# Preview Assignment 12.A

```Quantitative Reasoning
Preview Assignment 12.A
Preview Assignment 12.A
Preparing for the next class
In this lesson, you continue to explore mathematical models, extending to cyclical data
and periodic functions and relationships. You will need to plot points on the Cartesian
plane, look for patterns in the graphs, and describe the behaviors of real-world
scenarios.
1) Otis was sitting in the morning sun beside the dock on the
bay. He was watching the tide roll away. Being from
Georgia, he was unfamiliar with the tides in Galveston,
Texas, so he decided to record some observations.
At 9:00 am, he placed his chair 25 feet away from the
water’s edge (300 inches). Each half-hour thereafter, he
measured the distance from the chair to the water’s edge
except for when he took breaks for lunch and dinner. A
portion of the data is shown in the table at the right; you can
see that Otis took his lunch break after his 12:30 pm
measurement and did not return until 2:00 pm. The entire set of measurements is
If you are unfamiliar with tides, do some research on what causes them.
Create and print a graph using the spreadsheet, or sketch one in your notebook. Be
sure to place the break times on the horizontal axis, even though you do not have
measurements for those times. Draw a smooth, continuous curve through the points.
Part A: Use your graph to estimate the distance in inches from the chair to the
water’s edge at Time 13 (1:00 pm).
Part B: Estimate the distance in inches at Time 13.5 (1:30 pm).
Part C: Estimate the distance in inches at Time 18.
Part D: Estimate the distance in inches at Time 18.5.
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Preview Assignment 12.A
Part E: Write a paragraph about the graph you created, relating what you see in the
graph to the context. Include sentences that describe how this curve is
similar to and/or different from previous curves that you have studied.
Predict what you think would happen if Otis continued to collect data for
another 12 hours. Explain your reasoning.
2) To effectively plan and use your time wisely, it helps to think about what you know
and do not know. For each of the following, rate how confident you are that you can
successfully do that skill. Use the following descriptions to rate yourself:
5—I am extremely confident I can do this task.
4—I am somewhat confident I can do this task.
3—I am not sure how confident I am.
2—I am not very confident I can do this task.
1—I am definitely not confident I can do this task.
Skills needed for Lesson 12, Part A
Skill or Concept: I can . . .
Questions used for
understanding
Create a graph of data.
Rating
from 1 to 5
1
Use the trends in the given data
to approximate missing data
using a smooth, continuous
curve.
1, Parts A–D
Describe a mathematical model
in terms of the context of the
problem.
1, Part E
Now use the ratings to get ready for your next lesson. If your rating is a 3 or below,
you should get help with the material before class. Remember, your instructor is
going to assume that you are confident with the material and will not take class time to
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Student Pages 12.A, How long is the longest day?
Lesson 12, Part A
How Long Is the Longest Day?
A periodic function is a relationship with a repeated pattern. Graphs of periodic
functions can have many shapes, as long as the graph repeats itself. You saw these
cyclical patterns in ocean tides in Preview Assignment 12.A and in the predator-prey
populations in Lesson 11.
1) Locate the graph of the tides that you
created in the Preview Assignment.
Discuss how the behavior of the graph
relates to the tides in Galveston, Texas.
Credit: ThinkStock
Objectives for the lesson
You will understand;
&uml; What a periodic model is and the vocabulary that describes this model.
You will be able to:
&uml; Sketch a graph that depicts a periodic phenomenon.
&uml; Identify the period and amplitude of a periodic function.
&uml; Compare and contrast the graphs of different periodic models.
Applications of periodic behavior can be found in fluid flow, wave motion (e.g., sound
and current), economics, and business (e.g., crop yields and prices).
In a periodic function, the period 𝑃 is the length
of time that it takes for the cycle to repeat itself
(example shown at the right).
The amplitude of a graph is: 𝐴=
!
!
maximum height − minimum height
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Student Pages 12.A, How long is the longest day?
2) Consider the graph of the high and low tide in the Preview Assignment.
Part A: What is the period of the graph?
Part B: What is the amplitude?
3) The number of daylight hours in a particular location depends upon latitude. The
variation is caused by the tilt of the Earth’s axis of rotation with respect to the ecliptic
plane of the Earth around the Sun. The graph below shows a prediction of the
number of daylight hours for 2016 and part of 2017 in New Orleans, Louisiana.1.
Length of Day (minutes) 900 The Length of a New Orleans Day Sunrise to Sunset -&shy;‐ 2016 840 780 720 660 600 0 30 60 90 120 150 180 210 240 270 300 330 360 390 420 450 480 510 540 Day of the Year (Day 1 = January 1, 2016) Part A: How many minutes pass between two consecutive sunrises?
Part B: Estimate the length of the shortest day (sunrise to sunset) in New Orleans in
2016. On approximately what day will it occur?
Part C: Estimate the length of the longest day. On approximately what day will it
occur?
Part D: What is the length of the average day in New Orleans? Draw a horizontal
line through the graph to represent the average length of a day.
Part E: There are at least two ways to determine the amplitude of this periodic
function. Choose one way, and find the amplitude.
Part F: The curve exhibits a repeating pattern. Place a point on the curve at Day 80.
Place a second point on the curve when the length of the day is the same
as it was on Day 80 and the length of the day is changing in the same way
that the length of the day is changing on Day 80. How many days are there
between these two points?
1
The predictions for the time of sunrise and sunset are fairly reliable. See
http://www.sunearthtools.com/solar/sunrise-sunset-calendar.php.
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Student Pages 12.A, How long is the longest day?
Part G: The length of time that it takes the curve to repeat is called the period of the
curve. What is the contextual significance of the period of this curve?
Part H: The number of daylight hours changes more rapidly during parts of the year
and more slowly during other parts of the year. When is the curve changing
rapidly? When is it changing slowly?
4) The graph showing the length of a New Orleans day is repeated below.
980 Length of Day (minutes)
920 860 800 740 680 620 560 500 0 30 60 90 120 150 180 210 240 270 300 330 360 390 420 450 480 510 540 Day of Year (Day 1 = January 1, 2016)
Part A: Do some internet research to determine the lengths of the longest and
shortest days of 2016 in Seattle, Washington, and when they occur. Sketch
a graph of the length of a day in Seattle on the same axes as the New
Orleans graph. Use a different color or a dashed line.
Part B: What is the period of the Seattle graph? What is the amplitude?
Part C: Research information about the lengths of the longest and shortest days of
2016 in Santiago, Chile, and when they occur. Sketch a graph of the length
of each day in Santiago on the same axes as the New Orleans and Seattle
graph. Use a different color or a dashed line.
Part D: Compare and contrast the New Orleans graph with the Santiago graph.
•
http://www.smithsonianmag.com/smart-news/weird-blips-randomly-change-thelength-of-earths-days-for-months-on-end-10661780/
•
•
vimeo.com/24756892
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Student Pages 12.A, How long is the longest day?
Pendulum Behavior
A pendulum is a bob (a weight) suspended from a pivot so that it can swing freely.
When the weight is pulled to one side and released, the bob is subject to a restoring
force due to gravity that will accelerate it back toward the equilibrium position. The
restoring force combined with the mass of the bob causes it to oscillate about the
equilibrium point. To see a video of the behavior of an interesting group of pendulums,
Part A: Construct a pendulum using string or a shoelace, and something you can tie to
one end, such as a washer or small superhero action figure. (A long string will
be easier to work with than a short one.) Tie the other end to something stable
that will allow the bob to swing back and forth easily. If possible, use your
cellphone to take a picture of the pendulum at rest.
Part B: Pull the bob to a certain point and let go. If possible, take a picture of the
pendulum in action.
Notice how long it takes to make a complete cycle and about how far the pendulum
travels from one side of equilibrium to the other. Repeat the experiment several times,
better estimates of the period and amplitude of the pendulum.
Part C: Record the time and distance in the table of values. You need at least five data
points: the point when the bob is releases (𝑡 = 0); the approximate time when
the bob moves through the equilibrium point the first time (when 𝑑 = 0); the
approximate time and distance the bob reaches a maximum distance from
equilibrium on the other side; the approximate time when the bob moves
through equilibrium the second time (when 𝑑 = 0 again); and the approximate
time when the bob returns to its original position. Record the distance from
center to the right of center as positive values and to the left of center as
negative values.
Time in
Seconds
Distance from
Center in
Inches
0
0
0
Part D: Create a graph to show the movement of the pendulum bob.
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Suggested Instructor Notes 12.A, How long is the longest day? (Optional)
Lesson 12, Part A
How Long Is the Longest Day? (Optional)
Overview and student objectives
Overview
Lesson Length: 25 minutes
This lesson extends the exploration of cyclical data and
periodic functions begun in Lesson 11. Because you may
have omitted those optional lessons, the vocabulary is
repeated in this lesson.
Prior Lesson: Lesson 11,
Part E, “Reindeer and
Lichens”
Objectives
Constructive Perseverance
Level: 1
Students will understand:
•
Next Lesson: Lesson 12,
Part B, “What’s My Sine?”
(Optional) (25 minutes)
Theme: Civic Responsibility
What a periodic model is and the vocabulary that
describes this model.
Students will be able to:
Goal: Reasoning
Quantitative Reasoning
Outcomes: M1, M2
Related Foundations
Outcomes: N8, PR3, A6
•
Sketch a graph that depicts a periodic phenomenon.
•
Identify the period and amplitude of a periodic function.
•
Compare and contrast the graphs of different periodic models.
Suggested resources and preparation
Materials and technology
•
Computer, projector, document camera
•
Preview Assignment 12.A
•
Student Pages for Lesson 12, Part A
•
Practice Assignment 12.A
•
Globe or world map
•
Consider whether you want to provide information about the daylight hours in
your city, Seattle, and Santiago, rather than have students do the research
during class.
Prerequisite assumptions
Students should be able to:
•
Plot points on coordinate axes.
•
Analyze the shape of a graph and find coordinate values on a graph.
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Suggested Instructor Notes 12.A, How long is the longest day? (Optional)
Making connections
This lesson:
•
Connects back to previous lessons on modeling and graphical displays.
•
Connects forward to additional periodic models.
Background context
In Preview Assignment 12.A, students were given data (Spreadsheet 12.A) about the
tides in Galveston, Texas. However, the measurement was given as “the distance from
a chair to the water’s edge.” Some of the data points were missing.
Students were asked to create a graph to bring to class; the graph will be used in
questions 1 and 2 of this activity. They were also asked to estimate the missing data
values and to extrapolate for the next 12 hours.
Suggested instructional plan
Frame the lesson
(4 minutes)
Group
Work
•
•
Ensure students understand that the graph from the Preview
Assignment is not a picture of the tides (high and low). Rather, the
maximum of the graph represents low tide since the water is the
farthest from the chair at that time; the minimum of the graph
represents high tide.
•
You may wish to discuss the numbering of “time” on the graph.
•
Because the point would have been in the same place as the original
9am data point.]
•
Transition to the lesson activities by briefly discussing the Objectives
for the lesson.
Lesson activities
(18 minutes)
Group
Work
Question 2
•
If you omitted the predator-prey activities in Lesson 11, you may need
to allow additional time to ensure that students understand the
vocabulary of period and amplitude.
•
Consider displaying the given sample graph to illustrate the
vocabulary before applying the terms to tides and daylight hours.
o Ask, “What are some other phenomena that are periodic?”
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Suggested Instructor Notes 12.A, How long is the longest day? (Optional)
[Sample answers: A swing on a porch; a weight bouncing on a
spring; the flickering of a fluorescent light; the vibration on a guitar
or piano string; electricity bills.]
•
Transition to the lesson context by thinking about reasons behind the
cyclical nature of electricity bills (e.g., temperature cycles, the length
of a day).
•
The number of daylight hours changes each day as we change
seasons, but some students may not know that the length of a day
changes as you move north or south from where you are (or they may
think of it backwards). Mentioning Alaska is often sufficient for putting
them on the path to understanding that the farther north from the
equator, the longer the summer day.
Questions 3 and 4
•
Circulate through the room to facilitate as students complete these
questions.
•
Part A is designed to contrast with Part B to help students understand
what the graph is illustrating. For example, some students may have
a vague idea that the graph is showing that the sun is going up and
down.
•
Ensure that students can pick two points on the graph and compare
them to each other in context.
•
Students should know why the number of minutes of daylight
changes, but some may have misconceptions. Consider using a
globe to show the Earth as it circles the sun tilted, causing the
seasons. The videos referenced at the end of the activity may be
•
Part H asks about the important concept of rate of change. Have
students point to the graph to show how the curve changes—from
increasing relatively rapidly to increasing more slowly.
•
It is not necessary to wait for all students to complete question 4
before debriefing the class on question 3.
•
For question 4, Part A, if your city is located at a latitude that is very
different from New Orleans, consider replacing Seattle with your city.
•
The main point of question 4 is to illustrate that a city north of New
Orleans is going to have longer days in the summer and shorter days
in the winter because of the latitude (but that the days of the equinox
will be the same). Chile, in the southern hemisphere, will be
approximately 183 days out of phase (horizontally shifted) with the
northern hemisphere.
•
Ask students to display their graphs.
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Suggested Instructor Notes 12.A, How long is the longest day? (Optional)
Wrap-up/transition
(3 minutes)
Wrap-up
•
Ask students to reflect on this lesson and to write their reflections on a
3x5 inch card. List one concept that is not clear after the lesson.
•
Have students refer back to the Objectives for the lesson and check
the ones they recognize from the activity. Alternatively, they may
check objectives throughout the lesson.
•
“In the next activity, we will investigate the symbolic form of these
cyclical models.”
Transition
Suggested assessment, assignments, and reflections
•
Give Practice Assignment 12.A.
•
The last page of the Student Activity Page contains a problem about pendulum
behavior. Consider assigning the problem as a group project or as optional
credit.
•
Give the Preview Assignments, if any, for the lesson activities that you plan to
complete in the next class meeting.
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Suggested Instructor Notes 12.A, How long is the longest day? (Optional)
Lesson 12, Part A
How Long Is the Longest Day? – ANSWERS
A periodic function is a relationship with a repeated pattern. Graphs of periodic
functions can have many shapes, as long as the graph repeats itself. You saw these
cyclical patterns in ocean tides in Preview Assignment 12.A and in the predator-prey
populations in Lesson 11.
1) Locate the graph of the tides that you
created in the Preview Assignment.
Discuss how the behavior of the graph
relates to the tides in Galveston, Texas.
Credit: ThinkStock
Objectives for the lesson
You will understand;
&uml; What a periodic model is and the vocabulary that describes this model.
You will be able to:
&uml; Sketch a graph that depicts a periodic phenomenon.
&uml; Identify the period and amplitude of a periodic function.
&uml; Compare and contrast the graphs of different periodic models.
Applications of periodic behavior can be found in fluid flow, wave motion (e.g., sound
and current), economics, and business (e.g., crop yields and prices).
In a periodic function, the period 𝑃 is the length
of time that it takes for the cycle to repeat itself
(example shown at the right).
The amplitude of a graph is: 𝐴=
!
!
maximum height − minimum height
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Suggested Instructor Notes 12.A, How long is the longest day? (Optional)
2) Consider the graph of the high and low tide in the Preview Assignment.
Part A: What is the period of the graph?
Part B: What is the amplitude?
Answer: 120 inches or 10 feet
3) The number of daylight hours in a particular location depends upon latitude. The
variation is caused by the tilt of the Earth’s axis of rotation with respect to the ecliptic
plane of the Earth around the Sun. The graph below shows a prediction of the
number of daylight hours for 2016 and part of 2017 in New Orleans, Louisiana.1.
Length of Day (minutes) 900 The Length of a New Orleans Day Sunrise to Sunset -&shy;‐ 2016 840 780 720 660 600 0 30 60 90 120 150 180 210 240 270 300 330 360 390 420 450 480 510 540 Day of the Year (Day 1 = January 1, 2016) Part A: How many minutes pass between two consecutive sunrises?
!&quot; !&quot;#\$%&amp;'
&times;24 ℎ𝑜𝑢𝑟𝑠 = 1440 minutes. Some days it
!!&quot;#
is a little more, and some days it is a little less.
1
The predictions for the time of sunrise and sunset are fairly reliable. See
http://www.sunearthtools.com/solar/sunrise-sunset-calendar.php.
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Suggested Instructor Notes 12.A, How long is the longest day? (Optional)
Part B: Estimate the length of the shortest day (sunrise to sunset) in New Orleans in
2016. On approximately what day will it occur?
Sample answer: The shortest day of the year occurs around Day 355 (December 21)
and is about 611 minutes (10 hours, 11 minutes) long.
Part C: Estimate the length of the longest day. On approximately what day will it
occur?
Sample answer: The longest day of the year occurs around Day 171 (June 20) and
is 846 minutes (14 hours, 6 minutes) long. The longest day is more than a third
longer than the shortest day.
Part D: What is the length of the average day in New Orleans? Draw a horizontal
line through the graph to represent the average length of a day.
Answer: The average of 611 and 846 is 728.5 minutes.
Part E: There are at least two ways to determine the amplitude of this periodic
function. Choose one way, and find the amplitude.
Method 1: 𝐴 =
!
!
!
maximum height − minimum height = 846 − 611 = 235 =
!
!
117.5
Method 2: 𝐴 = (answer to Part D) – minimum = 728.5 – 611 = 117.5
Method 3: 𝐴 = maximum – (answer to Part D) = 846 – 728.5 = 117.5
!
Part F: The curve exhibits a repeating pattern. Place a point on the curve at Day 80.
Place a second point on the curve when the length of the day is the same
as it was on Day 80 and the length of the day is changing in the same way
that the length of the day is changing on Day 80. How many days are there
between these two points?
between the two points.
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Length of Day (minutes)
Quantitative Reasoning
Suggested Instructor Notes 12.A, How long is the longest day? (Optional)
Part G: The length of time that it takes the curve to repeat is called the period of the
curve. What is the contextual significance of the period of this curve?
Answer: The period of the curve is the length of one year.
Part H: The number of daylight hours changes more rapidly during parts of the year
and more slowly during other parts of the year. When is the curve changing
rapidly? When is it changing slowly?
Answer: The length of the day changes more rapidly in the middle of the curve—
near the average day length. The day length changes more slowly at the maxima or
minima.
4) The graph showing the length of a New Orleans day is repeated below.
980 Length of Day (minutes)
920 860 800 740 680 620 560 500 0 30 60 90 120 150 180 210 240 270 300 330 360 390 420 450 480 510 540 Day of Year (Day 1 = January 1, 2016)
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Suggested Instructor Notes 12.A, How long is the longest day? (Optional)
Part A: Do some internet research to determine the lengths of the longest and
shortest days of 2016 in Seattle, Washington, and when they occur. Sketch
a graph of the length of a day in Seattle on the same axes as the New
Orleans graph. Use a different color or a dashed line.
980 Length of a SeaIle Day Length of a SanJago Day Length of a New Orleans Day 920 860 800 740 680 620 560 500 0 30 60 90 120 150 180 210 240 270 300 330 360 390 420 450 480 510 540 Part B: What is the period of the Seattle graph? What is the amplitude?
Answer: The period is about 365 days and the amplitude is approximately:
!
!
!
𝐴 = maximum height − minimum height = 960 − 505 = 455 = 227.5
!
!
!
minutes
Part C: Research information about the lengths of the longest and shortest days of
2016 in Santiago, Chile, and when they occur. Sketch a graph of the length
of each day in Santiago on the same axes as the New Orleans and Seattle
graph. Use a different color or a dashed line.
Answer: Shown above with Part A.
Part D: Compare and contrast the New Orleans graph with the Santiago graph.
Answer: Santiago has approximately the same amplitude and period as the New
Orleans graph. It must be the case that Santiago is about as far south of the equator
as New Orleans is north of the equator. However, the graphs are mirror images of
one another along the horizontal axis because of the tilt of Earth; when New Orleans
is having long days, Santiago is having short days and vice versa.
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Suggested Instructor Notes 12.A, How long is the longest day? (Optional)
•
http://www.smithsonianmag.com/smart-news/weird-blips-randomly-change-thelength-of-earths-days-for-months-on-end-10661780/
•
•
vimeo.com/24756892
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Quantitative Reasoning
Suggested Instructor Notes 12.A, How long is the longest day? (Optional)
Pendulum Behavior
A pendulum is a bob (a weight) suspended from a pivot so that it can swing freely.
When the weight is pulled to one side and released, the bob is subject to a restoring
force due to gravity that will accelerate it back toward the equilibrium position. The
restoring force combined with the mass of the bob causes it to oscillate about the
equilibrium point. To see a video of the behavior of an interesting group of pendulums,
Part A: Construct a pendulum using string or a shoelace, and something you can tie to
one end, such as a washer or small superhero action figure. (A long string will
be easier to work with than a short one.) Tie the other end to something stable
that will allow the bob to swing back and forth easily. If possible, use your
cellphone to take a picture of the pendulum at rest.
Part B: Pull the bob to a certain point and let go. If possible, take a picture of the
pendulum in action.
Notice how long it takes to make a complete cycle and about how far the pendulum
travels from one side of equilibrium to the other. Repeat the experiment several times,
better estimates of the period and amplitude of the pendulum.
Part C: Record the time and distance in the table of values. You need at least five data
points: the point when the bob is releases (𝑡 = 0); the approximate time when
the bob moves through the equilibrium point the first time (when 𝑑 = 0); the
approximate time and distance the bob reaches a maximum distance from
equilibrium on the other side; the approximate time when the bob moves
through equilibrium the second time (when 𝑑 = 0 again); and the approximate
time when the bob returns to its original position. Record the distance from
center to the right of center as positive values and to the left of center as
negative values.
Time in
Seconds
Distance from
Center in
Inches
0
0
0
Part D: Create a graph to show the movement of the pendulum bob.
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Suggested Instructor Notes 12.A, How long is the longest day? (Optional)
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Practice Assignment 12.A
Practice Assignment 12.A
1) Alternating current (AC current), delivered via electrical wires in your house, has
voltage that alternates from positive to neutral to negative and back to positive.
Neutral is the state when no current is flowing. Specifically, the voltage changes over
and over. The graph that represents the voltage over time is shown below. Use the
graph to estimate each value requested below.
Part A: How many times does the curve repeat in one second?
Part B: What is the period of the curve? Round your answer to the thousandths
place.
Period = ________second
Part C: What is the amplitude of the curve?
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Practice Assignment 12.A
2) Sound waves, like those from a note played on a piano, have a wave form similar to
the phenomena you investigated in the lesson. The graph below depicts a wave
formed by playing the note “middle C” on a piano. To see an image of a piano
keyboard with Middle C marked, visit http://en.wikipedia.org/wiki/C_(musical_note).
Part A: Estimate the number of periods of the wave in one second.
Part B: Use your answer for Part A to estimate the period of the wave. Round to the
thousandths place.
Period = _________second
Part C: What is the amplitude of the note played?
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Practice Assignment 12.A
3) The figure below depicts a wave formed by playing the note one octave below
Middle C on a piano.
Part A: Estimate the number of periods of the wave in one second.
Part B: Use your answer to Part A to estimate the period of the wave. Round to the
thousandths place.
Period = _________second.
Part C: What is the amplitude of the note played?
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Practice Assignment 12.A
4) The graph below simulates the attendance numbers at a small, local amusement
park. Although this graph is not a series of smooth curves that are exact copies, like
the musical notes in questions 2 and 3, it does exhibit cyclical behavior.
Part A: Write a few sentences describing what you know about park attendance
based on the graph. Be specific.
Part B: If we extended the graph three more days, what do you predict that it would
look like?
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Preview Assignment 12.B
Preview Assignment 12.B
Preparing for the next class
In the next lesson, you continue to explore mathematical relationships with a study of
1) Before you use a spreadsheet to compute specific values of some functions, let’s
review the definitions of five functions. Match the definitions below with the
appropriate term.
Terms:
• Absolute value
• Factorial
• Square root
Part A: Given that 𝑦 ! = 𝑎, the value of 𝑦 is called a ___________of 𝑎.
Part B: The non-negative value of a number, without regard to its sign is called the
___________. It may also be thought of as the number’s distance from
zero.
Part C: Given any non-negative integer 𝑛, the product of all positive integers less
than or equal 𝑛 is called the ___________ of 𝑛 and is denoted by 𝑛!
For example: 5! = 5&times;4&times;3&times;2&times;1 = 120. By convention, the value of 0! is 1.
2) The image below shows the location of the function toolbar in a spreadsheet. There
are a variety of functions available in that menu, many of which we have not studied
and will not study in this course.
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Preview Assignment 12.B
Open Spreadsheet_12B_Preview_Assignment and click on the tab labeled
Functions And Arguments to see the five functions that you are going to
investigate. A function in a spreadsheet has to have an input. The input can be a
number or a cell reference. You may change the argument (the input value, set off
by in parentheses) by entering some other number for any number in red.
The point to understand with this task is that each function requires an input. The
output is based on what the function does mathematically to the input value.
Use the first two functions in the spreadsheet to find the values.
Part A: ABS( –9 ) =
Part B: ABS( 5 ) =
Part C: EXP( 0 ) =
Part D: EXP( 1 ) =
Part E: EXP( 2 ) =
3) Now use the remaining functions in the spreadsheet to find the values.
Part A: FACT( 6 ) =
Part B: FACT( 9) =
Part C: LN( 33 ) =
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Preview Assignment 12.B
Part D: LN( 2.72 ) =
Part E: LN( 1 ) =
Part F: SQRT( 121 ) =
Part G: SQRT( 77 ) =
4) Now let’s explore some built-in spreadsheet functions that require more than one
number in the argument. In fact, you probably used some of these functions in
previous lessons or courses. For example, the output of the AVERAGE function is
the average of the numbers in the argument (separated by commas or defined by a
cell reference). The “PMT” function calculates the payment for a loan based on
constant payments and a constant interest rate. The PMT function requires three
specific input values, separated by commas.
Click on the tab in the spreadsheet labeled Practice. Notice that the worksheet is
blank. This sheet will be your practice space to determine the value of each of the
functions below.
OR
Type commands into a cell
Part A: =AVERAGE(4,5,6,7,10)
Part B: =AVERAGE(4,6,10,15,30)
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Preview Assignment 12.B
Part C: =PMT(0.02,60,1000)
Part D: =PMT(0.02,80,1000)
Part E: =PMT(0.03,80,1000)
5) The function that we will examine in the next lesson is the sine function. Click on
the tab labeled Sine Function, which contains a table of values. The input is a
degree measure and the output is a number.
Part A: For inputs between 0 and 90 degrees, is the sine function increasing or
decreasing?
a) Increasing
b) Decreasing
Part B: For inputs between 90 and 180 degrees, is the sine function increasing or
decreasing?
a) Increasing
b) Decreasing
Part C: For inputs between 180 and 270 degrees, is the sine function increasing or
decreasing?
a) Increasing
b) Decreasing
Part D: For inputs between 270 and 360 degrees, is the sine function increasing or
decreasing?
a) Increasing
b) Decreasing
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Preview Assignment 12.B
6) Have you ever been on a Ferris wheel? If you wish, you can read about the top 10
wheels at http://listverse.com/2011/07/12/top-10-ferris-wheels.
The first Ferris wheel was built in 1893 in Chicago. Let’s use the dimensions of the
original wheel to investigate the behavior of height as a function of time. The original
wheel was 264 feet tall. Let’s say that 4 feet of that amount was the loading platform,
and the actual wheel diameter was 260 feet. A simplified sketch is shown below.
One non-stop rotation of the original Ferris wheel took 9 minutes1 but to make your
estimates easier, let’s speed up the wheel and complete the rotation of the wheel in
8 minutes. Fill in the table, and then create a graph showing the distance between
the chair and the ground over two complete rotations (from 𝑡 = 0 to 𝑡 = 16
minutes). Bring your sketch to class.
Time 𝒕
(in seconds)
0
1
2
3
4
5
6
7
8
Distance from the Ground
(in feet)
1
Source: http://en.wikipedia.org/wiki/Ferris_wheel
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Preview Assignment 12.B
7) To effectively plan and use your time wisely, it helps to think about what you know
and do not know. For each of the following, rate how confident you are that you can
successfully do that skill. Use the following descriptions to rate yourself:
5—I am extremely confident I can do this task.
4—I am somewhat confident I can do this task.
3—I am not sure how confident I am.
2—I am not very confident I can do this task.
1—I am definitely not confident I can do this task.
Skills needed for Lesson 12, Part B
Skill or Concept: I can . . .
Questions used for
understanding
Understand a working definition
of absolute values, factorials,
and square roots.
Rating
from 1 to 5
1
Calculate values using
2–5
Determine whether a function
increases or decreases by
5
Sketch a curve to represent
periodic behavior.
6
Now use the ratings to get ready for your next lesson. If your rating is a 3 or below,
you should get help with the material before class. Remember, your instructor is
going to assume that you are confident with the material and will not take class time to
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Student Pages 12.B, What’s my sine?
Lesson 12, Part B
What’s My Sine?
In the previous lesson, we saw that a periodic model is appropriate for a variety of
phenomena. This lesson introduces you to an equation involving the sine function,
whose graph is also periodic. (Sine rhymes with “nine.”)
1) How can a Ferris wheel be related to
the periodic model? Answer in words or
with a labeled graph.
Credit: ThinkStock
Objectives for the lesson
You will understand that:
&uml; The sine function is periodic.
You will be able to:
&uml; Describe the effect that changing one or more parameters has on the graph of a
sine function.
&uml; Change the parameters of the sine curve to match given criteria.
The sine equation can be used to model smooth periodic behavior, such as the tides
and day length. The shape of the curve depends on four parameters. You will
investigate the effect that each parameter has on the shape of the curve.
Open Spreadsheet_12.B_Investigating_the_Sine_Curve. The worksheet contains a
graph similar in shape to the graphs in the last lesson. There are four parameters to
investigate: A, B, C, and D. Look at their location in the equation. Also notice that sin is
used as an abbreviation for sine in the equation. You still pronounce it as a rhyme with
“nine.”
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Student Pages 12.B, What’s my sine?
2) Consider the function 𝑦 = 𝐴(𝑠𝑖𝑛(𝐵𝑥 + 𝐶) + 𝐷:
Part A: What is the input (independent variable)?
Part B: What is the output (dependent variable)?
Part C: What letters represent parameters?
3) Since you already know periodic curves have a period and amplitude, let’s find out
which parameters control those attributes of the sine curve and how the other
parameters change the shape of the curve.
Part A: Use the scroll bar to change the value of 𝐴. Examine the graph as you do so
to determine the effect of the value of 𝐴 on the shape of the graph.
Part B: Use the scroll bar to change the value of 𝐵. Examine the graph as you do
so to determine the effect of the value of 𝐵 on the shape of the graph.
Part C: Use the scroll bar to change the value of 𝐶. Examine the graph as you do so
to determine the effect of the value of 𝐶 on the shape of the graph.
Part D: Use the scroll bar to change the value of 𝐷. Examine the graph as you do
so to determine the effect of the value of 𝐷 on the shape of the graph.
4) Now click on the tab for the worksheet SIN( ). Recall from the Preview Assignment
that the argument is the expression inside the parentheses. The argument for a sine
function can be a value that represents the degree measure of an angle.
Part A: Use the sheet SIN( ) to complete the table. The first row is done for you.
Recall that ∆ represents change and, in this case, is the change in the curve
from 𝑠𝑖𝑛(𝑥) to 𝑠𝑖𝑛(𝑥 + 1).
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Student Pages 12.B, What’s my sine?
𝒙
(degrees)
𝒔𝒊𝒏(𝒙)
𝒔𝒊𝒏(𝒙 + 𝟏)
0
𝑠𝑖𝑛 0 =
0.0000
15
𝑠𝑖𝑛 15 =
0.2588
𝑠𝑖𝑛 0 + 1 =
𝑠𝑖𝑛 1 =
.0175
𝑠𝑖𝑛 15 + 1 =
𝑠𝑖𝑛 16 = ?
30
0.5000
45
∆
(column 3 – column 2)
0.0175
0.7193
60
0.8660
75
0.9848
89
1.0000
Part B: As you noticed in the last activity, periodic curves change value at different
rates. That is, some parts of the curve increase (or decrease) more rapidly
than other parts of the curve. For what values of 𝑥 does the sine curve
change relatively rapidly? When is it changing more slowly?
5) Return to the first tab, SINE_Curve. What would the values of the parameters need
to be in 𝑦 = 𝐴𝑠𝑖𝑛 𝐵𝑥 + 𝐶 + 𝐷 in order to generate the graph of 𝑦 = sin (𝑥)?
A=
B=
C=
D=
6) Change the parameters as indicated in question 5 to see the graph of 𝑦 = 𝑠𝑖𝑛(𝑥).
Be ready to show how the graph you see supports the findings from the table in
question 4.
7) Classify each scenario as most likely to be periodic, exponential, logistic, or linear.
Part A: The amount of electricity used each day by a street lamp that comes on at
dusk and turns off at dawn.
Part B: The amount of electricity used by a light that stays on continuously.
Part C: The water level on a pylon (pole) for a pier in the Gulf of Mexico near New
Orleans.
Part D: The water temperature measured each day throughout the year in the Gulf
of Mexico near New Orleans.
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Student Pages 12.B, What’s my sine?
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Suggested Instructor Notes 12.B, What’s my sine? (Optional)
Lesson 12, Part B
What’s My Sine? (Optional)
Overview and student objectives
Overview
Lesson Length: 25 minutes
In the previous lesson, students investigated data that are
cyclical and graphs that are periodic. This lesson introduces
them to the sine function, a mathematical function that is
often used to model periodic behavior.
Prior Lesson: Lesson 12,
Part A, “How Long Is the
Longest Day?”
Next Lesson: Lesson 12,
Part C, “SIR Disease”
(Optional) (25 minutes)
To explore the behavior of this function and explain the
notation, students should relate the sine function to their
understanding of the way in which algebraic functions use an
input value to obtain an output, hence the notation “sin(?)”.
Constructive Perseverance
Level: 2
This concept can also be related to spreadsheet functions in
which the input value is either a number or a cell reference.
Goals: Reasoning,
Technology
A spreadsheet template is provided that includes scroll bars
to investigate the effects of the four parameters on the graph
of 𝑦 = 𝐴𝑠𝑖𝑛(𝐵𝑥 + 𝐶) + 𝐷.
Quantitative Reasoning
Outcome: M2
Theme: Civic Responsibility
Related Foundations
Outcomes: N8, A6
Objectives
Students will understand that:
•
The sine function is periodic.
Students will be able to:
•
Describe the effect that changing one or more parameters has on the graph of a
sine function.
•
Change the parameters of the sine curve to match given criteria.
Suggested resources and preparation
Materials and technology
•
Computer, projector, document camera
•
Preview Assignment 12.B
•
Student Pages for Lesson 12, Part B
•
Practice Assignment 12.B
•
•
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Suggested Instructor Notes 12.B, What’s my sine? (Optional)
Prerequisite assumptions
Students should be able to:
•
Plot points on a graph.
•
Understand that constants (parameters) in an equation control the relationship
between the dependent variable and independent variable.
Making connections
This lesson:
•
Connects back to previous lessons on modeling and graphical displays.
•
Connects forward additional lessons on periodic modeling.
Background context
In Preview Assignment 12.B, students worked with a spreadsheet to evaluate functions
requiring one or more inputs. In addition, they explored the distance from the ground
during rotation of the original Ferris wheel in 1893.
Suggested instructional plan
Frame the lesson
(5 minutes)
Whole
Class
Discussion
•
Consider discussing the graph and table of the original Ferris wheel
from the Preview Assignment. You may wish to point out that the
height at Time 1 is not halfway between the heights at Time 0 and
Time 2.
•
Project the sketch of the Ferris wheel. Slide your finger along the
bottom of the circle and ask:
o “How much is the distance from the ground changing when the
chair and rider are in this portion of the wheel?” [Answer: Very
little; when the chair is at the bottom, the height is changing very
slowly.]
o “How much is the distance from the ground changing when the
chair and rider are here (pointing at Time 2)?” [Answer: A lot;
along this portion of the wheel, the height is changing rapidly.]
•
“The difference in behavior along the bottom and along the left side
illustrates why the height at Time 1 is not halfway between the
heights of Time 0 and Time 2.”
•
“What is the period of the Ferris wheel graph?” [Answer: 8 minutes]
•
“What is the amplitude?” [Answer: (264 – 4)/2 = 130 feet]
•
Ask students if they have ever graphed a curve with periodic
behavior in a previous course. This very different-looking curve
requires a different-looking equation—the purpose of this lesson.
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Suggested Instructor Notes 12.B, What’s my sine? (Optional)
•
Transition to the lesson activities by briefly discussing the
Objectives for the lesson.
Lesson activities
(18 minutes)
Questions 2–6
Guiding
Questions
•
Display the spreadsheet and show students the model 𝑦 =
𝐴𝑠𝑖𝑛(𝐵𝑥 + 𝐶) + 𝐷. Discuss input, output, parameters, and argument
of the function. Point out that the portion of the model that
determines the basic shape is the function 𝑦 = sin (𝑥), shown at the
end of these Instructor Notes. When the original argument or input,
𝑥, is modified to 𝐵𝑥 + 𝐶 , the shape of the graph is modified. When
the output of sin 𝐵𝑥 + 𝐶 is multiplied by A, and the result is added
to D, the shape of the graph is modified some more.
•
Although unlikely, some students may recognize that this is actually
a composite function: 𝐵𝑥 + 𝐶 is one function, with an output that
becomes the input of the sine function. It is not necessary to
mention this if none of your students do.
•
If possible, ask students to explore question 3 in groups. If you have
only a demonstration computer available, slowly manipulate the
value of A while students analyze the effect on the graph. Repeat
for B, C, and D.
•
For question 4, you may wish to save time by projecting the values
in the table. The main purpose of the question is the analysis of the
values that takes place.
•
For question 5, it may help to write the two forms one above the
other:
𝑦 = 𝐴𝑠𝑖𝑛(𝐵𝑥 + 𝐶) + 𝐷
𝑦 = 𝑠𝑖𝑛(𝑥)
o “Compare the two equations. What must be the value of 𝐴?”
[Answer: 1] Repeat for B, C, and D.
•
Ask questions to probe whether students understand this notation:
o “What is it about the location of 𝐴 in the equation that makes the
curve have a greater amplitude?” [Answer: The value of 𝐴 is
being multiplied by 𝑠𝑖𝑛(𝐵𝑥 + 𝐶); the larger 𝐴 is, the greater the
result in both directions. When 𝐴 &gt; 1, the basic curve is
stretched vertically. For example, notice that the basic curve has
a maximum value of 1 and a minimum value of –1. If 𝐴 = 2, then
the maximum becomes 2 and the minimum becomes –2.]
o “Why does 𝐷 make the curve move vertically?” [Answer: The
value of 𝑦 gets larger or smaller depending upon the value of D
since it is being added to the result of the rest of the equation.]
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Suggested Instructor Notes 12.B, What’s my sine? (Optional)
o “What is it about the location of B that makes the curve change
the period?” [Answer: A significant amount of background is
required to illustrate how B changes the period—this is beyond
the scope of this course. It is an opportunity to say how much
more there is to study about the sine function.]
Questions 7
•
If students are working in groups, it is not necessary to wait for all
students to finish question 7 before beginning the whole class
discussion.
Wrap-up/transition
(2 minutes)
Wrap-up
•
Ask students to describe the characteristics of a sine function and
the parameters A, B, C, D in their own words, on a 3x5 inch card.
•
Have students refer back to the Objectives for the lesson and
check the ones they recognize from the activity. Alternatively, they
may check objectives throughout the lesson.
•
“Now let’s explore some more serious examples of periodic
functions.”
Transition
Suggested assessment, assignments, and reflections
•
Give Practice Assignment 12.B.
•
Give the Preview Assignments, if any, for the lesson activities that you plan to
complete in the next class meeting.
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Suggested Instructor Notes 12.B, What’s my sine? (Optional)
Lesson 12, Part B
In the previous lesson, we saw that a periodic model is appropriate for a variety of
phenomena. This lesson introduces you to an equation involving the sine function,
whose graph is also periodic. (Sine rhymes with “nine.”)
1) How can a Ferris wheel be related to
the periodic model? Answer in words or
with a labeled graph.
Sample answer: When a rider goes
around on the Ferris wheel, her distance
from the ground is a function of time.
Credit: ThinkStock
Objectives for the lesson
You will understand that:
&uml; The sine function is periodic.
You will be able to:
&uml; Describe the effect that changing one or more parameters has on the graph of a
sine function.
&uml; Change the parameters of the sine curve to match given criteria.
The sine equation can be used to model smooth periodic behavior, such as the tides
and day length. The shape of the curve depends on four parameters. You will
investigate the effect that each parameter has on the shape of the curve.
Open Spreadsheet_12.B_Investigating_the_Sine_Curve. The worksheet contains a
graph similar in shape to the graphs in the last lesson. There are four parameters to
investigate: A, B, C, and D. Look at their location in the equation. Also notice that sin is
used as an abbreviation for sine in the equation. You still pronounce it as a rhyme with
“nine.”
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Suggested Instructor Notes 12.B, What’s my sine? (Optional)
2) Consider the function 𝑦 = 𝐴(𝑠𝑖𝑛(𝐵𝑥 + 𝐶) + 𝐷:
Part A: What is the input (independent variable)?
Part B: What is the output (dependent variable)?
Part C: What letters represent parameters?
3) Since you already know periodic curves have a period and amplitude, let’s find out
which parameters control those attributes of the sine curve and how the other
parameters change the shape of the curve.
Part A: Use the scroll bar to change the value of 𝐴. Examine the graph as you do so
to determine the effect of the value of 𝐴 on the shape of the graph.
Answer: The value of A stretches and compresses the graph vertically. It affects the
maximum and minimum values of the output, which determines the amplitude.
Part B: Use the scroll bar to change the value of 𝐵. Examine the graph as you do
so to determine the effect of the value of 𝐵 on the shape of the graph.
Answer: The value of B stretches and compresses the graph horizontally. It
determines the period, the time it takes for one cycle to complete.
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Suggested Instructor Notes 12.B, What’s my sine? (Optional)
Part C: Use the scroll bar to change the value of 𝐶. Examine the graph as you do so
to determine the effect of the value of 𝐶 on the shape of the graph.
Answer: The value of C shifts or slides the curve right and left.
Part D: Use the scroll bar to change the value of 𝐷. Examine the graph as you do
so to determine the effect of the value of 𝐷 on the shape of the graph.
Answer: The value of D shifts or slides the curve up and down.
4) Now click on the tab for the worksheet SIN( ). Recall from the Preview Assignment
that the argument is the expression inside the parentheses. The argument for a sine
function can be a value that represents the degree measure of an angle.
Part A: Use the sheet SIN( ) to complete the table. The first row is done for you.
Recall that ∆ represents change and, in this case, is the change in the curve
from 𝑠𝑖𝑛(𝑥) to 𝑠𝑖𝑛(𝑥 + 1).
𝒙
(degrees)
0
15
𝒔𝒊𝒏(𝒙)
𝒔𝒊𝒏(𝒙 + 𝟏)
𝑠𝑖𝑛 0 =
0.0000
𝑠𝑖𝑛 0 + 1 =
𝑠𝑖𝑛 1 =
.0175
𝑠𝑖𝑛 15 + 1 =
𝑠𝑖𝑛 16 = ?
0.2756
𝑠𝑖𝑛 15 =
0.2588
∆
(column 3 – column 2)
0.0175
0.0168
30
0.5000
0.5150
0.0150
45
0.7071
0.7193
0.0122
60
0.8660
0.8746
0.0086
75
0.9848
0.9877
0.0029
89
0.9998
1.0000
0.0002
Part B: As you noticed in the last activity, periodic curves change value at different
rates. That is, some parts of the curve increase (or decrease) more rapidly
than other parts of the curve. For what values of 𝑥 does the sine curve
change relatively rapidly? When is it changing more slowly?
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Suggested Instructor Notes 12.B, What’s my sine? (Optional)
Answer: Based on the table of values in Part A, the sine curve is changing the
quickest near 0 degrees. As 𝑥 gets larger, the values are still increasing but more
slowly than before.
5) Return to the first tab, SINE_Curve. What would the values of the parameters need
to be in 𝑦 = 𝐴𝑠𝑖𝑛 𝐵𝑥 + 𝐶 + 𝐷 in order to generate the graph of 𝑦 = sin (𝑥)?
A=
B=
C=
D=
Answer: A and B should be set equal to 1. C and D should be set equal to 0.
6) Change the parameters as indicated in question 5 to see the graph of 𝑦 = 𝑠𝑖𝑛(𝑥).
Be ready to show how the graph you see supports the findings from the table in
question 4.
Answer: The curve rises steeply from 𝑥 = 0 but is increasing at a decreasing rate
until it achieves a maximum value at 90 degrees.
7) Classify each scenario as most likely to be periodic, exponential, logistic, or linear.
Part A: The amount of electricity used each day by a street lamp that comes on at
dusk and turns off at dawn.
Part B: The amount of electricity used by a light that stays on continuously.
Part C: The water level on a pylon (pole) for a pier in the Gulf of Mexico near New
Orleans.
Part D: The water temperature measured each day throughout the year in the Gulf
of Mexico near New Orleans.
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Practice Assignment 12.B
Practice Assignment 12.B
1) Open Spreadsheet_12.B_Investigating_the_Sine_Curve and click on the tab for the
worksheet Temperature. The red curve represents the average daily temperature in
a certain region. Day 0 represents June 1. Find the values of A, B, C, and D so that
the blue curve is the same as the red curve.
A=
B=
C=
D=
2) Use Spreadsheet_Investigating_the_Sine_Curve to find the values of the
parameters A, B, C, and D in the formula 𝑦 = 𝐴𝑠𝑖𝑛(𝐵𝑥 + 𝐶) + 𝐷 to reproduce each of
the three curves below.
Part A:
A=
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C=
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D=
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Quantitative Reasoning
Practice Assignment 12.B
Part B:
A=
B=
C=
D=
B=
C=
D=
Part C:
A=
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Practice Assignment 12.B
3) Ovens have a thermostat that controls the cooking temperature. Much like the
thermostat that controls the temperature in your house, the thermostat in the oven
turns on when the temperature is too low and shuts off when the temperature is too
high. The dial temperature may be set at 350&deg;, but the temperature in the oven
fluctuates (cycles) from a little above that to a little below that, and then up again; the
temperature inside is rarely at exactly 350&deg;. The graph shows the actual temperature
inside an oven from 𝑡 = 15 to 𝑡 = 60 minutes.
Part A: What is the period of the curve?
Part B: What is the amplitude of the curve?
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Quantitative Reasoning
Practice Assignment 12.B
4) Four periodic functions are shown below. Each function has a story. The story could
be about bank accounts, water and a bucket, electronics, driving—you name it! Pick
one of the graphs and write the story behind the graph. Include details that explain
the behavior of the graph. Be prepared to share your creativity.
Graph 1:
Graph 2:
Graph 3:
Graph 4:
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