective bulbs than this box? Explain what this means in
terms of the given context.
Part B. If there are 60 boxes that contain between 12 and 15 defective
bulbs, how many total boxes were inspected?
Use words, numbers, and/or pictures to show your work.
183. A book editor was proofreading a draft of a novel. She found that the
number of errors on each page of the book was normally distributed,
with the mean number of errors on a page as 8 and a standard
deviation of 1. If 82 pages had between 7 and 9 errors, what was the
approximate total number of pages in the book?
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A. 56 pages
B. 68 pages
C. 120 pages
D. 202 pages
184. The distributions below represent the batting averages of the players on
two baseball teams, A and B.
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Which of these is true based on the graph?
I. The mean batting average for team A is less than the mean batting
average for team B.
II. The standard deviation for team A is less than the standard deviation
for team B.
A. I only
B. II only
C. both I and II
D. neither I nor II
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