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  • 8/18/2019 Graphing Sine and Cosine (1)

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    Lesson Plan

    Graphing Sine and Cosine

    Lynn Weiss

    Goal:

    Students will use technology to discover how to graph trigonometric functions. They will work

    cooperatively to model sine and cosine functions after real world applications.

     Prerequisites:

    Students should have a basic knowledge of the unit circle. They should know how to graph

     y = sin(x) and y = cos(x). They should understand basic trigonometric definitions such as

    amplitude, period, and shift.

     Materials:

    Students should have a T !raphing "alculator (T#$%, T#$&, or T#$').

     Procedure:

     n past years have used a Spaghetti ab where students work in groups to discover the graphs

    of y = sin(x) and y = cos(x). They use butcher paper and spaghetti along with their knowledgeof right triangle trig and the unit circle to graph the functions. This lesson will begin on the

     school day following the completion of this lab.

     Day 1:.) *eview graphing y = sin(x) and y = cos(x). +e sure that students know the amplitude,

     period, domain and range of the functions. iscuss how to graph the functions in both

    radian and degree mode.%.) istribute the worksheet -!raphing "alculator xploration, !raphing Sine and

    "osine/. 0ave students work individually to complete the worksheet. They should

    discover how given the e1uation y = a sin b(x#c)2 d, changes in a, b, c, and d effect the

     graph.&.) !o over the answers to the worksheet as a class. 0ave students volunteer to explain

    their solutions.

     Day 2.) !ive a pop partner 1ui3 to assess students4 knowledge of graphing sine and cosine. The

    1ui3 will ask them to graph a function and to determine the e1uation of a given graph.

    %.) istribute index cards to all students in the class. ach classroom table will have a signwith an -answer/ on it. The students will use their knowledge of the unit circle to find

    other members of their group. 5or instance a student with -sin &67/ and a student with

    -cos 867/ will each sit at the table with 9 as the answer. There will be a maximum of

     four students per group.

    &.) istribute the 5erris :heel pro;ect to each student. 0ave them work with their group to

    complete the pro;ect.

     Day 3

    1.) Students will need additional time to work on the pro;ect. "ollect the pro;ects. Studentswill be assigned both a group and individual grade based on their effort in completing

    the pro;ect.

    2.)  istribute the 5erris wheel extra credit assignment to those groups who desire achallenge.

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    Algebra Trig

    Graphing Calculator Exploration

    Graphing Sine and Cosine

    Today we are going to use our graphing calculators to experiment with changing the

    values of a, b, c and d in the general equation:y a ! sin "b"x # c)) $ d.

    %ur goal is to discover how a, b, c, and d effect the graph.

    &irst, use what you learned during the 'paghetti (ab, and graph y sin"x)

    and y cos"x) from * to +* below.

    y sin"x)

    -hat are the values of a, b, c, and d

    a b c d

    y cos"x)

    -hat are the values of a, b, c, and d

    a b c d

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    !/efore we begin0

    To what mode should our calculator be set

    emember: The general equation is

    y a ! sin "b"x # c)) $ d

    1.) 'et your window to view x from to + and y from #+ to +.

    raph

    1

    2

    +

    sin" )

    2sin" )

    "132)sin" )

    < x

    < x

    < x

    =

    =

    =

    4s the absolute value of a increases, 0

    4s the absolute value of a decreases,0

    5n the general form of the equation, changes in a impact the

     666666666666666666 of the graph.

    2.) 'et your window to view x fro to 72 and y from #2 to 2.

    raph

    1

    2

    +

    sin" )

    sin"2 )

    sin"13 2)" )

    < x

    < x

    < x

    =

    =

    =

    4s b increases00

    4s b decreases00.

    4 change in b alters the 666666666666666 of the graph of the equation.

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    +.) 'et your window to view x from #8 to 9 and y from #2 to 2.

    raph

    1

    2

    +

    sin" )

    sin" 8 )

    sin" 9 )

    < x

    < x

    < x

    =

    = −

    = +

    o

    o

    -hen c is positive"x # c), the graph 0.

    -hen c is negative"x $ c), the graph0.

    4 change in c causes a 66666666666666666666666 666666666666666666666in the graph.

    .) 'et your window to view x from to + and y from # to .

    raph

    1

    2

    +

    sin" )

    sin" ) 9sin" ) +

    < x

    < x< x

    =

    = +

    = −

    -hen d is positive, the graph0

    -hen d is negative, the graph0

    4 change in d causes a 6666666666666666666666666 6666666666666666 in

    the graph.

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     ;ow, use what you learned to complete the following.

    1.)

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    -rite a sine and cosine equation for each given graph.

    5n general, when given a graph, how do you determine the values of a and b

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    Algebra Trig. Good Luck to _______________________

    Weiss

    Pop Quiz on Sine and Cosine Graphs Period: ________________

    12 points – NO CALCULATOR

    For #1 and #2, graph one full cycle. Be sure to label your axes and the five critical

    points.

    1. 2sin"+ ) y x=

    4mplitude =eriod

    2. cos" ) 2 y x= −

    4mplitude =eriod

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    For #3 and #4, write a sine and cosine equation to match the given graph.

    3.)

    4.)

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    Ferris Wheel Comparison Project

    The 1>8+ ?hicago -orld@s &air is considered the birthplace of the classic amusement

     parA ride, the &erris wheel. The architectural wonder was created by an 4mericanengineer named eorge &erris. The original &erris wheel no longer exists. /ut, in 188,

    a new &erris wheel was built at ;avy =ier in ?hicago to resemble the original. -hile the

     ;avy =ier &erris -heel is a beautiful ?hicago landmarA, its grandeur actually pales incomparison to Br. &erris@ creation.

    The &erris wheel built for the -orld@s &air had a diameter of 29 feet. 5t stood 1 feet off the ground. 5t had + wooden boxcars that were the siCe of train cars. Dach car could

    hold peopleE The wheel would load cars in such a way that each rider could enFoy a

    full rotation that lasted about 1 minutes.

    The &erris wheel at ;avy =ier has a diameter of 1 feet. 5t stands 1 feet off the ground.

    The wheel has gondolas that seat six passengers each. 5t taAes about minutes for the

     ;avy =ier &erris -heel to complete one rotation.

    /elow is a picture of the first &erris wheel next to the &erris wheel at ;avy =ier.

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    5n the space below is a diagram of the -orld@s &air &erris wheel and the boarding

     platform. &ill in the necessary information.

     ;ow, complete the table below. (et h represent your vertical position "height) at time t

    where t is given in minutes. emember, you need 9 critical t#axis values.

    t h

    d

    h

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    /efore you sAetch the cosine and sine graphs, answer each of the following.

    1.) -hat is the amplitude of the graph 66666666666666666666666666666 

    a 6666666666 

    2.) -hat is the period of your curve 6666666666666666666666666666666 

    "emember2

    b period 

    π  = )

     b 6666666666 

     

    +.) -hat are your 9 critical t#values 666666666666666666666666666666666 

    .) -hat are the maximum and minimum values Bax 66666666 Bin 66666666 of your curve "/e carefulE)

    d 6666666666 

    9.) -hat is the height at time "y#intercept) 66666666666666 

    -rite a cosine equation for your curve.

    "emember there are an infinite number of possible answers)

     6666666666666666666666666666666666666 

    raph the equation.

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    -rite a sine equation for your curve.

     6666666666666666666666666666666666 

    raph the equation.

     ;ow, answer the following questions. Gou may want to graph your equation on your

    graphing calculator.

    1.) -hat is the circumference of the wheel

    2.) 4t what speed is the wheel traveling =lease give your answer in feet3second.

    +.) 5f you begin your ride at the base of the wheel, what is your height after0

    a.) 1 minute b.) minutes

    .) 4t what approximate time"s) will you reach the following heights

      a.) 1 ft. b.) 2 ft.

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     ;ow, repeat each step for the ;avy =ier &erris -heel.

    5n the space below is a diagram of the ;avy =ier &air &erris -heel and the boarding

     platform. &ill in the necessary information.

     ;ow, complete the table below. (et h represent your vertical position "height) at time t

    where t is given in minutes. emember, you need 9 critical t#axis values.

    t h

    d

    h

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    /efore you sAetch the cosine and sine graphs, answer each of the following.

    1.) -hat is the amplitude of the graph 66666666666666666666666666666 

    a 6666666666 

    2.) -hat is the period of your curve 6666666666666666666666666666666 

    "emember2

    b period 

    π  = )

     b 6666666666 

     

    +.) -hat are your 9 critical t#values 666666666666666666666666666666666 

    .) -hat are the maximum and minimum values Bax 66666666 Bin 66666666 of your curve "/e carefulE)

    d 6666666666 

    9.) -hat is the height at time "y#intercept) 66666666666666 

    -rite a cosine equation for your curve.

    "emember there are an infinite number of possible answers)

     6666666666666666666666666666666666666 

    raph the equation.

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    -rite a sine equation for your curve.

     6666666666666666666666666666666666 

    raph the equation.

     ;ow, answer the following questions. Gou may want to graph your equation on your

    graphing calculator.

    1.) -hat is the circumference of the wheel

    2.) 4t what speed is the wheel traveling =lease give your answer in feet3second.

    +.) 5f you begin your ride at the base of the wheel, what is your height after0

    a. 1 minute b.) minutes

    .) 4t what approximate time"s) will you reach the following heights

      a.) 1 ft. b.) 2 ft.

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    Ferris Wheel Extra Credit

    1.) 5magine the ;avy =ier and the -orlds &air &erris -heel being built beside eachother. 5f both wheels begin turning at once, over a 2 minute time period, at what

    times are the wheels at the same height

    2.) -hat is the length of the arc traveled by the ;avy =ier &erris wheel from the

    o@clocA to the 7 o@clocA position