scale changes and translations of circular...

Scale changes and translations of circular functions

Math B/Grade 115 Day Unit Plan

Technology:TI 83 Graphing Calculator

Green Globs and Graphing EquationsGeometer’s Sketchpad

TI View ScreenComputers

Overhead projector

Written by:Michelle Loncar

Loncar –

OverviewThis unit on trigonometry focuses on scale changes and transformations of

circular functions.Day 1

fi Review of right triangle trigonometryfi Examine the relationship between sine and cosine on the unit circle.

fi Find exact values of trigonometric functions that are multiples of 6p and

4p .

fi Using exact values students will discover behaviors of the sine and cosinefunctions.

Day 2fi Learn the basics of Green Globs and Graphing Equationsfi Examine the graphs of y = asin(bx) and y = acos(bx).fi Discover amplitude and period of sine and cosine from its equation and from

its graph.fi Graph sine and cosine functions with various amplitude and period.

Day 3fi A demonstration with an oscilloscope will be done to model amplitude and

period in sound waves.fi Using Green Globs and Graphing Equations students will identify vertical

translations of the sine and cosine functionsfi Students will identify phase shifts and properties they hold.

Day 4fi Review of sine and cosine functions.fi Fill in a table of exact values of the y=tan(x).fi Plot the tangent function from 0 to 2pi.fi Investigation of the tangent function using TI 83 Graphing Calculators.fi Students will discover amplitude, period, phase shift and vertical stretching of

the tangent function.

Day 5fi A review of the first four days will take place.fi Using a TI 83 Graphing Calculator students will functions in stagesfi A quiz will also sum up the beginning of the week.

ResourcesForesman, Scott. Functions, Statistics, and Trigonometry: Teacher’s Editions. Illinois:

Scott Foresman and Company. 1992. pages 292 – 303 and 339-351

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Day by Day overview of objectives, standards, and materials.DAY 1

Objectives Using right triangle trigonometry the students will investigate the unit circle.They will evaluate circular functions outside the first quadrant. They will be able to labelthe exact values of the sine and cosine for values of q that are multiples of p/6 or p/4.

New York State Key IdeasModeling/Multiple Representation – Students use mathematical

modeling/multiple representation to provide a means of presenting, interpreting,communicating and connecting mathematical information and relationships.

Measurement – Students use measurement in both metric and English measure toprovide a major link between the abstractions of mathematics and the real world in orderto describe and compare objects and data.

NCTM StandardsAlgebra Standard

-Using patterns, relations, and functionsGeometry standard

-Analyze characteristics and properties of two- and three- dimensional geometricshapes and develop mathematical arguments about geometric relationships

MaterialsOverhead projector, worksheets, Computer access, class set of TI 83 calculators,

Geometers Sketchpad 4.0

DAY 2Objectives

The students will be investigating the functions y =a sin(bx) and y = a cos(bx).They will find the amplitude and period of sine and cosine functions from its equationand from its graph. They will graph sine and cosine with functions with variousamplitudes and periods. Also, students are expected to check their predictions using acomputer.




Patterns/Functions – Students use patterns and functions to develop mathematicalpower, appreciate the true beauty of mathematics, and construct generalizations thatdescribe patterns and compare objects and data.


-using patterns, relations, and functionsGeometry Standard

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- Apply transformations and use symmetry to analyze mathematical situations.Materials

Computers, Green Globs and Graphing Equations, Worksheet.

DAY 3Objectives

Students will identify vertical translations of the sine and cosine functions alongwith the properties they hold. They will also investigate phase shifts of the sine andcosine wave from their equations and graphs. They will use the computer program GreenGlobs and Graphing Equations.






- Apply transformations and use symmetry to analyze mathematical situations.Connections Standard

-Recognize and apply mathematics in contexts outside of mathematics.Materials

Oscilloscope, computers, Green Globs and Graphing Equations, musicalinstrument, worksheet.

DAY 4Objectives

The students will investigate the tangent function. They will determine theproperties of the tangent function and be able to graph the function with a graphingutility. They will also investigate vertical stretching, phase shifts, amplitude, period, andvertical shifts. Students will be able to determine the equation of function from lookingat its graph.






- Apply transformations and use symmetry to analyze mathematical situations.

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MaterialsOverhead projector, transparencies 1 and 2, class set of TI 83 Graphing

Calculators, worksheets

DAY 5Objectives

Students will find the amplitude, period, phase shifts, and vertical shifts of sineand cosine functions. They will also graph sine and cosine functions with variousamplitudes, periods, phase shifts and vertical shifts. They will also find the period, phaseshift and vertical sifts of tangent functions.





- Apply transformations and use symmetry to analyze mathematical situations.Materials

Class set of TI 83 Graphing Calculators, TI 83 View Screen.

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Day 11. Using right triangle trigonometry the students will investigate the unit circle.

They will evaluate circular functions outside the first quadrant. They will be ableto label the exact values of the sine and cosine for values of q that are multiples ofp/6 or p/4.

2. Concepts include: sine, cosine, tangent, right triangle, radians, degrees, unit circle,opposite, adjacent, hypotenuse, reflection

3. Generalizations include: right triangle trig definitions, properties of triangles

4. Procedures include: find angles and sides using right triangle trigonometry,determining sine and cosine values on the unit circle

5. Procedures build off generalizations, which build off concepts.

Day 1Lesson Summary

Using right triangle trigonometry the students will investigate the unit circle.They will evaluate circular functions outside the first quadrant. They will be able to labelthe exact values of the sine and cosine for values of q that are multiples of p/6 or p/4.

MaterialsTeacher: overhead projector Student: worksheets

Computer access class set of TI 83 Geometers Sketchpad 4.0 calculators

StandardsModeling/Multiple Representation – Students use mathematical modeling/multiplerepresentation to provide a means of presenting, interpreting, communicating andconnecting mathematical information and relationships.

Measurement – Students use measurement in both metric and English measure to providea major link between the abstractions of mathematics and the real world in order todescribe and compare objects and data.

Patterns/Functions – Students use patterns and functions to develop mathematical power,appreciate the true beauty of mathematics, and construct generalizations that describepatterns and compare objects and data.

Consider the warm upThe warm up is a review of right triangle trigonometry that students should alreadyknow. The warm up is to get the kids familiar with using the terms sine, cosine, andtangent.

T: Has anyone ever heard of sine, cosine or tangent before? Where?

S: Yes. When talking about trig ratios.

S: When talking about a right triangle.

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T: When talking about ratios what does sin, cos, and tan stand for?

S: sin q = side opposite q Hypotenuse

cos q = side adjacent to q Hypotenuse

tan q = side opposite qside adjacent to q

T: Does anyone have a way to memorize those that they’d like to share?

S: Soh Cah Toa

Write the above ratios along with the following example on the overhead.

T: GivenC

12 13

B 5 AHow would I find the sin A?

S: The sine of A is equal to hypotenuse

oppsiteside

Plug in the numbers. So the sinA = 1312

T: Since this is review find the rest of these answers.(write the questions on the overhead). Walk around the room and observestudents work.b) sin C c) cos Ad) tan C e) tan A

Afterward ask the students what they found for each problem. If they seem to being havetrouble review another problem.

T: What if you were given a triangle and wanted to find two of its sides. How wouldyou do it?

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23°

F

D

First find side DE.

T: What do we know that can help us?

S: We know that angle and the side opposite it.

T: What are we looking for?

S: Side DE. The hypotenuse

T: So we have opposite and hypotenuse. What is the trig ratio that we can set up?

S: Sin(23) = x12.

T: Solve for x. Who has an answer?

S: 30.71

S: -14.18

T: Which answer is right? The angle we used was 23. Was that in radians ordegrees?

S: It was 23 degrees.

T: So we have to put our calculator in degree mode. Know which answer iscorrect?

S: 30.71.

S: The other answer couldn’t work anyway because it was negative and the angle ispositive.

Hand out worksheet of more review of these trig problems. Allow time to complete andthen go over the answers. Walk around the room and help students if necessary.

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LessonPrerequisite Knowledge

The students should be familiar with radians. They should know the basics ofcoordinate plane.

The beginning of the lesson is an introduction to the unit circle.

T: Lets look at the unit circle on the xy plane. (draw on overhead and have thestudents write along with you in their notes.)

(-1,0)

(0,-1)

(1,0)

(0,1)

b

aX

Y

Side a is the initial side. Side b can be rotated anywhere around the circledepending upon the angle. It is called the terminal side. (label each) When the angle isgreater than zero the terminal side is rotated counterclockwise. When the angle is lessthan zero the terminal side is rotate clockwise.

T: Lets look at quadrant one. Given an angle q, follows that we have a terminal side.

p

1(-1,0) (1,0)

(0,1)

(0,-1)

T: Can we build a right triangle in the unit circle with our terminal side?

S: Yes. Draw a line down from the point on the circle of the terminal side to theradius.

T: Since that forms a right triangle, what do we know about the terminal side?

S: It is the hypotenuse of the triangle.

S: It has length one because it is another radius.

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T: Lets use our right triangle definitions to break down this unit circle. What is thesin(q)?

S: Sine is opposite over hypotenuses.T: Do we know the value of either the side opposite q or the hypotenuse? Plug them

into the equation.

S: We know the hypotenuse.

S: So sin(q) = 1

opposite = opposite (write it on the overhead)

T: Label the side opposite q, sin(q) on your diagram. What about the side adjacentto q? Can you find the length of that?

S: We can use cosine.

S: cos(q) = hypotenuseadjacent =

1adjacent = adjacent. (write it on the overhead)

T: Label the adjacent side of q on the diagram cosine.

T: Lets forget about right triangle trigonometry for a minute. Point p has an xcoordinate and a y coordinate. What are they?

S: (cosq, sinq)

T: For q between zero and p/2, we can find the coordinates for the intersection of theterminal side of q and the unit circle.

p = (cos q,sinq)

1cos

1

qsin

q=0

q=3p

2

q=p

q=p

2

T: What happens if we reflect the triangle over the y-axis? Do it on your paper.

S: We get another point on the circle.

T: Label that point r.

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r

cos

sin1

p = (cos, sin)

1(-1,0) (1,0)

(0,1)

(0,-1)

T: Since we reflected the triangle over the y-axis, what are the coordinates or r?

S: (-cosq, sinq)

T: Take a minute and using reflections, see if you can find the coordinates of a pointbetween p and 3p/2, and 3p/2 and 2p. Label them on your unit circle.

Allow a few minutes to complete.

T: Who found the coordinates of a point between p and 3p/2?

S: (-cosq, -sinq).

T: How about the coordinates of a point between 3p/2 and 2p.

S: (cosq, -sinq).

T: We can find exact values of sine and cosine using right triangle trig.Lets look at a 45-45- 90 triangle. Lets label the angles in radians.

p

4

p

2

p

4

What kind of triangle is this?

S: Isosceles.

T: If it is and isosceles triangle and the two angles are congruent, what can weconclude about the two sides opposite them.

S: The two sides are congruent.

T: The length of each of the two sides is one. Since we know two sides of a righttriangle, how can we find the third side?

S: Pythagorean theorem. The third side has length equal to 2 .

T: Label the triangle.

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2

1

1

p

4

p

2

p

4

Using the right triangle trig definitions find the cos(p/4) and sin(p/4)

S: cos(p/4) = adjacent/hypotenuse = 1/ 2 .sin(p/4) = opposite/hypotenuse = 1/ 2 .

T: Lets look at 30-60-90 triangle. Label the angles in radians. One leg has lengthone.

1

p

3

p

6

p

2

How can we find the other lengths of the legs of the triangle?Lets make this into an equilateral triangle.

1

p

6

p

3

p

2

1

p

3

p

6

p

2

T: What is the length of each side of the triangle? Why?

S: Each leg has length two because when we just had half the triangle the bottom leghad length one. Know it is doubled.

T: Good. And what about the length of the line that bisects the triangle?

S: Use the Pythagorean theorem. It has length 3

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T: Using the 45-45-90 and 30-60-90 triangles we can find the exact values of thepoints of intersection of the angle and unit circle between 0 and p/2. Once we have thosevalues we can reflect the angles over the unit circle to find the exact values of sine andcosine for the rest of the unit circle that are multiples of p/6 and p/4.

Hand out worksheet. Give time to complete. Walk around the room and help whereneeded. After five minutes or so put the answers on the overhead. Lets students comparetheir work with the key.

Have the students look at their unit circle diagram and make the following connections.

T: If we start on the initial side where that line intersect the unit circle, what is thevalue of the sine?

S: The sine is equal to zero.

T: If we increase the angle to p/6, what is the value of sin(p/6)? What happened toit?

S: sin(p/6) = 12

It increased.

T: What happened to the value of sin when we increase the angel up to p/2?

S: The value of sin increases from 0 to p/2.

T: How about from p/2 to p? What happened to the values of the sine?

S: The sine values are decreasing.

T: How about from p to 3p/2?

S: The sine values are still decreasing.

T: And from 3p/2 to 2p?

S: The values are increasing.

T: Take a look at the values of cosine as they go around the unit circle. Whatconclusions can you make?

S: The cosine decreases from 0 to p.

S: The cosine increases from p to 2p.

T: In a sense the values of sin and cosine are wrapped around the unit circle. Whenif we were to unwrap the unit circle and lay it out on to the plane? What do youthink the values of sine would look like?

At this point use the computer and GSP 4.0 to show the wrapping function of sine andcosine. Before you actually show the wrapping move drag point B around and ask thestudents to describe what happens to point C. It moves horizontally across the line

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segment. Drag point A around the circle and ask what happens to point C. It moves upand down. Tell them that point A is going to move around the circle and point B is goingto move along the x-axis. Ask them to guess the path that point C will take.

6

4

2

- 2

- 4

- 6

- 8

- 10

5 10 15 20

C

B

A

Line up points A and B at the point (1,0) on the unit circle. Animate those points. Highlight point C and chose trace. Then line up point C at (1,0). Press the animate button.You should get the following picture.

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6

4

2

- 2

- 4

- 6

- 8

- 10

5 10 15 20

Animate Points

C

B

A

Ask the students to describe what was happening as point A went around the unit circle.

T: What should happen if point A continued to wrap around the circle and neverstop?

S: The line, the sine function would continue and never stop.

T: When a function has that property it is called periodic.

To show the wrapping cosine function, line up point B on (1,0) and drag point C as highas it can go. Then animate. You should get the following picture.

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6

4

2

- 2

- 4

- 6

- 8

- 10

5 10 15 20

Animate Points

C

B

A

T: Now what would happen if point A continued to wrap around the circle and neverstop?

S: It would go on forever just like the sine function did.

T: So what can we conclude about eh sine function and cosine function?

S: They are both periodic.

Allow students to ask questions.

Show another sine curve using a Java script. Tell the students the address of the page sothat they can access it themselves from home. It should look like this.

3

2

1

- 1

- 2

- 3

- 4 - 2 2 4 6 8 10

Animate Points

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Wrap class up by answering any questions. To end the class, pose a final question to sumup today ‘s lesson. Write it on the board. Have the students turn it in as there ticket outof the room. Also pass out homework activity.

T: In order to leave the room each of you need to write out on a piece of paper toturn in 1) Given an angle p/3 what are the sine and cosine values? 2) For the followingangles tell whether the sine is positive or negative. A) q = p/2 B) q = 5p/3 C) q = 5p/4and D) q = p/6.You may use the unit circle to help you.

T: Also complete the worksheet for homework.

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Worksheet - Review

A

B c

C B a

Given right triangle ABC1) a = 11, b = 6, c = 12.53. Calculate cos B

2) a = 2, c = 2, c = 2.828. Calculate tan B

3) a = 1, b = 1, c = 1.414 Calculate sin A

4) a = 2, b = 6, c = 6.325. Calculate cos A

5) a= 3, b = 9, c = 9.487. Calculate tan B

Using the given right triangle find the legs of the triangle.

c b

a

1. b= 5, b = °20 ; find a and c2. a = 6, b = °40 ; find b and c

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Worksheet

Given the angle find the exact value of sin and cosine. Use the 30-60-90 and 45-45-90triangles to help you.

q=3p

2

q= p (1,0)

q=5p

3

q=7p

4

q=11p

6

q=4p

3

q =5p

4

q =7p

6

q =5p

6

q =3p

4

q =2p

3

q =p

6

q=p

4

q=p

3

q =p

2

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Answer Key

(-1

2,

3

2)

(-1

2,- 3

2) (

1

2,- 3

2)

(1

2,

3

2)

(0,-1)

(-1,0)

(0,1)

(- 2

2,

2

2)

(- 2

2,- 2

2) (

2

2,- 2

2)

(2

2,

2

2)

(3

2,-1

2)(

- 3

2,-1

2)

(- 3

2,1

2) (

3

2,1

2)

q=11p

6

q=7p

4

q=5p

3q=

3p

2

q=4p

3

q=5p

4

q=7p

6

q=p

q=5p

6

q=3p

4

q=2p

3

q=p

2 q=p

3

q=p

4

q=p

6

(1,0)

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HomeworkFind the exact values for problems 1-5. You may use your unit circle.

1. sin 45p

2. cos 45p

3. sin 23p

4. sin 4p

5. cos 67p

6. Using the angles from your unit; circle, plot the sine curve by hand.

7. Using the angles from your unit circle, plot the cosine curve by hand.

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Day 21. The students will be investigating the functions y =a sin(bx) and y = a cos(bx).

They will find the amplitude and period of sine and cosine functions from itsequation and from its graph. They will graph sine and cosine with functions withvarious amplitudes and periods. Also, students are expected to check theirpredictions using a computer

2. Concepts include: sine, cosine, cycle, amplitude, period, periodic, vertical

3. Generalization include: determining the amplitude, determining the period

4. Procedures include: graphing functions, translating functions

5. Procedures build off generalizations which build off concepts

Day 2Lesson summary

The students will be investigating the functions y =a sin(bx) and y = a cos(bx).They will find the amplitude and period of sine and cosine functions from its equationand from its graph. They will graph sine and cosine with functions with variousamplitudes and periods. Also, students are expected to check their predictions using acomputer.

MaterialsTeacher: computer Students: computers

Green globs Green globs Work sheet

StandardsModeling/Multiple Representation – Students use mathematical

modeling/multiple representation to provide a means of presenting, interpreting,communicating, and connecting mathematical information and relationships.



Warm upIntroduce students to Green Globs and Graphing equations. Show them how to

open to. Make sure they pick the graphing grid from (-2p, -5) to (2p,5). Ask students torecall that sine and cosine have a special feature. They go on forever. What is thatcalled? Periodic. Work along with the students and have them click set bounds. Changethe lower x to zero. Plot the function y = sin(x). Explain that what they see if a piece ofthe function. It is one cycle of the sine curve. Ask them how many cycles do they thinkthe sine function has. Hopefully, since the curve goes on forever, they will say infinite.Do the same with y = cos(x).

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LessonHave the students work in pairs on the following worksheet. The teacher needs to

circulate the room and ask students questions to get them to reflect on the data.Anticipated student answers are in red.

Wrap up the class by having the students close the program Green Globs and GraphingEquation. Ask them to take out a piece of paper put their name on it and have themanswer the following questions as their ticket out of the room for the day.1. What is amplitude? Write an equation that has an amplitude of 4.2. What is period? Given the function y = 2sin(2x) determine the period.Handout homework activity.

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Worksheet

Siney=a sin(x)

Make sure your graphing grid is from (-2p, -5) to (2p,5).

Graph the function y = sin(x). On the same set of axes graph y = 3sin(x).Do the graphs look the exact same, or different? They look different.

Erase your graph. On the same set of axes graph y = sin(x) and y = 4sin(x).What is different between the two graphs of these functions? Y = 4sin(x) looks like it is stretched vertically. The have different maximum andminimum values.

What stays the same when you graph the two functions?The two functions have the same x and y intercept. They follow the same direction.

The vertical stretching you see is called amplitude. Amplitude is half the differencebetween the maximum and minimum values attained by the function.

Graph the equation y = 2sin(x). Locate the minimum and maximum values of thefunction. Fill in the chart below.

Equation Maximum value Minimum value Amplitude = 21 (max-min)

Y = 2sin(x) 2 -2 (1/2) * (2—2)

2

Erase the graph. Graph the following functions one at a time. Copy it to the graph paperbelow and label each function. Then fill in the chart.

Equation Maximum value Minimum Value Amplitudey = sin(x) 1 -1 1y = 3sin(x) 3 -3 3

y =21sin(x)

1/2 -1/2 1/2

y = 5sin(x) 5 -5 5

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What pattern do you see?The amplitude is always positive. The amplitude is the number in front of sin(x).

Graph the following equations. Determine the amplitude of each function.

What pattern can you find here?The amplitude is the absolute value of the number in front of sin(x).

Predict the amplitude of the following functions and graph them. You may need tochange the bounds to see the complete graph.

y = 8sin(x) amplitude = 8

y = -(1/4)sin(2x) amplitude = (1/4)

y = (1/2)sin(4x) amplitude = (1/2)

Write an equation in the form y = a sin(x) where the amplitude is 5.y = 5sin(x)

Equation Maximum value Minimum Value Amplitudey = - sin(x) 1 -1 1y = -3sin(x) 3 -3 3

y = -21sin(x)

1/2 -1/2 1/2

y = - 5sin(x)5 -5 5

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Take a look at cosine.y = a cos(x)

Graph y = cos(x) and y = 2 cos(x) on the same set of axes. Do these two graphs look theexact same or different?They look different.

What is different?Y = 2cos(x) looks like it was stretched vertically. They have different min and max

values.

What is the same?The have the same x and y intercepts. They go in the same direction.

Graph the equation y = 3cos(x). Fill in the table to find the amplitude.Equation Maximum value Minimum value Amplitude

21 (max – min)

y = 3cos(x) 3 -3 (1/2) (3- -3) = 3

Erase the graph. Graph the following functions at one time. Fill in the table.Equation Maximum value Minimum value Amplitudey= cos(x) 1 -1 1y = 4cos(x) 4 -4 4y = -8cos(x) 8 -8 8

y = 21cos(x)

(1/2) -(1/2) (1/2)

What pattern do you see?The amplitude for sine and cosine can be found the same way. The amplitude is theabsolute value of the coefficient of sine or cosine.

Predict that amplitude of the following functions.

y = -3cos(x) amplitude = 3

y = )x41cos(

32 amplitude = (2/3)

y = 7cos(8x) amplitude = 7

Graph y = 3sin(x) and y=3cos(x). on the same axes. What do these two graphs have incommon?They have the same amplitude. The two graphs look the same but they have different xand y intercepts.

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In your own words write down what you think amplitude is and how you find it. Tell itto your partner and see if he/she agrees.Amplitude stretches a function vertically. It is the absolute value of the number in frontof the sin(x) or cos(x).

Graph the equation y = sin(x) along with y = sin(2x).Do these graphs look the exact same or different?They look different.

Erase and graph the equation y = sin(x) along with y = sin(4x) on the same axes. Copythe graphs below.

What is different between the two graphs?y = sin(4x) crosses the x axis many more times then y = sin(x).

What is the same on the two graphs?They have the same amplitude.

Graph y = sin(x). What is the length of one complete cycle in the function y = sin(x)? 2p

Graph y = sin(2x). What is the length of one complete cycle in that function? p

The length of one complete cycle is called the PERIOD.

Erase graphs. Graph the following equations one at a time. Copy them onto the graphpaper below and determine the period of each. Fill in the table.

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Equation Periody = sin(x) 2p

y = sin(2x) py = sin(4x) p/2

y = sin(21x)

4p

Given y = sin(bx) the period of a function can be found, ie the length of one cycle, bythey cycle of the sin(x) divided by b. Using that find the period of the followingfunctions. Fill in the table.

Equation Periody = sin(bx) 2p/by = sin(3x) 2p/3

y = 31sin(

41x)

8p

y = sin(bx) p

What can you conclude about y = sin(bx)?The number in front of x is used to determine the period.

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Given y = y = sin(-bx). This equation can be written as y = -sin(bx). Rewrite thefollowing equations with out a negative argument. Then find the period.

Equation Rewrite Equation Period

y = sin(-31x) y = -sin(

31x)

6p

y = 2sin(-21x) y = -2sin(

21x)

4p

y = 4sin(-4x) y = -4sin(4x) p/2

What can you conclude about the period of any function?The period is always positive.

Do you think the same generalizations about y = sin(x) can be made for y = cos(x)?Lets find out!

Graph the equation y = cos(x) along with y = cos(2x). Copy the graphs below.

What is different between these two graphs?y= cos(2x) crosses the x axis more times than y = cos(x).

What is the same between these two graphs?They have the same amplitude. They have the same y-intercept.

Look at the graph of y = cos(x). What is the length of one complete cycle? 2p

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Graph the following functions one at a time. Determine the period of each. Fill in thetable below.Equation Periody = cos(2x). p

y = cos(21x).

4p

y = cos(bx). 2p/by = 3cos(4x). p/2

y = 5cos(41x).

8p

What can you conclude about the period of a sine or cosine function?They can both be found the same way.

Summarize your results. When an equation is of the form y = asin(bx) or y = acos(bx)a tells you the amplitude of the function and b is used to determine the periodof the function.

Predict the amplitude and period for the following functions.Amplitude period

y= 2sin(2x) 2 p

y= -21sin(

21x) 1/2 4p

y = 4cos(x) 4 2p

State the equation of the graphs below.For the following graphsx min = -2p x max = 2py min = -5 y max = 5

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Given the graph of sin(x) find the equation of the other graph.

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g x( ) = sin x( )

Given the graph of cos(x), find the equation of the other graph.

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h x( ) = cos x( )

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Homework Activity1. Given the equation y = )

2xcos(

31

a) State the amplitude.b) Find the period.

2. Consider the function y = -3sin(x)a) What is the period?b) What is the amplitude?

3. Consider the function y = 21cos(x).

a) What is the period?b) What is the amplitude?

c) Sketch a graph of y = 21cos(x) and y = cos(x) on the same set of axes.

(axes on back)

4. Sketch one cycle of the graph. (axes on back)Y = 2sin(2x)

5. The following is the graph of sin(x) along with another equation. What is the equationcorresponding to the second graph?

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Day 3

1. Students will identify vertical translations of the sine and cosine functions alongwith the properties they hold. They will also investigate phase shifts of the sine andcosine wave from their equations and graphs. They will use the computer program GreenGlobs and Graphing Equations.

2.Concepts include: congruent, vertical, amplitude, period, sine, cosine, maximum,minimum, phase shift.

3.Generalizaitons include: properties of translations

4. Procedures include: graphing functions, translating functions

5. Procedures build off generalizations, which build of concepts

Day 3Lesson Summary

Students will identify vertical translations of the sine and cosine functions alongwith the properties they hold. They will also investigate phase shifts of the sine andcosine wave from their equations and graphs. They will use the computer program GreenGlobs and Graphing Equations.

MaterialsTeacher: Oscilloscope Students: computers

Green Globsworksheetmusical instrument

StandardsModeling/Multiple Representation – Students use mathematical

modeling/multiple representation to provide a means of presenting, interpreting,communicating, and connecting mathematical information and relationships.



Consider the warm upLead a discussion about the results from yesterday in the computer lab. Begin by talkingabout how sound is produced. Make sure to talk about how different kinds offluctuations cause us to hear different kinds of sounds. Bring out an oscilloscope.(borrow from science department) Have a student who plays an instrument bring it toclass (or the teacher can) and play a few notes. Vary the pitch and volume of a tone toproduce changes in the period and amplitude of the tone. Ask students question such aswhat do you notice when a high pitched sound is made? The period is shorter. What doyou notice when a sound is loud? The amplitude becomes greater. Etc. Point out that

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yesterdays lesson about finding the amplitude and period is the foundation for all musicalsound. Also, the rest of the week will be dealing with the mathematics of the changes inthe sine wave.

LessonThe students should be on the computers working in pairs on the following

worksheet. The teacher should circulate the room and ask students questions to get themto reflect on the data.

To wrap class up close the Green Globs program and have students take out a piece ofpaper. They need to answer the following questions to turn in as their ticket out of thedoor. Write them on the board.1. When you translate a function vertically what properties hold2. Write an equation of a sine function that has a vertical shift of 2.3. Write an equation of a cosine function that has a phase shift of p/2.

Hand out homework activity.

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Worksheet

Graph y = sin(x). On the same set of axes graph y = sin (x)+2. Copy it to thegraph paper below.

Graph y = sin(x) and y = sin(x) – 3 on the same axes.

What happens when you add or subtract a number from the sin(x)?

The graph of the function raises or lowers vertically from its normal position.

Look at the graph you copied of y = sin(x) and y = sin(x) + 2. Fill in the chart below.Minimum

valuesMaximum

valuesAmplitude Period

sin(x) 1 -1 1 2psin(x) + 2 1 3 1 2p

What can you conclude about a sine function and its image when you translate itvertically?When you shift the graph vertically, the amplitude and period of the function and itsimage remain the same. The graphs still have the exact same shape.

Since the two graphs under a translation have the same shape, amplitude and period wecan call these two curves congruent.

Graph the following function one at a time. Fill in the table.

Equation Describe position of the graphy = sin(x) + 1 The graph is raised one unit above its normal positiony = sin(x) + 4 The graph is raised four units above its normal positiony = sin(x) + 3 The graph is raised three units above its normal position

y = sin(x) + 21 The graph is raised half a unit above its normal position

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Can you find a pattern?The amount you add to the sin(x) is the number of units the graph will rise above itsnormal position.

Erase the graphs. Graph the following equations and fill in the table.

Equation Describe position of the graphy = sin(x) - 1 The graph is lowered one unit below its normal positiony = sin(x) – 5 The graph is lowered five units below its normal position

y = sin(x) - 25 The graph is lowered five halves below its normal position

y = sin(x) - 2 The graph is lowered two units below its normal position

Can you find a pattern?The amount subtracted from the sin(x) is the number of units the graph will lower fromits normal position.

Predict the amount of units the graph will be raised or lowered for the followingequations. Graph each equation to check your answer.

y = sin(x) stay at 0y = sin(x) + 3 raised 3y = sin(x) + -2 lowered 2y = sin(x) –(3/4) lowered (3/4)y = sin(x) +(1/2) raised (1/2)

What about cosine?Graph y = cos(x), y = cos(x) +1, and y = cos(x) –1. Copy it to the graph paper below.

What happens when you add or subtract a number from cos(x)?The graph raises or lowers vertically from its normal position.

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Look at the graph you just copied. Fill in the chart below.Minimum

valuesMaximum

valuesAmplitude Period

cos(x) -1 1 1 2pcos(x) - 1 0 -2 1 2p

What can you conclude about a cosine function and its image when you translate itvertically?When you shift the graph vertically, the amplitude and period of the function and itsimage remain the same. The graphs still have the exact same shape.

Graph the following function one at a time. Fill in the table.

Equation Describe position of the graphy = cos(x) + 1 The graph is raised one unit above its normal positiony = cos(x) + 3 The graph is raised three units above its normal position

y = cos(x) + 23 The graph is raised three halves above its normal position

y = cos(x) + 21 The graph is raised half a unit above its normal position

What pattern do you see?The amount you add to the cosine of x is the number of units the function will raisevertically.

Erase the graphs. Graph the following equations and fill in the table.Equation Describe position of the graph

y = cos(x) - 1 The graph is lowered one unit below its normal positiony = cos(x) - 5 The graph is lowered five units below its normal position

y = cos(x) - 25 The graph is lowered five halves below its normal position

y = cos(x) - 4 The graph is lowered four units below its normal position

Can you find a pattern?The amount subtracted from the cos(x) is the number of units the graph will lower fromits normal position.

Predict the amount of units the graph will be raised or lowered for the followingequations. Graph each equation to check your answer.

y = cos(x) – 5 lowered 5y = cos(x) + 5 raised 5y = cos(x) – 2 lowered 2y = cos(x) +3 raised 3y = cos(x) + -(3/2) lowered (3/2)

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What can you conclude about shifting the graph vertically for sine and cosine functions?The number added or subtracted to a sine or cosine function determines the about of unitsthe graph will be raised or lowered.

Graph the function y = 2sin(2x) + 2. What is the direction and amount the graph wasshifted vertically from y = 2sin(2x)?It was raised 2 units.

Graph the following functions: y = sin(x) and y = sin(x +2p ). Draw on the graph paper

below. Describe what you see.

The graph was shifted to the left by(p/2). The graph was translated horizontally -(p/2)units.

Erase the graphs. Graph y = sin(x) and y = sin(x - p/2). Describe what happened.The graph was shifted horizontally to the right by p/2 units.

Graph the following equations. Identify which way the graph shifts and by how much.

Equation Shifty= sin(x + p/3) -p/3

y= sin(x -p) py= sin(x - p/2) p/2y= sin(x + 2p) 2p = 0

When the graph shifts horizontally either in a positive or negative direction it is called aPHASE SHIFT.

Predict the phase shifts for the following equations.

Equation Phase ShiftY= sin(x - p/3) p/3Y=sin(x + 3p/2) -3p/2

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Y=sin(x - p/4) p/4Y=sin(x +p) -p

How did you predict the phase shift?The number added to or subtracted to x determines the phase shift. The phase shift is theopposite sign of the number that is added or subtracted.

Take a look at cosine.Graph y = cos(x) and y = cos(x - p/2). Draw on the graph paper below. Describe the twographs.

y = cos(x-p/2) looks like it was shifted to the right p/2 units. It is a horizontal shift.

Erase the graphs. Graph y = cos(x) and y = cos(x + p/2). Describe what happened.y = cos(x + p/2) shifted horizontally p/2 units to the left. That is -p/2 units.

Graph the following equations. Identify which way the graph shifts and by how much.

Equation Phase Shifty = 2cos(x + p/6) -p/6y = 4cos(3x) 0y = 2cos4(x + 3p/4) -3p/4y = 3cos2(x - p/2) p/2

Can the phase shift of the sine and cosine be found the same way?Yes.

What is the phase shift of the graph of y = 2cos(x + p) – 1?-p

Write the equation for an image of y = cos(x) under a phase shift of -p/3. Graph theimage.

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Y = cos(x +p/3)

Summarize your results. Fill in the blanks.When an equations is written in the form y = a sin(bx + c) + da tells you the amplitude , b is used to determine the period, c determines the phase shiftand d shifts the graph vertically.

State the equation of the green graphs below.

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h x( ) = sin x( )

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f x( ) = cos x( )

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Homework Activity

In problems 1- 5 describe the shifts that takes place.

1. y = cos(x) +1

2. y = sin(x) – 2

3. y = sin(x -p/2)

4. y = cos(x + p/3)

5. y = sin(x-p/4) – 1

Sketch a graph of the function.6. y = cos(x-p/2) –17. y = sin(x - p)

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Day 4

1. The students will investigate the tangent function. They will determine theproperties of the tangent function and be able to graph the function with agraphing utility. They will also investigate vertical stretching, phase shifts,amplitude, period, and vertical shifts.

2. Concepts include: amplitude, period, vertical stretching, phase shifts, verticalshift, asymptote

3. Generalizations include: Atan(Bx + C) + D. determining period, determiningamplitude,

4. Procedures include: graphing functions


Lesson summaryThe students will investigate the tangent function. They will determine the

properties of the tangent function and be able to graph the function with a graphingutility. They will also investigate vertical stretching, phase shifts, amplitude, period, andvertical shifts. Students will be able to determine the equation of function from lookingat its graph.

MaterialsTeacher – overhead projector Students – class set of TI 83 Calculators

Transparency 1 & 2 worksheets

StandardsModeling/Multiple Representation – Students use mathematical modeling/multiplerepresentation to provide a means of presenting, interpreting, communicating, andconnecting mathematical information and relationships.


Consider the warm upLead a discussion about the results from yesterday in the computer lab. Focus on

phase shifts and vertical shifts. Put transparency one of the sine curve on the overhead.Using a marker draw another sine curve that has a vertical shift. Ask the students todetermine the equation of that curve and to explain their reasoning. Erase that curve anddraw one that has only a phase shift. Again have the students determine that equationand give a reason. Then draw one that has both a vertical shift and phase shift. Askstudents to determine an equation. Use transparency two of the cosine function and dothe same thing. Before you go on make sure the students can tell the difference betweenamplitude, period, phase shift, and vertical shift. Also, they should be able to concludethat vertical shifts and phase shift affect only the location of the graph, while amplitudeand period affect the shape of the graph.

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LessonIntroduce the tangent function a little differently. Hand the students worksheet one andgraphing calculators. Have them work with a partner to complete it. Allow a fewminutes to complete.

T: If we have a function and we know the value of x and the value of y can we plotit?

S: Yes.

T: Lets graph the function tan(x). Did anyone notice anything strange when theywere filling out the table?

S: Yes. When x = 2p and

23p the calculator says undefined.

T: What does it mean when the calculator says undefined?

S: A certain number doesn’t exist.

T: Lets try to graph this function. Draw the axes. This time we need a lot of roomon the x-axis. Label all the points on the x-axis. When you get to a point wherethe tan is undefined draw a vertical line.

Draw the tangent curve on the overhead along with the kids. Using the worksheet justcompleted ask them what is the tangent of a certain angle, and then plot it. Continue withall the angles in the same manner. Then connect the points to form the curve.

T: What can you tell me about the vertical lines where the graph is undefined?

S: The tangent curve looks like it gets closer and closer to those lines but nevertouches them.

T: That vertical line that you see is called an asymptote. How often does andasymptote occur?

S: Every p units.

T: When you think of tangent what do you think of?

S: opposite/adjacent.

T: Another way to describe tangent is sin/cos.When again do we have an asymptote in the tangent function?

S: When q = 3p/2 and p/2.

T: Keep that in mind. Take out your unit circle from the other day. Look at the sineand cosine when q = 3p/2 and p/2. Do you notice anything?

S: In both angles the cosine is zero.

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T: Exactly. The tangent function does not exist when cos(q) = 0. This makes sense

because we said tan = cossin . When a 0 is in the denominator of a fraction, that

number does not exist. Hence, the asymptotes.

Hand out worksheet two to the students. Have them work in pairs using the graphingcalculator for the rest of the period. Walk around the room and ask the students questionsto get them to reflect on their findings.

Wrap up the class by collecting the graphing calculators. To end class pose a finalquestion to sum up today’s lesson. Write it on the board. Have the students turn it in astheir ticket out of the room. Also pass out homework activity.

T: In order to leave the room each of you need to write down on a piece of paper toturn in 1) What is the period of the tangent function? Write an equation of the tangentfunction that has a period of p/2 and a vertical stretch of 3. 2) Answer true or false. A)The amplitude of y = tan(x) is 2. B) The period of y = tan(x) is the same as y = sin(x).C) To shift the tangent function up to units the equation would look like y = tan(x) + 2.

T: Complete hand out for homework.

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Transparency 1

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Transparency 2

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Worksheet 1

q tanq q tanq0 0

67p

31

6p

31

45p 1

4p 1

34p 3

3p 3

23p

undefined

2p

undefined 35p 3-

32p 3-

47p -1

43p -1

611p -

31

65p -

31 2 p 0

p 0

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Worksheet 2What is amplitude?It is half the difference between the maximum and minimum values.Amplitude is a measure of the height of a function.On your calculator graph y = 2sin(x). What is the amplitude? 2

Try y = -4cos(x). What is the amplitude?4

Now graph y = tan(x). What is the amplitude? Fill in the table. Draw on graph below.Equation Amplitude (1/2) (maximum value – minimum value)y = tan(x) (1/2)(?-?)y = 5tan(x) (1/2)(?-?)y = -2tan(x) (1/2)(?-?)

What can you conclude about the amplitude of a tangent function? Why?The amplitude of a tangent function does not exist. There is no maximum or minimumvalue.

Why can’t you find a maximum or minimum value for y = tan(x)?The graph continues to go on forever in a vertical direction. It never ends, so there is nomaximum or minimum value. There is no height.

In the following equation A stands for amplitude. y = Asin(Bx + C) + D .In this function y = Atan(Bx + C) + D, A no longer stands for amplitude. When wesubstitute a number in for A, what happens to the function?

Look at the graph of y = tan(x) and y = 2tan(x) on the same axes. Describe what yousee.

The line y = 2tan(x) looks like it was pulled tight. It looks stretched.

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Look at the graph for the following equations and fill in the table.Equation Descriptiony=3tan(x) Stretches the graph verticallyy=5tan(x) Stretches the graph verticallyy=-2tan(x) Stretches the graph vertically

Give the amplitude of the following functions.y = 3sin(x) + 2 amplitude = 3y = 3tan(x) + 2 amplitude = does not existy = 3cos(x) + 2 amplitude = 3

What is the period of a function?The length of one complete cycle of a function.

Graph y = sin(x). What is the period of the function?2p

Graph y = cos(x). What is the period of the function?2p

Graph y = tan(x). What is the period of the function?p

Given that y = Atan(Bx), the period of a function can be found by the period of the tan(x)divided by B. Fill in the table to find the period of the following functions.Equation Period p/By = 2tan(3x) p/3y = tan(2x) p/2y = tan(1/2x) 2p

Predict the period of the following functions.

Equation Periody = tan(1/3x) 3p

y = tan(4x) p/4

y = sin(4x) p/2

Write an equation of the tangent function that has a vertical stretch of 3 and a period of p.y = 3tan(x)

Write an equation of the tangent function that has a vertical stretch of 1/2 and a period ofp/2.

y = (1/2)tan(2x)

Write an equation of the tangent function that has a vertical stretch of –5 and a period of2p.

y = -5tan(1/2x)

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Graph the equation y = tan(x) along with y = tan(x) + 2 on the same axes. Copy on thegraph below. Compare the graphs.

What stays the same?Both functions have the same asymptotes.What is different?y = tan(x) + 2 has a y-intercept of 2. The graph of y = tan(x) + 2 is two units above they = tan(x)

Graph the following equations. Fill in the table. Compare each equation to the tan(x).Equation Descriptiony = tan(x) + 1 The graph is one unit above the tan(x)y = tan(x) + 3 The graph is three units above the tan(x)y = tan(x) + 0 The graph is equal to the tan(x)y = tan(x) + -1 The graph is one unit below the tan(x)y = tan(x) – 2 The graph is two units below the tan(x)y = tan(x) - 4 The graph is four units below the tan(x)

In the above examples when you added or subtracted a number to the tan(x), whathappened to the graph?The graph shifted vertically.

Predict the vertical shift from the parent graph y = tan(x) that will take place in thefollowing equations. Graph it to check your answer.

y = tan(x) + 4 shifts up 3 unitsy = tan(x) + 10 shifts up 10 unitsy = tan(x) – 8 shifts down 8 unitsy = tan(x) – 6 shifts down 6 units

Graph y = tan(x) and y = tan(x + p/2). Copy it to the graph below. Describe the shift thatyou see.

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The graph was shifted to the left by p/2 units.

Graph y = tan(x - p/2). What happens the to graph?The graph was shifted p/2 units to the right.

What is this horizontal shifting called?Phase shifts

Graph the following equations and identify which way the graph shifts and by how much.Equation Shifty = tan(x + 3p/2) To the left 3p/2 unitsy = tan(x - p/4) To the right p/4 unitsy = tan(x + p/3) To the left p/3 units

Graph y = tan(x + p). What happens? Why?The function looks like the y = tan(x). p is the period. The graph moved over onecomplete cycle, so it lies on top of itself.

Summarize your results.The amplitude of the tangent function is does not exist

The period of the tangent function is pIn y = Atan(Bx + C) + D the D affects the vertical shift and the C affects the phase shift

The graph below is the translation image of the graph of y = tan(x). y= tan(x) is the blackfunction. Find an equation for the yellow function and then the green function.

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Homework Activity

1. Where do vertical asymptotes occur in the tangent function between -2p and 2p?

2. What is the y-intercept for y = tan(x)?

3. Given y = 3tan(2x), what is the amplitude?

4. Graph one period of the tangent function on the back of this hand out.

5. Given y = tan(x + 3p ) + 1, describe the phase shift and vertical shift. Then graph

the function.

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Day 5Putting It All Together

1. Students will find the amplitude, period, phase shifts, and vertical shifts of sineand cosine functions. They will also graph sine and cosine functions with variousamplitudes, periods, phase shifts and vertical shifts. They will also find the period, phaseshift and vertical shifts of tangent functions.

2. Concepts include: amplitude, period, phase shifts, vertical shifts

3. Generalization include: y = AsinB(x + C) + D, y = AcosB(x + C) + D

4. Procedures include: graphing functions in stages


Day 5Putting It All Together

Lesson SummaryStudents will find the amplitude, period, phase shifts, and vertical shifts of sine

and cosine functions. They will also graph sine and cosine functions with variousamplitudes, periods, phase shifts and vertical shifts. They will also find the period, phaseshift and vertical shifts of tangent functions. A quiz will also be administered for the restof lesson.

MaterialsTeacher: TI 83 Graphing Calculator Students: class set of TI83 Graphing

TI View Screen calculators

StandardsModeling/Multiple Representation – Students use mathematical modeling/multiplerepresentation to provide a means of presenting, interpreting, communicating, andconnecting mathematical information and relationships.


Consider the warm upThe warm up ties up amplitude, period, phase shifts, and vertical shifts. It checks

the student’s knowledge. Explain that graphing more than two elements of a sine orcosine function can become very confusing. The easiest way to graph these complicatedequations is to work in stages, especially if you were graphing them by hand. However,we are going to work on the TI83 to guide us.

T: The general formulas for any sine or cosine function looks like this:y = AsinB(x + C) + D

y = AcosB(x + C) + DWho can tell me what A affects?

S: A affects the amplitude

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T: What does B affect?

S: B affects the period

T: What does C affect?

S: C affects the phase shift

T: What does D affect?

S: D affects the vertical shift.

T: What if we look at y = Atan(Bx + C) +D. Is the tangent function affect in thesame way as sine and cosine when we multiply by A and B and add C and D?

S: No. A affects the tangent function by stretching it vertically. There is noamplitude.

S: B affects the period. However, the period of the tangent function is p.

S: C and D, the phase shift and vertical shift affect the location of the tangent justlike it does in sine and cosine functions.

T: We are going to graph some functions in stages. We’ll start with the basic sine orcosine function. If there is an A in the equation we’ll start there. If not we’ll skipthat step and move to B. C and D will follow. We are going to go through eachstage so you can really see what is happening.

The Teacher should be using the TI View screen on the overhead to work along withstudents. It may help to change the style of each line when you graph each function soyou can distinguish between them easily.

T: Lets graph the equation y = cos(2x - p) over the interval (0, 2p)Graph the basic function of the equation. What should I being graphing?

S: y = cos(x)

T: So graph y = cos(x). Remember our general formula for a cosine function is(have it on the board)

y = AcosB(x + C) + D

What does the A stand for? Does that affect our equation?

S: A affects the amplitude.

S: No. There is no amplitude change in our function.

T: Lets move on. Do we have a B in our equation? What does it tell us?

S: There is a B in our equation. It tells us there is a period change.

T: Before we graph y = cos(2x), who can tell me what they think is going to happen?

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S: Our period is going to be p.

S: The graph we originally had y = cos(x) will be squeezed so that instead of havingone period from 0 to 2p, it will have one period from 0 to p.

T: Great! Lets see if you’re right. You are!!! Do we have a C in our equation?

S: Yes. A phase shift of p/2 units.

T: Before we graph y = cos(2x - p/2), who can tell me what they think is going tohappen?

S: Each point on the graph will simply move p/2 units to the right.

T: Why to the right?

S: To the right, because we are subtracting.

T: Well done. That is all there is no vertical shift.

T: Lets do another one. Lets break apart the function y = tan(2x + p/2) - 1 from-2p to 2p

Again ask the students questions like what do they think will happen will happen to thegraph when you do a certain move. Make sure they tell you why.

Do one more problem. Y = 3sin(x - p/4) + 1. This time split the students into pairs.Have them go through each step and describe what is going on. Walk around the roomand listen to make sure students are using math language (amplitude, period, phase shift,vertical shift, etc).

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LessonA quiz takes the place as the rest of the lesson. Pass out quiz. Make sure enough time is

given to complete it. Collect quiz at end of period.

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Quiz

Name

Answer all questions. You may use your TI 83 graphing calculator.

Consider triangle ABC below.

C

17 8

A B 15

1. Find sin C exactly. 15/172. Find tan B exactly 8/153. Find m– B to the nearest tenth of a degree. 28.1°

4. Find the sin and cos of p/3. sin(p/3) = 3/2 and cos(p/3) = 1/2

5. What is amplitude? Vertical stretching of a function. The difference betweenthe maximum and minimum values.

Write an equation that has an amplitude of 3. y = 3sin(2x)

6. What is period? The length of one complete cycle. Consider the function y = (1/2)sinx. What is the period? 2p

7. Sketch one cycle of the graph. Y = 2sin2x

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2

h x( ) = 2?sin 2?x( )

8. Consider the function y = sin(x - p/2). What is the phase shift from its original curve y = sin(x). p/2

9. Name the phase shift and vertical shift of the graph y= 3cos(x + p) - 2phase shift = - p vertical shift = down 2

10. Explain why or why not tangent graphs have an amplitude.They do not have an amplitude because they go on forever in vertical directions,so they cannot have a height.

scale changes and translations of circular...

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