UNIT 6—Oscillation

Teacher Materials

CLICK THE  SYMBOL OF EACH SECTION HEADER TO RETURN HERE.
TEKS Support
Teacher Notes
	Context Overview
	Mathematical Development
	Unit Project—Swing Dance
	Teaching Suggestions—Preparation Reading
	Activity 1—Everyday Patterns
	Homework 1—Follow the Bouncing Ball
	Activity 2—What Goes Around Comes Around
	Homework 2—Going Nowhere Fast
	Activity 3—Spinning Your Wheels
	Homework 3—Going Around In Circles
	Activity 4—Learning A, B, C, and D
	Homework 4—Sine Up!
	Activity 5—Moonlight and Sunlight
	Homework 5—Modeling Ferris Wheel Rides
	Supplemental Activity 1—Directed Investigation
	Handout 1—Wheel Data
	Handout 2
	Supplemental Activity 1—Directed Investigation
	Transparency 1—Radian Measure
Annotated Student Materials
	Preparation Reading—Planning Cycles
	Activity 1—Everyday Patterns
	Homework 1—Follow the Bouncing Ball
	Activity 2—What Goes Around Comes Around
	Homework 2—Going Nowhere Fast
	Activity 3—Spinning Your Wheels
	Homework 3—Going Around in Circles
	Activity 4—Learning A, B, C, and D
	Homework 4—Sine Up!
	Activity 5—Moonlight and Sunlight
	Assessment—Modeling Ferris Wheel Rides
	Unit Project—Swing Dance
	Mathematical Summary
	Key Concepts
Solution to Short Modeling Practice
Solutions to Practice and Review Problems

TEKS Support

• Coordinate graphing.
• Geometry of circles.

• Periodic functions.
• Radian measure of angles.
• Sinusoidal functions.
• Regression of sine functions using technology.

• Pendulum, which may be constructed from a quart bottle and string.
• Acoustic motion detector
• CBL (if needed for your motion detector)
• Graphing calculator
• Several wheels of different sizes
• Meter sticks
• Programs Slinky or Pendulum

Teacher Notes

"Simple Model" — Context Overview

This unit is multicontextual. However, several major contexts are revisited more than once during the unit. Those include the portion of the moon’s visible surface that is illuminated versus time, the length of daylight versus time, and a Ferris wheel rider’s height versus time. Data for the first two contexts appear in the first activity as part of an introduction to periodic behavior. However, it is not until the last activity that students are able to create sinusoidal models describing each of these relationships. After creating these models, students use them to make predictions for fishermen about the amount of moonlight and for SAD sufferers about the length of daylight.

Mathematical Development

The unit begins with an introduction to periodic phenomena. Students generate descriptions of the characteristics of periodic data and learn vocabulary connected with periodic graphs (period, amplitude, axis of oscillation). Then they draw graphs describing a Ferris wheel rider’s height over time and learn how changing the Ferris wheel’s speed, diameter, or height above the ground affects the period, amplitude and axis of oscillation of their graphs. In order to gather more precise information on the shape of these height-versus-time graphs, students use bicycle wheels as scaled-down models for Ferris wheels. They roll their wheels and record the height of a dot on the tire and the distance the wheel has rolled.

Collection of the wheel data prepares students for a discussion of the unit circle, radian measure, and the sine function. This is followed by a student investigation of sinusoidal functions expressed in the form y = A sin(Bx + C) + D. Through their investigations students discover which graphical features are controlled by each of the constants A, B, C, and D. Then students apply what they have learned from their investigations and fit (by hand) sinusoidal functions to data. If fitting sinusoidal functions is a regression option on the calculators that you are using, students use that option to fit sinusoidal functions to data.

	Unit Project—Swing Dance

Students use motion detectors to collect data on a swinging pendulum. Then they fit sinusoidal models to their data. They also investigate what they can do to the pendulum in order to change the phase shift, amplitude, and period of its recorded motion.

If you have access to a computer on which students can store their data, you could spread this project out over the course of the unit. Students could begin collecting data on pendulum swings early in the unit and then determine models based on their data toward the end of the unit.

	Teaching Suggestions—Preparation Reading

Planning Cycles

This reading introduces students to three contexts that will be used in this unit: patterns in temperature over time, patterns in the length of daylight over time, and patterns in the amount of illuminated moon surface over time.

	Activity 1—Everyday Patterns

This activity introduces students to periodic patterns from a variety of contexts. In addition, it introduces the terms periodic, period, and amplitude.

Students should work in small groups on this activity. Because many of the contexts introduced in Activity 1 will be revisited later, advise students to save their work so that they can refer to it later.

Set aside some time for students to share their answers to Items 1 (b, f, and g). In particular, use Item 1 (g) to motivate the need for an equation to model the relationship between the illuminated portion of the moon’s surface and time. In the last activity, students will create such a model. So, set up the need for it here.

The U.S. Navy has two Internet sites that provide interesting resources that are relevant to Items 1 and 3:

Times of sunrise/sunset, moonrise/moonset, twilight, and other astronomical data are available from the United States Navy Observatory’s (USNO’s) Astronomical Applications Department at http://aa.usno.navy.mil/data/

The site http://tycho.usno.navy.mil/srss.html provides virtual reality moon- phase pictures. You have only to specify the century, year, month, day, hour, and time zone.

For example, Item 3 presents data on the length of daylight in Boston, Massachusetts. You can extend this Item using data relevant to your area. Use USNO’s site to get data for the city nearest you.

The data in Item 4 are not as perfectly periodic as the daylight data. Help students to understand that periodic behavior in the real world includes some "noise."

	Homework 1—Follow the Bouncing Ball

In this assignment students sort out which types of situations involve periodic patterns and which do not. The first six items all involve balls (baseballs, basketballs, tennis balls, etc.) The last item gives students an opportunity to add their own examples of periodic behavior to those already introduced.

Note that in this assignment students are asked to classify motion as periodic according to a mathmetician’s definition: f(t) = f(t + np) where n is an integer and p is the period. (students need not see this formal definition.) Under this definition, amplitude and period are constant. Engineers, on the other hand, accept a more relaxed definition of periodic. Basically any motion that is repeated in equal intervals of time would be considered periodic. This definition allows for changes in amplitude.

Give students an opportunity to share their answers to Item 8 before beginning Activity 2.

	Activity 2—What Goes Around Comes Around

In this activity, students begin creating graphs to model a Ferris wheel rider’s height over time. In addition, they make the connection between circular motion measured in revolutions per minute (rpm) and velocity.

Before students begin this activity, review the relationship between a circle’s circumference, C, and its radius, r: C = 2p r. After the review, assign students to small groups to work on the activity.

Item 4 involves the motion of a Ferris wheel ride. The development of the sine function will rely heavily on this physical model. For this item, students make a graph of the relationship between the rider’s height and time. If a group has trouble coming up with the graph, suggest that they create a data table similar to the one shown in Example 1.

Help students notice, without telling them, that 10-second intervals correspond to a quarter-turn on this Ferris wheel. Because of the periodic nature of these data, the heights corresponding to times 40, 50, 60, and 70 are a repeat of the height values in Example 1.

Conclude the activity with a discussion of Item 7. To facilitate this discussion, you may want to ask groups to re-draw one or more of their graphs from Item 6 on large sheets of paper and hang these graphs in the front of the room.

	Homework 2—Going Nowhere Fast

This assignment gives students an opportunity to practice material learned in Activity 2.

One method of getting a graph for Item 4(a) is to have students make a data table. (See Example 2.) A picture can help them determine the values for the distance from the entrance. Assume when time = 0, the rider is closest to the entrance. Because the carousel rotates 5 times each minute, it makes one complete revolution in 1/5 min or every 12 seconds. So, selecting times in 3-second increments correspond to quarter turns around the carousel.

Students can use the Pythagorean Theorem or the distance formula to find the distance between the rider and entrance at time = 3 sec. Using the Pythagorean Theorem and solving d² = 22² + 29² for d, gives a distance of approximately 36.4 ft when time = 3 sec.

	Activity 3—Spinning Your Wheels

The goal of this activity is to determine, through experimentation, the graph of height versus time as a person rides a Ferris wheel. Instead of a Ferris wheel, students roll a bicycle wheel (or some other round object) and collect height-versus-distance data from the wheel.

Begin this activity with the following question. This question gives students an opportunity to suggest details of how the experiment could be conducted before they are told the details. Here’s the question:

Students should work in groups of three or four. You’ll need one wheel or circular object per group. One student can roll the wheel, another one (or two) can measure the distance and height, and the last can record the data on Handout 1. Although this activity suggests that data be collected from rolling a bicycle wheel (or bicycle), you may substitute some other circular object such as a large garbage can lid or a hoola hoop. In fact, for the purposes of class discussion, plan to assign different groups circular objects of various sizes. Be sure to instruct students to keep the object that they roll perpendicular to the floor.

In Item 2 students should use the meter stick to measure the height of the dot and the tape measure on the floor to measure the distance the wheel has traveled. Check that students are collecting at least six measurements per revolution of their wheel over at least three revolutions. Also, make sure that they don’t take so many measurements that they can’t complete this activity in a reasonable amount of time. Finally, remind students to save their data for use in Activity 4.

In Item 3 students should capture the flow of their data with a smooth curve. Remind them that they want to catch the essence of the relationship between height and distance with their graph and not simply play "connect the dots."

Concluding Discussion

At the end of the activity, have groups share what they have discovered about their data. In particular, if groups used wheels of different radii, have students find the connection between the radius of their wheel and the period and amplitude of their graph. They should be able to reason that the period of their graph is 2p r and the amplitude is r, where r is the radius of their wheel.

	Homework 3—Going Around In Circles

This assignment introduces the concept of radians as a measure of directed arc length on a unit circle. It concludes by developing the sine function.

Students can work individually on this assignment or in pairs. You might want to assign it for homework and then allow additional class time to complete the assignment after students have had a chance to ask questions.

In Item 1 students use direct circumference calculations to generate periodic data for a hypothetical tire. The graph they produce is a translated version of y = sin(x). Make sure that they connect the period of their graph with the circumference of their wheel—2p ft. Encourage them to write exact answers as well as decimal approximations to Items 1(a–c) and 1(e).

Discussion of Radians

Item 2 Introduces students to radian measure. You should spend some time discussing this concept. Radian measure is a method of describing the size of an angle without resorting to an arbitrary unit. Degrees, grads, and mills are arbitrary units that were adopted for specific purposes. In contrast to this the size of an angle, expressed in radian measure, has no unit. The following figure shows how the size of an angle is determined.

The size of the angle, Ð ABC, is determined as follows:

Draw a segment of a circle, of any radius, with the center at the vertex of the angle and the segment crossing both rays that define the angle.

Measure the radius, R, and the length of the arc between the rays, S, using the same unit of linear measure for both.

The size of the angle in radian measure is the ratio of S to R.

Notice that this ratio will always be unitless, because S and R are measured in the same units of length.

Example: If R = 20 inches and S = 35inches, then Ð ABC = 1.75

It is correct to say that Ð ABC=100^o = 111 grad = 1.75. The first two expressions of the size of the angle are in terms of defined, but arbitrary, units. The last expression of the size has no units, so we know it is in radian measure.

The definition of the size of an angle in radian measure can be used to produce a relationship between the radius and the arc length of a segment of a circle that is subtended by the angle.

Arc length = (Angle_{in radian measure})(Radius)

In Item 3 students collect vertical-displacement-versus-angle data from a graph of a unit circle. A scatter plot of their data should resemble a portion of the graph that they drew in Item 1(d). Both graphs have the same period and amplitude. They will need to use their graphing calculators to overlay a graph of y = sin(x) on the data from part (a). Make sure that students check that their calculators are in Radian mode. To choose window settings that match the ones for their graphs in part (b), they will have to convert radian measure involving p to their decimal equivalents.

Items 5 and 6 work students work with the definition of radian measure of a central angle: q = s/r, where s is the length of arc swept out by the angle and r is the radius of the circle. Emphasize the linear units associated with s and r, and the fact that the size of the angle has no units. Do not use the word "radian" as a unit. After this assignment check to see that they understand this relationship.

Conclude this assignment by letting students share their answers to Item 5.

	Activity 4—Learning A, B, C, and D

This activity develops sinusoidal functions, y = A sin(Bx + C) + D, and prepares students to apply what they have learned so that they can fit sinusoidal functions to periodic data.

If possible, students should work in the same groups as they did in Activity 3.

The investigation to determine what the control numbers A, B, C, and D actually control may take students longer to complete than you expect. You have three options for Item 1 in this Activity. Options 1 and 2 are more open and require students to plan a careful investigation on their own. Option 3 is more directive.

Option 1: Students use their calculators for the investigation written in Activity 4. Warn students to be careful placing parentheses. For example, if students enter y = 5 sin(2(x – 3) + 7 into a TI-83, they will not get the same graph as that produced by y = 5 sin(2x – 3) + 7. All that’s different between the two equations is a single parenthesis.

Option 2: If you feel that the investigation as presented in Item 1 is too open to be meaningful to your students, replace Item 1 with Supplemental Activity 1, "Directed Investigation."

After students have completed their investigation, check that they have discovered the relationship between the control number B and the period: that is, period = 2p/|B|. It’s best if students discover this relationship without being told directly.

If students get stuck on Item 4, tell them to pay attention to the order of the transformations when the equation is in y = A sin(Bx + C) + D form. First, A and B control the vertical and horizontal stretch (or compression) of the graph, respectively; second, C and D slide the graph horizontally or vertically, respectively. In other words, the stretches (controlled by A and B) act before the slides (controlled by C and D). Take, for example, Item 4(a). To graph y = 3sin(x) + 2, begin with the graph of y = sin(x). First, stretch the amplitude by a factor of 3 and then slide the graph up 2 units, in that order.

Students will need their wheel data from Activity 3 for Item 5. Encourage them to create their model from the wheel context before graphing the data. Then they can make additional adjustments after they overlay the graph of their model on a scatter plot of their wheel data. You may want to discuss this item in class or have one or more groups present their solutions to the class.

	Homework 4—Sine Up!

This assignment provides practice matching sinusoidal functions with graphs.

Make sure that students understand the role of each of the constants in the model

If you find students’ calculator-produced graphs don’t match how you expect the graph to appear, check that they have entered the equations into their calculators using the correct number and placement of parentheses. Also check that they are in Radian mode and have chosen appropriate window settings.

Item 1 asks students to draw by hand the graphs of two equations and compare each to the graph of y = sin(x). These equations have different periods but the same amplitude. It is helpful if students use scaling that involves p so that the exact periods of each of the graphs can be ascertained from students’ hand-drawn graphs.

Items 2 introduces the term phase shift. Be sure to use this term in subsequent discussions. You may want to ask students to define, in their own words, what the phase shift of a sinusoidal function is.

Check that students were able to complete Item 6 before you move on to Activity 5. You may want to let one or more students present their methods for determining their models.

	Activity 5—Moonlight and Sunlight

In this activity, students determine models describing the moon data and length of daylight data first introduced in Activity 1. This gives them a chance to fit sinusoidal models to real data and then use their models to answer questions relevant to the given situation.

Initially, students should fit their models by hand. Later, provided your graphing calculators have sinusoidal regression as one of their regression options, students can use regression to fit sinusoidal functions to their data. Warning: If students want to check the adequacy of a model using a residual plot, they should compute the residuals manually rather than using the residuals automatically computed by the calculator. (Some calculators’ automatically-computed residuals for nonlinear regressions are incorrectly computed.)

In Item 1 students fit a sinusoidal model to the moon data. Have students note the order suggested by this item for fitting the sinusoidal equation. The period and amplitude are adjusted first (the stretches). Then students are left to their own devises to make the appropriate slides. The vertical shift, changing the axis of oscillation is fairly easy. However, students may need some help determining the phase shift. They can begin by noting that the first "peak" in the graph of their unshifted model occurs at t = period/4. From their moon data, they can approximate the t-value that corresponds to the first "peak" in these data. Then they should use these two pieces of information to determine the value of C.

Students will need Handout 2, a 1997 calendar, in order to complete Item 2.

In Item 3 students fit a sinusoidal function by hand to the length of daylight data introduced in Activity 1. Check that students by-pass the data and determine the value of B directly from their knowledge that there are 365 days in a year. In Item 3(b) they use their calculators to fit a sinusoidal model to these data. Omit this item if your graphing calculators do not have this feature. (An alternative would be to enter these data in a spreadsheet such as Excel and then use the spreadsheet’s regression capabilities to fit the sinusoidal model.)

	Homework 5—Modeling Ferris Wheel Rides

In this assignment, students create sinusoidal models to describe a Ferris wheel rider’s height above the ground over time.

Check that students are able to determine an equation modeling the rider’s height above the ground during the Ferris wheel ride. If students are unable to complete this assignment, review strategies for determining the constants A, B, C, and D in the sinusoidal model from the context.

	Supplemental Activity 1—Directed Investigation

This activity is a more directed version of Item 1, Activity 4. Students investigate how the constants in sinusoidal models of the form y = A sin(Bx + C) + D affect the graph.

Handout 1—Wheel Data

Handout 2

1997 Calendar

Supplemental Activity 1—Directed Investigation

1. First, investigate what happens when you multiply the output or input of the sine function by a constant: y = A sin(x) and y = sin(Bx). Parts (a–c) help you organize this investigation.



1. If your calculator has a trig viewing window, graph y = sin(x) in that window. If not, decide on a good viewing window for the sine function. Sketch a graph similar to the one in your calculator screen. What is the period and amplitude for y = sin(x)?






The period is 2p » 6.28 and the amplitude is 1.

2. Investigate y = A sin (x). Select several positive values for A, some greater than 1 and others between 0 and 1. What effect does the value of A have on the graph of sin(x)? Does A affect the period? the amplitude? What happens if you use negative values for A? Illustrate your conclusions by providing several sketches of graphs drawn on the same set of axes. Label each graph with the corresponding value of A.




The value of A does not change the period of the graph; however, it does change the amplitude. The amplitude of y = A sin (x) is A. The effect of multiplying by A where A > 1 is to stretch the wave vertically. The effect of multiplying by A where 0 < A < 1 is to compress the wave vertically. If the sign of A is negative, then the wave reflects over the x-axis. Upward loops of the wave turn into downward loops and vice versa.



3. Investigate y = sin(Bx). Select several positive values for B, some greater than 1 and others between 0 and 1. What effect does B have on the graph of sin(x)? How does it change the period? the amplitude? What happens if you use negative values for B? Again provide sketches to support your findings.




The value of B affects the period of the wave. The amplitude is not changed. If B > 1, the period gets smaller; if B < 1, the period gets longer. Instead of 2p the period becomes 2p /B. Below are some sample graphs.

2. Next, consider what happens when you add or subtract a constant to the input or output of the sine function: y = sin(x + C) and y = sin(x) + D. Illustrate your conclusions from parts (a) and (b) by providing several sketches of graphs drawn on the same set of axes. Label each graph with its formula.



1. Investigate the effect of subtracting C from the input: y = sin (x + C). Select several positive values for C and several negative values. What affect does the value of C have on the graph of sin(x)? How does it change the period? the amplitude?




When you change the value of C, neither the period or amplitude are changed. If C is positive, the graph moves C units to the right; otherwise the graph shifts left.





2. Investigate the effect of adding D to the output: y = sin (x ) + D. Select several positive values for D and several negative values. What affect does the value of D have on the graph of sin(x)? How does it change the period? the amplitude?




Adding D to the sine function shifts the graph up D units if D is positive and down if D is negative.

Transparency 1—Radian Measure

Annotated Student Materials

Preparation Reading—Planning Cycles

How do power companies predict the amount of energy consumers will be using in the winter? In the summer? How does a department store plan to buy the right quantities of sweaters and bathing suits? The answers to these and many other planning questions rely on predicting how temperatures will vary over time. In places where seasonal temperatures change dramatically, companies plan their budgets and predict consumer buying patterns accordingly. Predicting the seasonal pattern of hours of daylight is important for another very different reason. Seasonal Affective Disorder (SAD) affects approximately 10 million Americans. Many SAD sufferers get depressed in the winter when the hours of daylight are shorter. Understanding the cyclical nature of hours of daylight helps physicians treating SAD sufferers plan for helpful treatments.Even the phases of the moon affect our plans. As the moon moves about the earth, its appearance changes on a regular cycle. It changes from new moon when the moon is totally dark to full moon when the moon is totally illuminated (lit by the reflection of the sun). Since the phases of the moon affect the tides, people who fish need this information to plan. (A number of websites for people who fish have links to sites with information on the moon’s phases. The U.S. Naval Observatory hosts one of those sites.) The military also plans night operations based on the amount of light the moon casts.In this unit, you will use mathematics to model the behavior of quantities that change based on cycles such as those described above.

Activity 1—Everyday Patterns

= 1, 2, 3

1. When you look at the moon, the portion that you can see from the earth looks like a circular disk. Sometimes this disk is fully illuminated (lit by the sun’s light). At other times, it is only partly illuminated. The size of the illuminated portion changes from night to night. For example, Figure 1 shows how the moon looked at midnight on January 5th, 10th, 15th, and 20th, 1997.



January 5, January 10, January 15 and January 20

Figure 1. Illumination of the moon on four nights in January

1. Imagine that each night for two months you look at the moon at the same time each night. Describe how the size of its illuminated surface changes over these two months.
   
   

   Sample answer #1: The moon goes through phases: new moon (surface dark), crescent moon, first quarter (half of moon’s surface illuminated), full moon, third quarter (half of moon’s surface illuminated), crescent, new moon. Then the pattern repeats.

   Sample answer #2: The moon goes from fully illuminated to totally darkened then back to fully illuminated. This cycle repeats over and over.

2. Predict how much of the moon’s surface appeared illuminated on Valentine’s Day (February 14th ) 1997. Explain how you got your answer.




Student answers will vary depending on their estimates for the length of the moon’s cycle. At this point in the unit, the logic of student arguments is more important than correct answers.

Sample answer #1: The moon completes its cycle approximately once a month. On February 14th, the moon should look similar to the way that it looked on January 15th. So, roughly half of the moon should be illuminated.

Sample answer #2: From the pictures in Figure 1, it appears that the moon went from no illumination on January 10th to almost total illumination on January 20th. So it took a few more than 10 days, say 13 days, for the moon to complete a half-cycle; in other words, to go from total darkness to full illumination. From January 20 to February 14 is 25 days, just short of a complete cycle. So the Moon on February 14 should be almost (but not quite) a full moon.

3. Suppose that Jason and his father planned to go fishing on a Friday, Saturday, or Sunday night in April 1997. They picked a date when the moon was nearly a full moon (totally illuminated, similar to January 20, 1997). Predict the date of their fishing trip. (See Figure 2 for an April 1997 calendar.) On what did you base your prediction?




April 1997
Sun	Mon	Tue	Wed	Thu	Fri	Sat
		1	2	3	4	5
6	7	8	9	10	11	12
13	14	15	16	17	18	19
20	21	22	23	24	25	26
27	28	29	30

Figure 2. April Calendar, 1997.

Sample answer: Assuming that the moon’s cycle is roughly one month, Jason and his father should have chosen a date somewhere around April 20, which is a Sunday.

One group member, however, felt strongly that a complete cycle of the moon takes less than one month. So, she wanted to select Sunday April 13th.

How sure are you of your answers to parts (b) and (c)? Perhaps the additional data given in the next item will help.

4. Figure 3 shows the illuminated portion of the moon’s surface for the first three months in 1997. (Note: A portion of 0.5 means that half of the moon’s visible surface is illuminated. A portion of 1.0 means that all of the moon’s visible surface is illuminated.)




Date	1/5	1/10	1/15	1/20	1/25	1/30	2/4
Day of year	5	10	15	20	25	30	35
Portion illuminated	0.20	0.02	0.43	0.89	0.98	0.66	0.17





Date	2/9	2/14	2/19	2/24	3/1	3/6	3/11
Day of year	40	45	50	55	60	65	70
Portion illuminated	0.04	0.48	0.91	0.97	0.63	0.12	0.07





Date	3/16	3/21	3/26	3/31
Day of year	75	80	85	90
Portion illuminated	0.52	0.92	0.96	0.57

Figure 3. Illuminated portion of the moon.

Make a scatter plot of the data in Figure 3. For your plot, put "portion-illuminated" on the vertical axis and "day of the year" on the horizontal axis. Then draw a smooth curve (no sharp corners) through the points on your scatter plot.

Answer:



5. What does your graph tell you about how the illuminated part of the moon changes over time?




The portion of the moon’s visible surface that is illuminated increases from 0 (new moon) to 1.0 (full moon) and then decreases back to 0 (new moon). Based on these data, this pattern of increase, followed by decrease, (beginning about January 10th) repeats itself twice and starts to repeat a third time from 1/5/97 – 3/3/97.

6. Extend your graph so that you can use it to predict the portion of the moon’s surface that is illuminated during April, 1997. Use your graph to answer part (c) a second time. Then compare this answer to your answer in part (c). Which prediction do you think is more accurate? Explain.




Sample answer: The moon is full around day 114, April 24th, which is a Thursday. So, Jason and his father should have taken their fishing trip on Friday, April 25th..



The predicted date based on the graph is somewhat later than the one in part (c). Looking at the dates close to when the moon is full, 1/25, 2/24, and 3/26, it appears that the moon repeats its cycle about every 30 days. Based on Figure 3, on January 20 the moon is not fully illuminated (as we assumed in part (c)); instead, it is fully illuminated approximately 5 days later. This accounts for the earlier prediction in part (c). The answer based on the graph is more accurate because it is based on more data and on data that is more precise than the answer in part (c) which is based on only 4 pictures of the moon.

7. How useful would your graph be for predicting Friday, Saturday, or Sunday fishing dates in September? (Remember that Jason and his father choose dates when the moon is nearly full.) If you could find an equation relating the two variable in Figure 3, would that be more useful? Explain.




Sample answer: The graph is not useful because it has not been extended far enough to include September. You would have to get a really long paper to extend the graph that far. If you were drawing it by hand, you would probably lose some accuracy as you extend the graph.

On the other hand, if you could find an equation, then you could enter it into your calculator, draw the portion of graph that corresponds to day numbers September, and get your answer from the graph.

2. Suppose you pedal your bicycle so that the pedals revolve once a second. The pedal is 5 inches from the ground at its lowest. The pedal is 18 inches from the ground at its highest.



1. Make a plot showing the height of the right pedal from the ground every 0.25 second for 4 seconds. (In other words, make a plot of height versus time; which means that height is on the vertical axis and time is on the horizontal axis.)
   Sample answer:






2. Use these points to draw a smooth graph that you think best represents pedal-height versus time.




Students should draw a smooth curve through the points on the scatter plot in (b).

3. Over a year, the length of the day (the number of hours from sunrise to sunset) changes every day. Figure 4 shows the length of day every 30 days from 12/31/97 to 3/26/99 for Boston Massachusetts.




Date	12/31	1/30	3/1	3/31	4/30	5/30	6/29	7/29
Day Number	0	30	60	90	120	150	180	210
Length (hours)	9.1	9.9	11.2	12.7	14.0	15.0	15.3	14.6





Date	8/28	9/27	10/27	11/26	12/26	1/25	2/24	3/26
Day Number	240	270	300	330	360	390	420	450
Length (hours)	13.3	11.9	10.6	9.5	9.1	9.7	11.0	12.4

Figure 4. Data on length of day.
1. Draw a set of axes similar to Figure 5. Then plot the data from Figure 4.



Figure 5. Axes for "length of day" versus "day number."

Sample answer:



2. Draw a smooth curve through the points on your graph. Extend your graph to show how you think it should look over a period of two years.


Sample answer:




3. Look at the portion of your graph that represents the first year (days 1 to 365). Predict which of these days had about 12 hours of daylight.




Sample answer (student answers should be close to these): days 80 and 270.

4. Look at the portion of your graph that represents the second year (days 366–730). Predict which of these days had about 12 hours of daylight.




Sample answer (student answers should be close to these): days 445 and 635.

5. How are your answers in part (d) related to your answers in part (c)?




The two answers in part (d) should differ from the corresponding answers in part (c) by 365 days, 1 year.

6. Describe in words how your graph would look if you extended the day numbers to 1,460 days (4 years).




The graph would rise and then fall a total of four times. The highest point on the graph is approximately 15 hours and the lowest is around 9 hours.

4. The Average monthly temperature (° F) in Boston Massachusetts from January 1995 to January 1997 is presented in the scatter plot below. (January 1995 = month number 1.)



Figure 6. Average monthly temperatures (°F) for Boston.

1. What month tends to have the highest average temperature? For that month, about how hot does it get?




Months 7, 19, and 31 appear to be the hottest. These month numbers correspond to July 1995, July 1997, and July 1998, respectively. The average temperature appears to be about 74° F.

2. What month tends to have the lowest average temperature? About how cold does it get?




Sample answer: For these data February 1995 was colder than January 1995. However, January 1996 and January 1997 are the coldest months in each of these years. So, January tends to have the lowest average temperature. The average temperature for January appears to be about 29° F.

3. Describe the pattern that you see in the graph (Figure 6). What does this pattern tell you about Boston’s average temperature?




The graph has an up-and-then-down pattern that repeats every 12 months. The average temperatures are low in January, get progressively warmer until July, and then drop until around January. This pattern repeats itself every year.

5. Compare the graphs from Items 1–4. Describe similarities and differences in these graphs. What characteristics did you use in determining similarities and differences?




Sample answer: All of the graphs share an up-and-then down pattern. The time it took to complete one up-down cycle differed from graph to graph. The height difference between the lowest point on the graph and the highest point on the graph also differed from graph to graph.

6. In Item 5, you probably noted that all of the graphs in Items 1–4 repeated a pattern. Any function that repeats itself on intervals of a fixed length is called periodic. Figure 7 shows graphs of three periodic functions, A, B, and C. Each graph repeats some basic shape over and over.



Figure 7. Graphs of three periodic functions.

1. The shortest horizontal (sideways) length of a basic repeating shape is called the period. What are the periods for functions A, B, and C (approximately)?




A: period is » 2; B: period is » 6.3; C: period is » 3.

2. The amplitude is half of the fixed vertical (up-and-down) length of the basic repeating shape. (It’s also half of the difference between the highest and lowest point on the graph.) What is the amplitude for each of these functions?




A: amplitude is 0.5(5 – 3) = 1; B: amplitude is 0.5(2 + .4) = 1.2; C: amplitude is 0.5(–3 + 4) = 0.5.

7. Return to Items 1–4.



1. Estimate the period for each of the graphs in Items 1–4.




Item 1: A little less than 30 days.

Item 2: One second.

Item 3: 365 days.

Item 4: Every 12 months.

2. Approximately what is the amplitude for each of the graphs in Items 1–4. How sure are you of these answers?




Item 1: The amplitude is 0.5. Since the new moon has 0% of its surface illuminated and the full moon has 100% of its surface illuminated, the graph will oscillate between 0 and 1. So, the amplitude will be exactly 0.5. Very sure of my answer.

Item 2: 0.5(20 – 5) = 7.5 feet. Very sure of my answer.

Item 3: 0.5(15.3 – 9.1) = 3.1 hours. Not as sure of my answer because it is not known if 9.1 hours is the shortest day and 15.3 is the longest day.

Item 4: 0.5(74° – 29°) = 22.5°. Not as sure of my answer because the highest and lowest point were estimated from the graph. Also, because the pattern does not repeat itself perfectly.

3. Which of the repeating patterns in Items 1–4 do you think will continue forever (or at least for a very long time)? Which patterns will change over time? Which patterns will stop at some point? Explain.




Sample answer: The patterns in Items 1 and 3 will continue indefinitely or at least for a very long time. Changes in the Earth’s rotational speed may change these patterns very gradually over time. But that change won’t be noticeable in our lifetimes.

The pattern in Item 2 will continue until you tire of riding your bicycle and stop riding. Then the pattern will stop.

Because of global warming, the pattern in Item 4 will change. The pattern will drift upward causing hotter summers and warmer winters.

8. Jumper horses on carousels move up and down as the carousel spins. Suppose that the back hooves of such a horse are six inches above the floor at their lowest point and two-and-one-half feet above the floor at their highest point.



1. Draw a graph that could represent the height of the back hooves of this carousel horse during a half-minute portion of a carousel ride.




Sample answer:



2. Is your graph periodic? If so, what is its period and amplitude?




Yes, this graph is periodic. The same pattern repeats over and over as the horse goes rhythmically up and down during the ride. Based on the graph, the period is 10 seconds. (In other words, it takes the horse approximately 5 seconds to move from its lowest point to its highest point and then another 5 seconds to return to its lowest point.)

Homework 1—Follow the Bouncing Ball

Sometimes the motion of a ball produces a periodic pattern. Its motion repeats itself in equal intervals of time. Other times, the motion of the ball produces an oscillating pattern that varies between alternate extremes. Its back-and-forth or up-and-down oscillating pattern has some, but not necessarily all, of the characteristics of a periodic pattern. For example, perhaps the amplitude of the basic repeating shape diminishes over time. In Items 1–6, you are asked to decide whether or not a ball’s motion is approximately periodic.

1. Imagine that you are at the World Series. To begin the games, a celebrity tosses the ball into the field. Suppose that data on the ball’s height over time is collected from a video of this event.



1. Would a plot of height versus time be periodic? If so, how would you determine the period and amplitude? If not, why not.

No, the ball goes up and then comes down. This pattern does not repeat so the graph is not periodic.

2. Would the plot of the height versus time produce an oscillating pattern? Explain.

No, the ball goes up and then comes down once. It does not alternate between extremes of up and then down.

2. Suppose that you drop a basketball and allow it to bounce on the gym floor. Imagine that you used a motion detector to graph the height of the ball versus time. Is your graph periodic? If so, how would you determine its period and amplitude? If not, why not.

Although the ball repeats its up and down motion as it bounces, the heights of the bounces decrease over time. This means that the time between bounces also decreases. Because the up and down pattern changes as the ball bounces, the graph of height versus time is not periodic.

3. Instead, suppose you bounce a basketball. As you bounce the ball, you try to push on the ball with just enough force so that it bounces back to the same height bounce after bounce. If you plot the height of the ball versus time, would your graph be periodic? If so, how would you determine its period and amplitude? If not, why not?

Sample answer: The ball repeats its up and down motion as it bounces. The period is the time between two consecutive bounces on the floor. Perhaps you could ask other students to use stop watches to time how long each bounce takes. The amplitude is half of the height of each bounce. Because of the way the ball is being bounced, this height stays the same for each bounce.

4. At a tennis match, a video is used to gather information on the height of the ball during a long volley.



1. If you plot the ball’s height versus time, will your graph be periodic? If so, how would you determine its period and amplitude? If not, why not?

Sample answer: No, the graph would not be periodic. Although the height would go up and down, the time between highs and lows would not be the same. For lobs, heights would be higher than for long drives. So, provided the players mix their shots, the graph would not be periodic.

2. Suppose that the height-versus-time graph for the volley was periodic. What would this graph tell you about how the players were hitting the ball? Do you think it’s possible for the players to hit the ball this way?

Sample answer: The players do not mix their plays. Both players must hit each shot exactly the way they hit their previous shot. They would have to hit the ball at the same angle, on the same spot on their racket, with the same force each time. It would be impossible (or nearly impossible) for two players to duplicate their shots exactly play after play.

5. Suppose a paddle-ball champion hits a ball with a paddle 500 times without missing. Data on the ball’s distance from the paddle over time are collected from a video of the action. Would a distance-versus-time graph based on these data be periodic? If so, how would you determine its period and amplitude? If not, is there some way the champion could hit the ball so that the distance-versus-time graph would be periodic?

Sample answer: In order for the graph to be periodic, the champion would need to hit the ball at the same angle and with the same force each time. Using the graph, you could determine the period by measuring the time between two consecutive hits of the ball. The amplitude would be half of the farthest distance that the ball travels from the paddle.

6. Imagine that your brother is sitting on a rope swing. The swing consists of a rope attached to a large ball. (See Figure 8.) He asks you to push.



Figure 8: Rope swing.

1. If, as he swings, you plot his distance from you over time, will your graph be periodic? If so, how would you determine its amplitude and period? If not, will your graph be oscillatory?

Sample answer: The graph will not be periodic. If the swing is not pushed repeatedly, each of your brothers swings (away from you and then toward you) will be smaller. From swing to swing, his farthest distance from you will decrease and his closest distance to you will increase. So, the amplitude, 0.5(maximum distance-minimum distance), will change from swing to swing. However, his distance from you on each swing will oscillate back and forth from a (variable) closest point to a (variable) farthest point.

2. Is there some way you could push so that the distance-versus-time graph would be periodic? Explain.

If you push the swing with the same force each time, then the distance- versus-time graph should be approximately periodic. The period is the time between pushes and the amplitude is half the difference between his farthest distance from you and his closest distance from you.

Items 7 and 8 deal with contexts that do not necessarily involve playing ball.

Housing Starts

7. The Bureau of the United States Census publishes data on the number of housing starts each month. These data are of interest to many people including city planners, real-estate agents, building contractors, and construction workers. Housing-start data (in thousands) for one year beginning March 1983 are presented in Figure 9. (You can find more recent data on the Internet.)




Month	Mar 83	Apr	May	June	July	Aug
Starts	124.3	122.1	161.5	160.1	148	159.8
Month	Sept	Oct	Nov	Dec	Jan 84	Feb
Starts	139.6	147.8	122.1	103.2	102.7	120.2

Figure 9. Housing starts 3/83–2/84.

1. Graph these data. Then assume that the pattern is periodic and extend your graph to show housing starts for a two-year period.


Sample answer:



Base your answers to the items that follow on the graph that you have drawn for part (a).

2. What months have the fewest housing starts? Explain why you might expect these months to have fewer housing starts than other months.

December and January have the fewest. In many areas in the U.S., you could not begin building in December or January because the ground is frozen. So, you would expect fewer housing starts during winter months.

3. What months have the largest number of housing starts? Explain why you might expect these months to have more housing starts than other months.

May, June, and August have the highest number of housing starts. The largest number of housing starts occurred in late spring (May) and continued into the summer months of June and August. Particularly in parts of the country that have cold winters, it is important to begin houses in the spring and summer so that the house (or at least the outside of the house) will be completed before winter. Buyers with school-age children may want to start houses in the spring so that they can move into their houses before the next school year.

4. Why do you think the number of housing starts in July is less than for June or August? Do you think this might happen every year? Explain.

Sample answer: Most likely because the July 4th holiday cuts out several possible work days in which to start new homes. This is a trend that should be expected every year, since July 4th is a yearly holiday.

5. Is reasonable to assume that housing starts are periodic (or approximately periodic)? Why or why not? What information would you need in order to check this assumption?

Sample answer: If housing starts are seasonal, then it seems reasonable to expect the same patterns to occur year after year. You’d expect more housing starts in the spring and summer and fewer in the fall and winter. This trend may be due, in part, to weather conditions.

However, it also seems reasonable that housing starts are tied to the economy. When the economy is good, you would expect an increase in the number of housing starts for all months. When the economy is depressed, then housing starts will probably decline for all months. So, it seems reasonable that the seasonal up-and-down pattern in housing starts would change and would not be purely periodic.

In order to check the whether housing starts are approximately periodic, you would need more data.

Your Choice

8. Describe a situation which would produce periodic data. How would you determine the period and amplitude of such data? What would a scatter plot of these data look like? (Be ready to share your answer with your class.)

Sample answer: Consider the motion of a yo-yo moving up and down along a string. If you plot the height from the ground of the center of the yo-yo over time, you will get a repeating up and down pattern. The period would be the time between consecutive downward throws of the yo-yo and the amplitude would be approximately half the length of the yo-yo string.

Activity 2—What Goes Around Comes Around

= 4, 5, 6, 7

Nauta-Bussink, a company based in Holland, sells amusement park rides. Among its most popular rides are the giant 33-meter, 44-meter, and 55-meter Ferris wheels. (These sizes refer to the height of the top of the wheel.)

The wheels turn slowly at the start of the rides, then rotate at a fairly constant speed until they slow for the stop. Details about each of these rides appear in Table 1.

Imagine that one amusement park has all three Ferris wheels and that your friend rides each one.

1. Suppose that your friend decides to ride the 33-meter Ferris wheel first.



1. Once the wheel is up to speed, how many seconds does it take for it to make one complete revolution?

If the wheel makes 2.6 turns in one minute, it takes 1/2.6 » 0.3846 min to make one complete turn. Convert this time to seconds to get approximately 23 seconds.

Note: The diameter of the 33-meter Ferris wheel is 29.5 meters (See Table 1.)

2. How far does a your friend travel each time the wheel makes a complete turn around the circle?

Each time the wheel makes a complete revolution, the friend travels (29.5m.)(p ) » 92.7 m.

3. How fast does your friend move, in meters per second, around the wheel?

The friend travels at approximately 92.7 m./23 sec » 4.0 meters per second.

4. Suppose that you start a stop watch when your friend is at the bottom of the wheel, how many seconds will it take for him to travel 510 meters? About how high will he be above the ground?

It will take about 127.5 seconds for the friend to travel 510 meters. During this time he will turn approximately 5.5 times around the wheel. He’ll be at the top of the wheel or 33 meters above the ground.

2. Next, your friend boards the 44-meter Ferris wheel. Does he move faster or slower on the 44-meter Ferris wheel than he did on the 33-meter Ferris wheel? In terms of meters per second, how much faster or slower?

Riders on the 44-meter Ferris wheel would travel at approximately 3.2 meters per second, about 0.8 meters per second slower than the riders on the 33-meter Ferris wheel.

3. Finally, your friend rides the 55-meter Ferris wheel. Without doing any calculations, does he travel faster or slower than he did on the 44-meter Ferris wheel? How do you know?

The riders on the 55-meter Ferris wheel travel faster than those on the 44-meter Ferris wheel. The two wheels make the same number of revolutions per minute. However, the riders on the 55-meter Ferris wheel travel a greater distance each time they circle the wheel. Therefore, they must be going faster.

4. Imagine that your friend is the last rider to board the 55-meter Ferris wheel. Assume that the ride starts and that the wheel reaches its constant speed of 1.5 revolutions per minute very quickly. (This modeling assumption allows you to ignore the brief non-constant speed at the ride’s start.)



1. About how long does it take for your friend to travel once around the wheel?

It takes 1/1.5 or 2/3 min or 40 seconds.

2. Plot your friend’s height above the ground every 10 seconds for the first two minutes of the ride. Then draw a smooth curve (no corners) connecting the points on your graph.

Answer:



equation of graph: y = 26sin((2p/40)(x – 10)) + 29

3. At approximately what time during the first half-minute of the ride is your friend 55 meters above the ground?

20 sec

4. Use the periodic nature of your graph to determine the approximate times when your friend is 55 feet above the ground during the entire ride. How are these times related to each other?

   20, 60, and 100 sec. Two consecutive times differ by 40 sec, one period.

5. Your graph in part (b) oscillates up and down, meaning its pattern goes up and down repeatedly. Your graph’s oscillation can be divided into halves:



• the top half—the part of the graph that represents the rider’s height versus time as he moves around the top half of the wheel and



• the bottom half—the part of the graph that represents the rider’s height versus time as he moves around the bottom half of the wheel.

Draw a horizontal line that divides your graph into these two parts. This line is called the axis of oscillation. What equation describes the axis of oscillation for the 55-meter Ferris wheel?

Height = 29 m. or y = 29

5. What would be the period and amplitude of the height-versus-time graphs for the 33-meter and 44-meter Ferris wheels? What would be their axes of oscillation?

33-meter Ferris wheel:

This wheel makes 2.6 revolutions in one minute. So the period is (1/2.6)(60) » 23 sec. The amplitude is 14.75 m. (half of the diameter). The axis of oscillation is height = (0.5)(29.5) + (33.0 – 29.5) = 18.25 m. (or y = 18.25). (This is same as the wheel’s height above the ground plus the wheel’s radius.)

44-meter Ferris wheel:

This wheel makes 1.5 revolutions per minute. Thus, it takes 2/3 min. or 40 sec. to complete a single revolution. So, the period is 40 sec. and the amplitude is 20.35 m. (half of the diameter). The axis of oscillation is height = 0.5(40.7) + (44.0 – 40.7) = 23.65 m. (or y = 23.65).

6. Return to the 55-meter Ferris wheel in Item 4. How would your graph in Item 4(b) change under each of the following conditions? Overlay a graph of height versus time for the altered situation on a copy of your answer graph in Item 4(b).



1. The Ferris wheel is slower and turns 1.3 revolutions per minute.

Sample answer: Student graphs should be similar to the one below.



equation of graphs: y = 26sin((p/20)x – p/2) + 29

y = 26sin((p/40)x – p/2) + 29

2. The Ferris wheel turns twice as fast at 5.2 revolutions per minute.

Sample answer:



equation of graphs: y = 26sin((p/20)x – p/2) + 29

y = 26sin((p/10)x – p/2) + 29

3. The Ferris wheel is turning at the original speed, but is 10 meters off the ground instead of three.

Sample answer:



equation of graphs: y = 26sin((p/20)x – p/2) + 29

y = 26sin((p/20)x – p/2) + 36

4. The Ferris wheel is turning at the original speed, is three meters off the ground, but has a diameter of 57 meters instead of 52.

Sample answer:



equation of graphs: y = 26sin((p/20)x – p/2) + 29

y = 26sin((p/20)x – p/2) + 27

7. The graph of height versus time for a Ferris-wheel rider is periodic.



1. What feature of the Ferris wheel controls the period of this graph?

The speed at which the wheel turns.

2. What feature of the Ferris wheel controls the amplitude of this graph?

The diameter of the wheel.

3. What feature of the Ferris wheel controls the axis of oscillation of this graph?

The height of the wheel’s axle.

Homework 2—Going Nowhere Fast

The only carousel owned by the U.S. government is located in Glen Echo Park, Washington D.C. The carousel is 48 feet in diameter. It turns counter-clockwise at a top speed of five revolutions per minute. Suppose that you have chosen to ride a horse that is in the outer row. Your horse is about two feet from the edge.

1. Each time the carousel makes a complete turn, how far (in feet) have you ridden?

The radius of the circle made as you ride on an outside horse is 22 feet. The circumference of a circle with radius 22 feet is 44p ft » 138.2 ft.

2. How fast, in feet per minute, are you riding? How fast would this be in miles per hour? (Remember there are 5,280 feet in 1 mile.)

About 691 ft/min or approximately 7.9 miles per hour.

3. Imagine that your friend is riding on an inside horse that is around 6 feet from the edge. In terms of miles per hour, how much faster are you moving than your friend?

The friend is traveling around a circle with circumference 36p ft » 113.1 ft. She is traveling 565.5 ft/min or approximately 6.4 miles per hour. So, you are riding 1.5 miles per hour faster than your friend.

4. Suppose the entrance to the ride is five feet from the outer edge of the carousel.



1. During the ride, your distance from the entrance oscillates from 7 ft to 51 ft. Draw a graph that could represent your distance from the entrance during your ride on the carousel. What assumptions did you make when you drew your graph?

Sample answer:



equation of graphs: y = 22sin((p/6)x – p/2) + 29

Assumptions: the carousel is moving at its top speed; my horse is 5 ft from the entrance when time = 0.

2. Draw a graph that could represent your friend’s distance from the entrance.

Sample answer:



equation of graphs: y = 18sin((p/6)x – p/2) + 29

3. Compare your graph from parts (a) and (c). How are these graphs the same and how are they different?

The periods are the same, 12 seconds. However, the amplitudes differ. The amplitudes for the graphs in parts (a) and (d) are 22 m. and 18 m., respectively.

Activity 3—Spinning Your Wheels

= 8, 9, 10

For Item 4, Activity 2 you were asked to graph the height of a rider at various times during a Ferris wheel ride. In order to make a precise plot of the rider’s height versus time, you need more data. However, you can’t fit a full-sized Ferris wheel into your classroom. You’ll need something smaller to model the motion of a Ferris-wheel rider. Instead of an actual Ferris wheel, you’ll collect data from a bicycle wheel (or some other round object).

Part I: Data Collection

1. Set up the experiment as follows:
1. Lay the tape measure on the ground in front of the bicycle wheel. Tape it to the floor with masking tape. (Later, you will roll your wheel along side the tape measure.)
2. Use another piece of tape, placed perpendicular to the tape measure, to clearly mark the zero-reading on the tape measure. Roll the wheel up to and directly on top of this marker. The tire should sit on the masking tape; its axle should be centered directly above the tape.
3. Place a dot on the tire (a sticker or piece of tape will do) at the point where the tire sits on the masking tape.

2. Roll the wheel forward along side the tape measure. Stop at regular intervals. When you stop, measure and record two things: the height of the dot above the floor and the distance the wheel has traveled. (See Figure 10.) Record your data in Figure 1, Handout 1.

Be sure to save your data for use in Activity 4.

Part II: Modeling the Motion

1. Plot your data on a piece of graph paper. Because you want to predict height, height should be the response variable. That means that height should be placed on the vertical axis.

See part (b) for the pattern that these points will follow.

2. On your plot from part (a), sketch a smooth curve (no corners) that you think best represents the relationship between height and distance.

Sample answer:



Save your graph for use in Activity 4.

4. The graph in Item 3 should appear periodic. (If not, recheck your measurements.) What is the (approximate) period and amplitude? Explain how you could have predicted the period and amplitude before you started your experiment.

In general, student data should be periodic with period 2p r (where r is the radius of the wheel or object used) and amplitude of 2r.

Sample answer based on sample data for Item 4:

These data were based on a 26-inch wheel diameter. The period is 26p in. (the circumference of the wheel) » 81.7 in. and the amplitude is 13 in., half of the wheel’s diameter.

1. Complete the pattern in your graph for negative distance-values by extending your graph to the left.

Sample answer:



2. In this situation, what do negative values for distance mean? How would you gather data corresponding to this situation?

To get height values corresponding to wheel movement in the negative direction, reposition the wheel at zero and roll it in the opposite direction.

Use your the extended version of your graph from part (a) to answer the remaining items.

3. For what distances is the dot on the wheel at the top of the wheel? How can you get this answer from your graph?

Sample answer: approximately –40, 40, 120, 205. You get these answers from your graph by determining the x-values that correspond to the peaks in the graph.

4. For what distances is the dot horizontally level with the wheel’s axle? How can you get this answer from your graph?

Sample answer: approximately –62, –11, 20, 62, 102, 143, 183, 225. The dot is horizontal to the axle when it is 13 in. high (half of the wheel’s diameter). Draw the line y = 13 and then find the x-values that correspond the points of intersection between the graph and line.

Homework 3—Going Around in Circles

1. In Activity 3, you and your classmates drew periodic graphs by plotting the heights of dots on tires as you rolled your wheels. Although wheels come in only one shape - circles, they come in many sizes. If different sized wheels were used by different groups, then different height-versus-distance graphs were draw for Item 3(b).



1. How would increasing the radius of the wheel affect your height-versus-distance graph from Item 3(b)? What if the radius of the wheel were decreased?

If the radius of the wheel is increased, then both the period and amplitude of the height-versus-distance graph would increase. If the radius of the wheel were decreased, then both the period and amplitude of the wheel would decrease.

For parts (b) – (e), imagine reworking Activity 3 using a wheel that has a one-foot radius.

2. How far would you have to roll the wheel for the dot on the tire to make one complete turn? Give both an exact answer (involving p ) and a decimal approximation.

2p ft » 6.28 ft

3. How far would you have to roll the wheel for the dot on the tire to move from the bottom of the wheel to the top of the wheel for the first time? and then the second time? Give both exact and approximate answers.

p » 3.14 ft; 3p » 9.42 ft

4. Draw a smooth curve that you think represents the pattern of height-versus-distance data for this wheel. (The shape of your curve should be similar to the one that you drew for Item 3(b), Activity 3.) Complete the graph for distances from 0 to 25 meters.

The shape of the curve should be similar to the one drawn for Item 3(b) but should oscillate between heights of 0 ft and 2 ft.



5. What is the period and amplitude of the graph that you drew in part (d)?

The period is 2p » 6.28 ft and the amplitude is 1 ft.

Calculations related to circles of radius 1 (such as in Item 1) are usually simpler than calculations related to circles with different radii. In an effort to keep things simple, mathematicians often base their study of periodic behavior on a unit circle, a circle with a one-unit radius that is centered at (0,0). We will use a unit circle with a radius of 1 foot for this discussion.

Imagine an ant walking along the edge of a unit circle. Suppose that as the ant walks, it pushes a straight blade of grass anchored at the circle’s center. The ant starts at the point (1,0). The ant can choose to walk either in the positive direction—counterclockwise (as shown in Figure 12) or in the negative direction—clockwise. Suppose after walking for a while the ant stops. The initial position of the blade of grass (on the horizontal axis) and its new position forms an angle q, called a central angle. Note also, that as the ant moves, the blade of grass sweeps through a portion of the circle’s circumference forming an arc. (See shaded portion of circle’s circumference in Figure 12.)

One convenient way to measure this angle is by the ratio of the arc length, with " + or – " to show direction, to the radius of the circle.

q = (± arc length)/radius

The size of the angle that describes to location of our ant is:

q = (+ 1.13 ft)/1 ft = 1.13

The unit circle makes the numerical part of the calculation easy, because we are dividing by 1. However we must always carry out the division to make the unit of measure behave properly. The size of the angle is expressed by a number that has no units. The size of an angle described this way is expressed in radian measure, but it has no units. The size of the angle, q, in the drawing is 1.13. Mathematicians have agreed to call an arc length measured in the counter clockwise direction "+", and one measured in the clockwise direction "–".

1. Suppose the ant (in Figure 12) walks so that the blade of grass it pushes forms an angle of 2p. This means the ant starts its walk at the point (1,0), walks in the counterclockwise direction, and stops after walking an arc of 2p ft. Where is the ant after its walk? How many times has it traveled around the circle?

The ant is back at (1,0). It has walked once around the circle.

2. If the ant walks so that the angle formed by the blade of grass measures –4p, how many times has it walked around the circle? Does the ant walk counterclockwise or clockwise?

It has walked twice around the circle in a clockwise direction.

3. Suppose that the ant’s walk makes an angle of p , how far around the unit circle has it walked? In what direction has it walked? Where is it on the circle?

It has walked half-way around the unit circle in the counterclockwise direction. It is at the point (–1,0).

3. Figure 13 shows two different walks. (Each arc represents one walk.)
   
   
   1. During Walk #1, what fraction of the circle did the ant walk? What is the radian measure for the central angle corresponding to this walk? How did you get your answer?

   

   Figure 13. Two arcs representing two walks

   Answer:

   For Walk #1, the ant traveled about 3/8th of the way around the circle. A trip once around the circle is 2p. So, the ant walked (3/8)( 2p) = 3p /4. This answer is positive because the ant was walking in the counterclockwise direction.

   2. During Walk #2, what fraction of the circle did the ant walk? What is the radian measure for the central angle corresponding to this walk?

   The ant walked 1/2 the circle. The radian measure of the angle that describes the walk is:

   q = –p ft/1 ft = –p

3. Figure 14 shows a unit circle superimposed on a coordinate plane. This will allow you to estimate the ant’s height (positive when the ant walks on the top half of the circle and negative when the ant walks on the bottom half of the circle) as it walks around the circle.



Figure 14. Graph of unit circle.

1. Copy the table below. Use the graph in Figure 14 to complete the entries in your table. (It may help to convert the angle to the fraction of a complete turn around a circle. For example, if the ant walks p/2, it has walked 1/4 of the way around the circle. That’s because (1/4)(2p) = p/2.)




Radian measure of central angle	–p	–3p/4	–p/2	–p/4	0	p/4	p/2	3p/4
Height (vertical displacement from horizontal axis) in ft
Radian measure of central angle	p	5p/4	3p/2	7p/4	2p	9p/4	5p/2	11p/4
Height (vertical displacement from horizontal axis) in ft

Table 2. Height-versus-angle data from a unit circle

Student answers should be similar to those in the table below.





Radian measure of central angle)	–p	–3p/4	–p/2	–p/4	0	p/4	p/2	3p/4
Height (vertical displacement from horizontal axis) in ft	0.0 ft	–0.7 ft	–1.0 ft	–0.7 ft	0 ft	0.7 ft	1.0 ft	0.7 ft
Radian measure of central angle	p	5p/4	3p/2	7p/4	2p	9p/4	5p/2	11p/4
Height (vertical displacement from horizontal axis) in ft	0	–0.7	–1.0	–0.7	0	0.7	1.0	0.7




2. Graph the height-versus-angle data from your table in (a) using the scaling shown in Figure 15.



Figure 15. Axes with scaling used for Item 3(b).




3. Draw a smooth curve through your plotted points. Compare your graph to your answer graph in Item 1(b). How are your graphs alike and how are they different?

The graph in Item 1(b) has y = 1 for its axis of oscillation. The graph in part 3(b) has y = 0 as its axis of oscillation. Both graphs have the same period and amplitude.

As the ant (or rotating dot or Ferris wheel rider) moves from its starting position, the point (1, 0) on the unit circle, a graph of its height versus its central angle (use radian measure) results in the oscillating graph that you drew in Item 3(c). The new function that produces this graph is called the sine function.

5. To graph the sine function on your calculator, set your viewing window to match the one that you used in Item 3(c). Next check that your calculator is in Radian mode, and turn off all scatter plots. Finally, use your calculator’s SIN key to enter and graph the function y = sin(x). Note, the central angle q will be represented by x.) Compare the graph in your calculator’s screen to the one that you sketched for Item 3(c).

Calculator’s screen:



The reason that you are able to determine the central angle on a unit circle is because you know that the radius of the unit circle is one unit, whatever that unit happens to be. So if the ant’s walk corresponds to a central angle of 3, that means that the ant has traveled an arc equivalent to three radii (each of one unit) placed end-to-end. Using radian measure to describe the size of angles, allows you to generalize the concept to larger circles. As you can see from Figure 16, a central angle of 3 on a unit circle, corresponds to an arc length that measures (3´radius). This arc extends nearly halfway around the circle, regardless of the circle’s size. Similarly, a central angle of –3p/4 » –2.4 corresponds to an arc that goes 3/8 of the way around the circle in the opposite direction, regardless of the circle’s size.



Figure 16. Central angles measuring 3 and –3p/4 or
–2.36 on circles of different radii.

6. A wheel of radius 10 feet is centered at the origin. Before turning the wheel, a dot is placed at (10, 0).



1. If the dot rotates once around the wheel, how far has it traveled? What is the radian measurement of the angle corresponding to this turn? (Hint: How many radii fit into the arc made by the circle’s circumference?)
   
   

   The dot has traveled (2p)(10 ft) = 20p ft. The central angle corresponding to this turn is 2p.

2. Suppose that the wheel is turned counterclockwise so that the dot travels 10p ft. What is the radian measurement of the central angle, q, corresponding to this turn?

The distance the dot has traveled, the arc length, is 10p ft q = 10p ft /10 ft = p. The dot has turned half of the way around the circle and is located at (–10,0).

3. After returning the dot to its original position, suppose that the wheel is turned counterclockwise so that the dot travels s ft. What formula describes the radian measurement of the central angle, q, corresponding to this turn?

q = s ft/10 ft = s/10.

4. After returning the dot to its original position, suppose that you turn the wheel so that the dot turns through an angle of -p/4 . How far and in what direction has the dot traveled? Draw a picture showing where the dot is located after this turn.

The dot has traveled in the clockwise direction. The formula

–p/4 = 5/10 ft

s = (–p/4)(10 ft) = (–5p/2) ft.

The dot travels a distance of 5p/2 ft. The dot is located midway on the arc connecting (0,–10) and (10,0).

7. A wheel of radius 20 ft. is centered at the origin. A dot is placed at (20,0).



1. If the dot rotates once around the wheel, how far has it traveled? What is the radian measurement of the angle corresponding to this turn? (Hint: How many radii fit into the arc made by the circle’s circumference?)

The dot has traveled (2p )(20 ft.) = 40p ft. The central angle corresponding to this turn is 2p.

2. Imagine turning the wheel so that the dot rotates through an angle of –p/4. How far and in what direction has the dot traveled?

Use the relationship p/4 = s /20. The dot has traveled 5p ft in the clockwise direction.

3. Draw a picture showing where the dot is located after this turn. Compare your picture to the one that you drew for Item 4(d). What’s the same about the two pictures and what’s different?

The dot is located midway on the arc connecting (0,–20) and (20,0). It makes the same angle with the positive horizontal axis as does the dot from Item 4(d).

4. What if the wheel radius is r ft and the dot is placed at (r,0). Suppose that the wheel is turned so that the dot rotates through an angle of -p /4. How far and in what direction has the dot traveled?

Use the relationship p/4 = s /r. The dot has traveled (p/4)r ft in the clockwise direction.

8. In Activity 3, you rolled a wheel and collected height versus distance data. How could you have rolled the wheel differently or measured differently in order that the graph of your data would match your graph in Item 4?

Sample answer. Place the wheel so that it’s axle is directly over the 0 marker on the tape measure. Then place a dot level horizontally with the wheel but opposite to the direction in which you will roll the wheel. Make all measurements, both height and distance-rolled in radius units. For example, if the wheel has a 26-inch diameter, then 13 inches constitutes a unit.

Activity 4—Learning A, B, C, and D

= 11, 12

Your graph in Item 3b, Activity 3, based on your wheel data, had the same shape as the sine curve. So, did the graph of the moon data (Item 1(d),Activity 1) and your graph of a Ferris-wheel rider’s height over time (Item 4(b),Activity 2). The periods, amplitudes and axes of oscillation, however, were different from the graph of y = sin(x).

Perhaps a sinusoidal function, a function that can be expressed by an equation of the form y = A sin(Bx + C) + D, could be used todescribe each of these situations. First, you’ll need to find out what each of the control numbers A, B, C, and D control before you can fit equations of this form to data.

1. Use your graphing calculator to investigate equations of the form y = A sin(Bx + C) + D.

Begin with the equation y = sin(x) (here, A = 1, B = 1, C = 0, and D = 0). Then change the value of one control number at a time to determine how changing this number affects the graph. Write a summary of the results of your investigation. Include in your summary sketches of graphs that show how changing the values of A, B, C, or D affects the sine graph.

(Warning: Make sure that your calculator is in Radian mode for this investigation.)

Sample answer:

Changing the value of A changes the amplitude of the sine wave. The effect of multiplying by A where A > 1 is to stretch the wave vertically. The effect of multiplying by A where 0 < A < 1 is to compress the wave vertically. If the sign of A is negative, then the wave reflects over the x-axis. Upward loops of the wave turn into downward loops and vice versa. Here are some graphs that support these conclusions.



The value of B affects the period of the wave. If B > 1, the period gets smaller; if 0 < B < 1, the period gets longer. The period and B are related by period = 2p/B or B = 2p/period. Below are some sample graphs.



When you change the value of C, neither the period nor amplitude are changed. If C is positive, the graph moves C units to the right; otherwise the graph shifts left.



Adding D to the sine function shifts the graph up D units if D is positive and down if D is negative.

2. Based on what you have learned from your investigation, answer the following questions. (If you can’t these questions, return to Item 1 and continue your investigation.)



1. Which control number, A, B, C, or D, shifts the graph horizontally (left or right)?

C

2. Which control number shifts the graph vertically (up or down)?

D

3. Which control number affects the graph’s amplitude? How is the graph’s amplitude related to this number?

A; the amplitude = A

4. Which control number affects the graph’s period? How is the graph’s period related to this number?

Period = 2p/B

3. The period of the graph of y = sin(x) is 2p. How would you modify this equation so that the period of the modified equation is:



1. 4p?

y = sin(0.5x)

2. 3p?

y = sin((2/3)x)

3. 2?

y = sin(px)

4. 10?

y = sin(0.2p x)

4. In parts (a) – (c) below, the equation of y = sin(x) has been changed. Describe how each of the changes to the sine function affects the graph. (In other words, explain how the change affects the amplitude, period, vertical or horizontal position of the sine wave.) Then sketch the graph of the new equation and y = sin(x) on the same set of axes.



1. y = 3 sin(x) + 2

The sine wave is stretched vertically to three times its original height and then shifted up two units. The period is still 2p. The amplitude changes from 1 to 3.



2. y = 0.5 sin(x – p/2)

The graph shifts to the right p/2 units and gets compressed to half its original height. The period does not change. The amplitude is 1/2.



3. y = sin(2x + p)

The graph shifts to the left p/2 units and is compressed horizontally by a factor of 2.

1. Use what you know about the geometry of your wheel and sinusoidal graphs to find a sinusoidal model for your wheel data in Activity 3. Explain your reasoning.

Sample answer:

The wheel used in Activity 3 had a 26-inch diameter. That means that A = 13 and D = 13. You get the period from the wheel’s circumference which is 2p(13) = 26p. So, B = 2p/26p = 1/13 » 0.077. So far, the model is y = 13 sin(0.077 x) + 13.

However, for the wheel data, the first data point was collected when the dot was at the bottom of the wheel, and x = 0. The angle is 3p/2 so [B(0) + C] = 3p/2, then C + 3p/2. The problem starts with x = 0 and the angle between the horizontal and a radii through the dot is (3p/2). The phase constant, C, is: 3p/2. Our final model is y = 13 sin(0.077x – 3p/2) + 13.

2. Enter your wheel data from Item 2, Activity 3 into your calculator. Make a scatter plot of height versus distance. In the same viewing window graph your model from part (a). Does your model do a good job of describing your wheel data?

Sample answer:

Below is a scatter plot of the sample data from Item 2, Activity 3. The model from part (a) has been superimposed on the scatter plot. The model appears to fit these data very well.

Homework 4—Sine Up!

1. Without using a graphing calculator, sketch the graphs of y = 3 sin(2x) and y = sin(x) on the same set of axes. Choose a scaling for the x-axis that involves multiples of p. What are the amplitudes and exact periods of these two functions? (After you have completed your sketch, use your calculator to check that your graphs are correct.)



The amplitude of y = 3 sin(2x) is 3 and the period is p. The amplitude of y = sin(x) is 1 and the period is 2p.

2. Without using a graphing calculator, sketch the graphs of y = 3sin(0.5 x) and y = sin(x) on the same set of axes. Then use your calculator to check that your graphs are correct.



3. How are the graphs for y = 3 sin(2x) and y = 3 sin(0.5 x) the same? How are they different?

They have the same amplitude but the period of the first graph is one-fourth the period of the second.

2. In the model

y = A sin(Bx + C) + D,

the value of C controls the horizontal shift of the graph and is called the phase shift. One way to think of C is that C is the angle when we start at x = 0,

y = A sin(Bx + C) + D.

1. Graph y = sin(x) and y = sin(x – 1) in the same viewing window. What effect did the phase shift of 1 have on the graph?



2. Graph y = 3 sin(0.5x) and y = 3sin(0.5x – 1) in the same viewing window. What affect did the phase shift of 2 have on the graph?



3. Graph y = 3 sin(0.5 x) and y = 3 sin(.5x + 1) in the same viewing window. What is the value of the phase shift? What effect did the phase shift have on the graph?

3. Write equations of sinusoidal functions having the amplitude, period, phase shift, and axis of oscillation given below. Then sketch a graph of your function without the aid of a graphing calculator. Scale your x-axis in multiples of p. (After you have sketched your graphs, use your calculator to check that your graphs are correct.)
1. An amplitude of 3/4, period of 4p, phase shift of p/4, and axis of oscillation 0.

y = 0.75 sin(0.5x – p/4)



2. An amplitude of 1, period of 3p, phase shift of –p/3, and axis of oscillation 0.

y = sin(2/3x + p/3)



3. An amplitude of 2, period of 2p, phase shift of p, and axis of oscillation 2.

y = 2 sin(x – p) + 1

4. Write an equation that describes each of the graphs in Figures 17–19. Check your answers with your calculator.



Figure 17. Graph of a sinusoidal function

y = 2 sin(x) + 2.



Figure 18. Graph of a sinusoidal function

y = sin(2x) – 1



Figure 19. Graph of a sinusoidal function.

y = 2.5 sin(0.5x – 0.5)

5. The graph in Figure 20 represents the height above the floor of a carousel-horse’s back hooves versus time during a carousel ride.

1. What are the period and amplitude of the function graphed in Figure 20?

The period is 10 seconds and the amplitude is 1.

2. Find an equation that describes height versus time. After writing the equation for your model, use your calculator to check that its graph matches the one in Figure 18.

A = 1; B = 2p/10 » 0.63; C = 2.5.

Model: y = sin(0.63x – 1.57) + 15.

6. Suppose Anne is the last rider to board the 33-meter Ferris wheel before the ride starts. Recall that this ride has a wheel diameter of 29.5 meters and rotates at a rate of 2.6 revolutions per minute. Write a sinusoidal equation that describes her height above the ground over time during the ride. Explain how you determined your model. What assumptions did you make in your model?

Sample answer: Assumption: The wheel moves at a constant speed turning 2.6 revolutions per minute. (Thus, the fact that the wheel takes several seconds to get up to its constant speed is ignored.) Time starts when she leaves the ground the first time, in seconds.

Amplitude: A = 29.50/2 = 14.75 meters.

Period: The wheel makes 2.6 revolutions per minute, hence it takes approximately 23.08 seconds to make one complete turn. To get a period of 23.08 seconds, B = 2p/23.08 » 0.27.

Axis of oscillation: The wheel is 33.00 meters high with a diameter of 29.50 meters. This means that the bottom of the wheel is 3.50 meters above the ground. Therefore D = 3.5 + (29.50)/2 = 18.25

Phase shift: At t = 0, Anne is at the bottom of the circle. The value of the angle, measured counterclockwise from the horizontal is

3p/2

C = 1.57

The model is:

y = 14.75sin(0.27t + 2p/3) + 18.25.

Activity 5—Moonlight and Sunlight

Now it’s time to apply what you have learned about graphs of sinusoidal equations to situations that you first studied in Activity 1.

1. Figure 21 shows the illuminated portion of the moon’s surface for the first three months in 1997. (This is the same table as Figure 3 in Activity 1.)

Date	1/5	1/10	1/15	1/20	1/25	1/30	2/4
Day of year	5	10	15	20	25	30	35
Portion illuminated	0.20	0.02	0.43	0.89	0.98	0.66	0.17





Date	2/9	2/14	2/19	2/24	3/1	3/6	3/11
Day of year	40	45	50	55	60	65	70
Portion illuminated	0.04	0.48	0.91	0.97	0.63	0.12	0.07





Date	3/16	3/21	3/26	3/31
Day of year	75	80	85	90
Portion illuminated	0.52	0.92	0.96	0.57

Figure 21. Illuminated portion of the moon.

1. Enter these data into your calculator. Then make a scatter plot of the illuminated portion of the moon versus the day number. What are the approximate period and amplitude for these data? How did you determine your answer?

Sample answer:

On day 10, 0.2 of the moon’s surface is illuminated and on day 40, 0.04 of the moon’s surface is illuminated. So, the period is slightly less than 30 days; say, 29 days. The amplitude is 0.5, half way between 0 and 1.

2. Graph y = sin(x) in the same window as your data. Does y = sin(x) do a good job in modeling the pattern of the moon data?

No, y = sin(x) does a very poor job in modeling the pattern of the moon data. Here is a graph that displays what students might see in their calculator’s screen.



3. Modify the equation y = sin(x) so that the graph of the modified equation has a period and amplitude that match those from your answer to part (a). Then on your calculator, graph your modified equation on the same set of axes as the original data. Does your modified equation do a good job in describing the moon data?

The graph of y = 0.5 sin(2p/29x) or approximately y = 0.5 sin(0.22x) does not do a good job in describing these data. The graph needs to be shifted both vertically and horizontally to fit these data.

4. What changes do you need to make to your equation in (c) in order to model the moon data? Make these changes. What is your new equation? Explain how you decided on this equation.

Sample answer:



Students models will vary. Here is an example of what you might expect for an answer.



The moon data oscillate between 0 and 1. The graph of the equation in (c) oscillates between –0.5 and 0.5. So, first add 0.5 to the equation to shift the graph vertically. Next, the graph must be shifted horizontally. The function y = 0.5 sin(0.22 x) + 0.5 attains its maximum value for the first time at approximately 7.35. The data reach 0.98 (very close to the maximum value) for the first time on day 25. That means that the function needs to be changed so that the graph shifts approximately 17.65 days. Then C/B = 17.65/0.22 and C » 80. After we looked at the graph of y = 0.5 sin(0.22x – 80) + 0.5, it appeared that it was shifted slightly too far to the right. Through experimentation we decided that the model y = 0.5 sin(0.22x – 79) + 0.5 did a reasonably good job in describing these data. The data fell very close to the graph of this function.



5. Look at a residual plot for your model in (d). Based on the residual plot, does your model from part (d) do a good job in describing the moon data? Do you need to make further changes to your equation? If so, what is your new equation?

Sample answer: We probably should have adjusted our equation even further because more of the residuals were positive than negative; however, the average of residuals was 0.04 which is fairly close to zero. So, we decided to leave our equation from (d) alone.

You may find the 1997 Calendar on Handout 2 helpful in answering Item 2.

1. Recall the situation in Activity 1 where Jason and his father wanted to go fishing on a Friday, Saturday, or Sunday night in April 1997. They picked a date when the moon was close to being a full moon. Your task is to predict the date for this fishing trip using your model from Item 1. Explain how you got your answer.

Sample answer based on the model y = 0.5 sin(0.22x – 79) + 9:

The day numbers for days in April 1997 are from 91 to 120. Graph the model above using the interval from 91 to 120 for Xmin and Xmax. Use 2nd CALC Maximum to determine the coordinates for the curve’s maximum in this interval. This gives day number 110. Day number 110 corresponds to April 20 which turns out to be a Sunday.

2. In September 1997, Jason and his father planned to rent a cabin for a weekend and do more night fishing. Use the 1997 calendar on Handout 2 to determine the best date for them to rent the cabin. Choose the weekend when the moon is closest to a full moon (totally illuminated).

The day numbers for September 1997 are from 244 to 273. Graph the model y = 0.5 sin(.22x – 3.78) + 0.5 using the settings Xmin = 244 and Xmax=273. The moon reaches its maximum on about day number 253. This day corresponds to September 10th which is a Wednesday. So they should rent the cabin either for September 6–7 or September 13–14. The portion of the moon that is illuminated on September 7th is approximately 0.90; the portion of the moon that is illuminated on September 13 is 0.89. So, the September 6–7 weekend might be slightly better than the September 13–14 weekend, however, the difference probably would not be noticeable.

3. When planning a nighttime bombing mission, the military takes into account the phase of the moon. If possible, missions are planned during new moon, when the moon’s surface is dark. Imagine that as part of a military exercise, you had to plan a nighttime bombing mission sometime during November 1997. What date(s) would you suggest for this mission?

Sample answer: The day numbers for November 1997 are from 305 to 334. Graph the model y = 0.5 sin(0.22x – 3.78) + 0.5 using the settings Xmin = 305 and Xmax=334. The graph reaches its minimum on days 319 or 320. So the mission should take place on November 15 or 16.

3. Over a year, the length of the day (the number of hours from sunrise to sunset) changes every day. Figure 22 shows the length of day every 30 days for Boston, Massachusetts, from 12/31/97 to 3/36. (You first saw these data in Figure 4, Activity 1.)

1. Write an equation of the form y = A sin(Bt + C) +D expressing the number of daylight hours, y, as a function of time, t. Explain how you determined each constant in your model.

Sample answer:

The period should be 365 days (except for leap years): So B = 2p /365 » 0.017.

A » 0.5(15.3 – 9.1) = 3.1.

D = 0.5(15.3 + 9.1) = 12.2.

C = the horizontal shift required to translate y = 3.1 sin(0.017 t) + 12.2 to match these data as closely as possible. The graph of this model reaches a maximum at t = 92.4 whereas the data reaches its maximum at around 180 days. So, try using C = 1.53. After looking at a graph of the function using this phase shift, it appeared that the shift was slightly too large. After some experimentation, it appeared that using C = 1.40 produced a model that better fit the pattern of these data.

Model: y = 3.1 sin(0.17t – 1.40)

2. If your calculator has a regression feature that fits sinusoidal models to data, use that feature to fit a model to these data. Compare this model to the one that you determined for (a).

Using the SinReg on the TI-83 yields approximately:

y = 3.02 sin(0.017t – 1.40) + 12.21.

The constants chosen for A, B, C, and D by the calculator were fairly close to the ones determined for the model in (a).

3. Mary who lives in Boston suffers from SAD (seasonal affective disorder). During the winter she gets very depressed but by the first day of spring, March 21st, she feels wonderful. How many hours of daylight are on March 21st? (Use your model from part (a) to answer this question.)

March 21, 1999 corresponds to day 445. According to the model in (b), there are approximately 12.15 hours of daylight on March 21, 1999.

4. She has been advised to sit in front of a special light to make up for the shorter number of hours of daylight during the winter months. Assume that one hour of sitting in front of the light replaces one hour of natural daylight. How long should she sit in front of the light on December 21st in order to feel as good as she does on March 21st?

December 21, 1998 corresponds to day number 355. According to the model in (b), there are approximately 9.49 hours of daylight. She should sit in front of the light for 2.66 hours.

5. Write a model that Mary could use during the winter of 1998/99 (December 21, 1998 - March 21, 1999) to figure out how long she should sit in front of the light on a particular day. Assume that she wants to feel as good as she does on March 21st. Draw a graph of this model over the interval corresponding to the days when she would use this model.

The model to determine the number of hours of light therapy, T, is:

T = 12.15 – 3.02 sin(0.017T – 0.023) – 12.21. The graph of this function over the interval from T = 355 to T = 445 appears below.






6. Use your model to determine how long Mary should sit in front of the light on January 15, 1999.

January 15, 1999 is day 380. According to the model in (e), she should sit in front of the light for approximately 2.77 hours, or about 2 hours 46 minutes.

Assessment—Modeling Ferris Wheel Rides

Recall the details of the popular 33-meter, 44-meter, and 55-meter Ferris wheels.

1. Suppose Anne is the last rider to board the 33-meter Ferris wheel before the ride starts.



1. Write a sinusoidal function that describes her height above the ground over time during the ride. Explain how you determined your model. What assumptions did you make in your model?

Sample answer:

Assumption: The wheel moves at a constant speed turning 2.6 revolutions per minute. (Thus, the fact that the wheel takes several seconds to get up to its constant speed is ignored.) Time starts when she leaves the ground the first time, in seconds.

Amplitude: A = 29.50/2 = 14.75 meters.

Period: The wheel makes 2.6 revolutions per minute, hence it takes approximately 23.08 seconds to make one complete turn. To get a period of 23.08 seconds, B = 2p/23.08 » 0.27.

Vertical shift: The wheel is 33.00 meters high with a diameter of 29.50 meters. This means that the bottom of the wheel is 3.50 meters above the ground. Therefore D = 3.5 + (29.50)/2 = 18.25

Phase shift or horizontal shift: Assume that when t = 0 Anne is at the bottom of the wheel and that the wheel makes one complete turn every 23.08 seconds. Under these assumptions, Anne reaches the midpoint between the bottom of the wheel and top of the wheel when t » 23.08/4 » 5.77 seconds. So, the phase shift is 5.77 seconds. Then C = (5.77/23.0)(2p) = 1.57.

The model is y = 14.75sin(0.27x – 1.57)) + 18.25.

2. Sketch a graph of your model from (a) using a time interval wide enough to represent the first three revolutions of her ride.



3. Certainly one of the more exciting times on any Ferris wheel are the times when you are high above the ground. During the first three revolutions of the ride, at approximately what times is Anne more than 30 meters above the ground? How did you determine your answer?

Between 9.2 and 14.0 seconds; between 32.5 and 37.3 seconds; and between 55.7 and 60.5 seconds. The answer was determined by graphing y = 30 and using the intersect feature on the TI-83 to determine the points of intersection of the line and sinusoidal function.

4. After the initial start-up of the ride, the Ferris wheel spins at a constant rate of 2.6 revolutions per minute. This causes the rider to travel in a circle at a constant rate of approximately 4 meters per second. However, the rate at which Anne’s height above the ground is changing is not constant as the she circles. Explain how this information is conveyed by your graph in (b).

If the rate of change in height over time were constant, the graph would be a line. However, the graph is a smooth oscillating curve. Therefore, the rate of change of height with respect to time is not constant.

5. Indicate on your graph when Anne’s height is changing most rapidly. What feature of your graph tells you the times when this occurs. Draw a picture of the Ferris wheel and locate the position of Anne’s seat when her height above the ground is changing most rapidly.



The times that correspond to the vertical line indicate the times when Anne’s height above the ground is changing most rapidly: at approximately 6, 17, 29, 40, 52, and 64 seconds into the ride. This is where the absolute value of the "slope" of the graph is greatest.

At each of these times her seat is midway between the bottom and the top of the wheel as shown on the wheel below.

2. Suppose that next Anne boards the 44-meter Ferris wheel. Write a sinusoidal function that describes her height above the ground over time during this ride.

Amplitude is 20.35; period is 40 seconds per revolution and B = 2p/40 » 0.157; vertical shift is 3.30 + 20.35 = 23.65 meters; horizontal shift is 40/4 = 10 seconds, C = (10/40)(2p) = 1.57.

y = 20.35 sin(0.157x – 1.57) + 23.65

3. What if Anne decides to board the 55-meter Ferris wheel? What constants in your model from Item 2 must change to describe her height over time on this Ferris wheel? What constants stay the same?

Amplitude is 26.00; period is 40 seconds and so B is the same as in Item 2; vertical shift is 3.00 + 26.00 = 29.00; the horizontal shift is the same as in Item 2. Model: y = 26.00 sin(.157x – 1.57) + 29.

4. Use your calculator to graph the models for both the 44-meter and 55-meter Ferris wheels on the same set of axes. Use a viewing window that represents the oscillation in Anne’s height during three revolutions of the Ferris wheels. On which of these two rides will Anne be more apt to feel the thrill of the ride? (Anne likes heights and enjoys the sensation of rapidly rising and falling.) Explain what features of the two graphs translate into thrills. Will the most thrilling part of these two rides coincide time-wise? Explain.

The graph with the larger amplitude corresponds to the 55-meter wheel; the smaller amplitude to the 44-meter wheel.



Two features enhance the thrill of a ride: (1) how high the riders can go and (2) how fast the riders’ heights change. The 55-meter ride carries riders higher above the ground than the 44-meter ride. In addition, when riders are midway between the top and bottom of the wheel (that’s when their height is changing most rapidly), their height is changing more rapidly on the 55-meter ride than on the 44-meter ride. The times for reaching the highest point on the ride and the times when height changes most rapidly coincide for the two rides. The vertical lines on the graph above indicate the times when the riders’ heights are changing most rapidly.

Unit Project—Swing Dance

So far the only activities that generated periodic data have involved rolling round objects. In this activity you will use a motion detector setup to collect periodic data that do not come from rolling objects.

You will need the apparatus for the motion detector set-up, string, and a plastic water or soda bottle or some other object that can be used for a plumb. You will also need a stopwatch or clock with a second hand and a meter (or yard) stick.

1. Set up the equipment as follows:
1. Fill and cap a small plastic soda bottle one-quarter full with water.
2. Attach the string to the neck of the bottle.
3. Attach the free end of the string to the ceiling so the bottle will be at tabletop level.
4. Place a table about one meter from the bottle.
5. Set up the motion detector/calculator or computer. Place the motion detector on the edge of the table so it can measure the distance to the hanging bottle.



Figure 24. Pendulum set-up.

Experiment #1: Timing the swings

1. Measure the distance from the motion detector to the near side of the free-hanging bottle. Record this measurement.

Sample answer: 48 inches.

2. Pull the bottle toward the motion detector at a 10° angle from its free-hanging state. The 10° angle measure is approximate, but should result in a range of motion of about 0.5 to 1 meter. Measure the displacement from the center. Record this number.

Sample answer: 14 inches.

3. Start the stopwatch and release the bottle at the same time. Count the number of times the bottle returns to its starting position in 10 seconds.

Sample answer: 4 times.

4. Do you have enough information to determine an equation describing the bottle's motion? If so, write the equation on a piece of paper. If not, make more measurements until you can determine the equation. Explain how you got your equation.

Sample answer:

Yes there is enough information to write an equation.

Amplitude is 14 in. so, A = 14.

There were 4 swings in 10 seconds so the period is 2.5 seconds. B = 2p/2.5 » 2.51.

The axis of oscillation is 48, so D = 48.

The bottle reaches the level of its axes of oscillation when it is 1/4 of the way through a cycle, when t = 2.5/4 =0.625, so C = (0.625/2.5)(2p) = 1.57.

The equation is d = 14 sin(2.51t – 1.57) + 48 or y = 14 sin(2.51x – 1.57) + 48.

Experiment #2: Using the motion detector

1. Move the bottle 10° from the center again and hold it motionless. Execute the program Pendulum. This program will record the distance the bottle is from the detector every 0.1 second for 10 seconds. The actual data collection is started by hitting the Trigger key on the CBL. Release the bottle and press the Trigger key at the same time.

Sample results appear below. The program Slinky was used instead of Pendulum which explains why the vertical axis is labeled as height and not distance. (Slinky and Pendulum are similar programs and can be used interchangeably.)



2. Enter your equation from 2(d) as Y1 and plot the equation with the data set. How good is your match? If it is not good, get a better equation for the data set. Describe the adjustments you needed to make and why you think you needed to make them.

Sample answer: The equation from 2(d) did not fit well at all. The motion detector recorded readings in feet and we had measured distances in inches. After converting the constants A and D to feet, we re-expressed our model as

d = 1.67 sin(2.51t – 1.57) + 4 or y = 1.67 sin(2.51x – 1.57) + 4

That produced a graph that was out of phase with our data and had too large an amplitude. (We should have been more careful to pull the soda bottle back the same distance in both experiments.) There were four times in our data when the soda bottle appeared to be farthest from the motion detector. We averaged these readings to get an estimate of the farthest distance the soda bottle was from the motion detector (5.35 feet). We then did the same to arrive at a minimum distance (3.40 feet). That gave us this estimate for amplitude: A = 0.5(5.35 – 3.40) » 0.98.

This changed our axis of oscillation to 0.5(5.35 + 3.40) » 4.38. (So, we know we didn’t measure the distance from the motion detector to the free hanging bottle very carefully in the first experiment.

We averaged the four times to get from maximum to maximum to get an approximate period of 2.5. That was exactly what we got in our first experiment.

Finally, our graph was out of phase with the data. The first "peak" in the graph of a sinusoidal function with period 2.5 and phase shift 0 occurs at t = 0.625 sec. The first "peak" in our data occurred at approximately t = 2.20 sec. So, C = 2.20 – 0.625 = 1.575.

Our model: y = 0.98 sin(2.51t – 3.95)) + 4.38

This model fits these data beautifully!

4. What do you think you could do to the string or bottle to change the amplitude of the data collected by the motion detector? Predict the new amplitude and then run the experiment to test your guess. Did it work? Explain.

Sample answer: If you pull the bottle farther back before letting it swing, that should increase the amplitude of the data collected. We pulling the bottle back farther and discovered that the amplitude for our new data was 1.2 ft instead of 0.98 feet from Experiment #2.

Then we ran an experiment where we didn’t pull the bottle back as far. The amplitude for our data from this experiment was 0.65 ft.

5. What do you think you could do to the string or bottle to change the period of the curve that is plotted? Try making changes to the pendulum to get a new period.

If students shorten or lengthen the string that will change the period. Some students may suggest changing the weight in the bottle. If so, then that should be tested also.

6. What do you think you could do to change the phase shift of the curve that is plotted? Try making this change and describe what happened.

If the TRIGGER button is pressed exactly when the bottle reaches the bottom of its swing (when the string is vertical) then the phase shift should be 0. Even though we were able to get a smaller value for the phase shift, we never were able to time this perfectly and get a phase shift of 0. However, we did learn that we could affect the value of the phase shift by pressing the TRIGGER button at different points during the pendulum’s swing.

Mathematical Summary

This unit deals with functions whose graphs are periodic; their graphic patterns repeat over fixed intervals. The period of a periodic function is the shortest horizontal length of the basic repeating shape of its graph. The amplitude is half the vertical height (top and bottom) of the basic repeating shape. The sine function is an example of a periodic function.

The sine function is defined as the vertical displacement (height) versus the radian measure of an arc made by a dot rotating around a unit circle. Any function of the form

y = Asin(Bt + C) + D

belongs to the family of sinusoidal functions. The control numbers A, B, C, and D control the graph’s amplitude, period, phase shift, and axis of oscillation, respectively.

When fitting a sinusoidal model to data by hand, you need to use the information contained in your data to estimate the values of the control numbers A, B, C, and D. Here is one method of estimating values for the control numbers. Sometimes, however, a variant of this method will provide better results. So, don’t get locked into a single method.

• To estimate A: Average several of the maximum values in the data to get an estimate of the function’s maximum value. Then do the same for the minimums. Let A be half the difference between the estimated maximum and minimum values.
• Next, estimate D. To estimate D: Let D be half the sum between the estimated maximum and minimum values.
• To estimate B: First, estimate the period. Select two data points associated with "peaks" or maximums but separated by several cycles of oscillation. Divide the time difference of these two data points by the number of cycles separating these maximums. This gives you an estimate of the period. Let B = 2p /(estimated period).
• To estimate C: C is equal to the value of the central angle when t = 0.
• To shift the phase Dx to the left,
  
Dx = C/B (alternately, [(Dx/B) = (C/2p)]; C = DxB.
To shift right, make C negative. To shift left, make C positive.

Many calculators have sinusoidal regression as one of their regression capabilities. If you have such a calculator, you can use sinusoidal regression to fit sinusoidal models to data. However, you may need to adjust the model fit by your calculator if the period or amplitude of the data are predetermined by the context.

Key Concepts

Amplitude (of a periodic function): Half of the fixed vertical height of the basic repeating shape; that is, half of the difference between the maximum and minimum values of the periodic graph.

Axis of Oscillation (of an oscillating function): The center of oscillations, determined as the horizontal line midway beween the maximum and minimum values of the graph.

Frequency: The number of oscillations per second. The value of the frequency for a periodic function is 1/period.

Oscillating motion: Motion that alternates back and forth between two extremes about some mean value (the axis of oscillation).

Period (of a periodic function): The shortest horizontal length of the basic repeating shape.

Periodic function: Function that repeats itself on intervals of a fixed length (equal to the period).

Phase shift: A horizontal shift of a periodic function from a standard reference function. In particular, the phase shift for functions of the form y = A sin(Bx + C) + D is C. This is the angular shift required to translate the graph of y = Asin(Bx) + D so that it coincides with the graph of y = A sin(Bx + C) + D. The quantity, C, is the size of the phase shift.

Radian measure of an angle: The ratio of a directed length of an arc that begins at (1, 0) on the unit circle, to the length of the radius, when both are measured in the same units. If the arc turns is in the counterclockwise direction, the radian measure is positive; if it turns in the clockwise direction, the radian measure is negative.

Sine function: The vertical displacement, from the horizontal axis, of a point on the unit circle. The input for the sine function is the radian measure of the central angle.

Sinusoidal function: Any function that can be expressed in the form y = A sin(Bx + C) + D, where neither A nor B are zero.

Unit circle: Circle with radius one unit centered at the origin.

Solution to Short Modeling Practice

Solution to the Short Modeling Problem—Part 1

Use technology to graph the data of the motion in the x direction. Using only what you observe from the graph, estimate the values of A, B, C, and D in the general form of the sine function.

y = Asin(Bt + C) + D

Use the sin regression tools of the calculator to check your estimates of the values for A, B, C, and D.

Write an equation that models the motion in the x direction.

Part 2

Use technology to graph the data of the motion in the y direction. Using only what you observe from the graph, estimate the values of A, B, C, and D in the general form of the sine function.

For Y, A ~ 4.0cm, Period ~ 1.3 seconds, B » 2p/1.3 » 4.7

Use the sin regression tools of the calculator to check your estimates of the values for A, B, C, and D.

Write an equation that models the motion in the y direction.

Part 3

Use the sine functions that you have produced in Part 1 and Part 2. Graph the x and y positions at the same time using your calculator.

Look at the curve for the y position. Find where it first crosses the horizontal axis. Notice the alignment of this "zero crossing" with the first peak on the x curve. Then the sine function is at this first peak value, the argument of the sine function (the value of the complete expression within the parenthesis sin(argument)) is always (p/2). The phase difference between the x position and the y position is p/2.

A sine function that is phase shifted p/2 ~ 1.57, is called a cosine function. The cosine function is the same as the sine function, except for the phase shift. The x and y positions can be modeled using either two sine function, or one sine function and one cosine function.

x position: a = A_xsin(Bt)

y position: y = A_ysin(Bt + (p/2)) or y = A_ycos(Bt)>

Part 4

Use the pairs of x and y data that have matching times to form ordered pairs (x, y).

Graph these ordered pairs of (x, y) data on a sheet of graph paper to determine the path the moving tray is following.

Solutions to Practice and Review Problems

1. Graph 2
2. Graph 3
3. Graph 1
4. Graph 1

1. Graph 3
2. Graph 3
3. Graph 2
4. Graph 2

1. The students should be able to count 4 cycles within a span of 18 hours, which yields a temperature cycle period of 4.5 hours. The frequency is thus the reciprocal of the period, or 0.222 cycle per hour.
2. The humidity curve is changing more slowly, cycling once in a time span of 18 hours. Thus its frequency is 0.056 cycle per hour.
3. Since the curves do not have the same frequency, they are clearly NOT in phase.
4. The maximum and minimum temperatures correspond to the set point or the reaction points of the thermostat that controls the cycling of the refrigeration unit in the chamber. (It appears that the unit is able to control the temperature within a range of 2°. This means that the amplitude of the sine curve is 1°C.)

About 131.95 feet

About 201.06 m2

1. The diameter of the tire is about 23 inches.
2. The tire will move 72.2 inches (rounded) in one complete revolution.
3. Miles ´ Conversion factor = Inches
   10,000 miles ´ 6.336 ´ 10⁴ inches per mile = 6.336 ´ 10⁸ inches

Revolutions = Total inches ¸ Inches per revolution
Revolutions = 6.336 ´ 10⁸ inches ¸ 72.2 inches per revolution
Revolutions = 8,800,000 revolutions

The student’s sketch of the pool and sidewalk should appear generally as shown below.

The radius of the pool is 1/2 the diameter, or 16 feet. The radius of the outer edge of the sidewalk is an additional 5 feet, or 21 feet in all.

1. The frequency of your pedal strokes is 45 strokes per minute (or 0.75 stroke per second), as given in the exercise. The period of your strokes is the reciprocal of the frequency, 0.022 minute per stroke (that is, 1/45 stroke per minute) or 1.33 seconds per stroke (that is, 1/0.75 stroke per second).
2. Since we are considering a "stroke" to be one foot going "down" and "up," then you and your friend are half of a cycle out of phase. This is a phase difference of 180°.

1. The time between adjacent breaks is simply the time between the 12 breaks divided by 12, or 19/12 = 0.833 second per break.
2. The result of Part a is really the period of the "wave," so the frequency is just the reciprocal, or12/10 = 1.2 breaks per second (or 1.2 cycles per second).
3. As given in the text

Speed = Wavelength ´ Frequency

So, solving for the wavelength and substituting...

Wavelength = Speed/Frequency

Wavelength = 73.3 fps/1.2 cps

Wavelength = 61 feet per cycle (rounded)

The road breaks, corresponding to the bouncing of the car, are approximately 61 feet apart.

1. The amplitude is the height (the vertical distance) of the wave above the middle line of each wave.

The amplitude of wave 1 is 5 V/div´ 2 div = 10 V.

The amplitude of wave 2 is 5 V/div ´ 1 div = 5 V.

2. The period is the amount of time (the horizontal distance) between crests.

The period of wave 1 is 50 msec/div ´ 2 div/cycle = 100 msec per cycle.

The period of wave 2 is 50 msec/div ´ 2 div/cycle = 100 msec per cycle.

3. The frequency is found by the formula 1/Period.

The frequency of wave 1 is 1/0.100 sec per cycle = 10 Hz.

The frequency of wave 2 is 1/0.100 sec per cycle = 10 Hz.

4. Sine wave 1 leads wave 2 by 1¤ 2 division, or 25 msec. This amounts to 90° of phase.

1. The students’ tables should appear generally as shown below.

2. The students' graphs should appear generally as shown below.



3. The period is simply the length of time needed for the curve to repeat, in this case 0.1 second.



4. This is an awkwardly worded question, but the goal is for the students to see that the value of 0.1 sec in the equation represents the period of the wave. Changing this value in the equation to 0.25 second, for example, would change the period of the graphed wave to 0.25 second.

T = 2p ´

1. T = 2p ´
   T = 1.004 sec



2. T = 2p ´
   T = 1.002 sec

Yes, the period is closer to the desired value. (If the students use a rounded value for p, slightly different answers might occur.

EXERCISE 13

1. The students will need to encode the dates as x-values. They can simply enter the dates for May (8, 10, 11...), or use the first date as x = 0(0, 2, 3,...), or even use the day number since the beginning of the year (127, 129, 130,...). Whatever method they choose, after appropriately scaling their graph, they should see a distribution of points resembling that shown here.






2. The resulting regression equation should be approximately as follows:



3. The maximum y-value is a + d = 8.08 + 32.0 = 40.08, or 40.1°C.



4. In this context, the x-variable represents time in days. Hence, the parameter b represents 2p/T, where T is the period in days. Thus, T = 2p/b = 2p/105 = 5.98 days, meaning the temperature varies through a cycle about once every 6 days. This seems consistent with Joseph's observations of steam bubbling forth "three or four times a month." (An inconsistent result would have been something like T = 0.2 days or T = 140 days. Such tests of reasonableness should always be applied to regression results.)



5. Student answers may vary here since there are several ways to interpret the phase constant c. The constant c indicates that Joseph's first measurement did not fall exactly at the midpoint of the cycle of variation, that is, at x = 0 the temperature did not equal the value for d. This is evident from a graph of the regression equation (see above), where the data in fact appear to begin near a minimum in the periodic cycle. The calculator adjusts the phase of the sinusoidal regression equation to match the data by providing the phase adjustment c.

EXERCISE 14

1. No. The sine function varies with time, and the voltage will vary with it.



2. 170 volts, which is the value of the product when sin(377t) = 1.



3. Period = 2p/377
   Period = 0.0167 seconds

EXERCISE 15

1. Possible answers: The three graphs have different starting values at t = 0.



2. The amplitude and frequency are the same for all three.



3. 622 volts, the amplitude of the sinusoidal function.



4. 622 volts



5. Va reaches the first peak when t = period/4 or t = 0.00417 sec. At that time V_b = –309 volts and V_c = –311 volts.