Investigating Trigonometric Graphs by azvbG4

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									                                        Grade 10 Mathematics Extended
                                    Global Warming – Is it Really Happening?

Guiding Question: How can I use trigonometric models to examine global warming? (AOI: Environment)

Sine and cosine functions can be used to model cyclical trends. Examples include the seasonal variation of demand for
certain items, the cyclical nature of recession and prosperity, average daily temperatures, and tidal patterns.
(http://people.hofstra.edu/stefan_Waner/RealWorld/trig/trigintro.html, accessed Nov 6, 2007). This assignment will
consist of three parts, each marked separately. You will first be investigating trigonometric graphs and their
transformations using your graphing calculator. You will then use the knowledge you gain to produce a mathematical
model for real-life data. Finally, you will use trigonometric models to explore the effects of global warming.

Part 1 – Investigating Trigonometric Graphs.
You will be given 80 minutes in class to complete this assignment.
You will be assessed on MYP Criteria B – Investigating Patterns
                                          C – Communication in Mathematics
                                          D – Reflection in Mathematics (to be completed on moodle)
You will work through the activity to investigate each family of graphs and record your results.
You are then expected to write a short paragraph summarizing your results.
You will need to describe the effects of A, B, C, and D in the graph of:
                 y = A sin [B(x – C)] + D and y = A cos [B(x – C)] + D
Questions to consider:
     Have I explained all the patterns and described them as relationships or general rules?
     Have I given justifications or proofs as to why I think my generalizations work?
     Have I presented all my information clearly and logically?
     Have I clearly explained my reasoning?
     Have I used a wide range of mathematical symbols and language?

Part 2 – Using a Trigonometric Function to Model Data: Due Monday, January 18th.
You will complete this as a homework assignment.
You will be assessed on MYP Criteria A – Knowledge and Understanding
                                          C – Communication in Mathematics
                                          D – Reflection in Mathematics
You will be working with the following situation:
  The depth of water at your favourite snorkelling spot varies from 1.2 meters to 4.4 meters, depending on the time.
  Last Sunday, high tide occurred at 5:00 am and the next high tide occurred at 6:30 pm. Create a model to describe
  the depth of water as a function of time t in hours since midnight on Sunday morning. Create a second model using
  data from the depth of water at your other favourite snorkelling spot, where the tide last Sunday varies from a low
  of 1.4 meters at 4:00 am to a high of 3 meters at noon.
  (http://people.hofstra.edu/stefan_Waner/RealWorld/trig/trigintro.html, accessed Nov 6, 2007)
You will hand in a one to two page written report on your findings. You are expected to explain the situation and
the way you came up with your models. You will be assessed on your mathematical content as well as the clarity of
your explanations. Remember to write as if the reader does not know the problem. In particular, you should focus
on carefully reflecting and evaluating on all your work.
Questions to consider:
     Have I shown that I thoroughly understand the mathematics involved?
     Have I presented all my information clearly and logically?
     Have I clearly explained my reasoning and my solution to the problem?
     Have I used a wide range of mathematical symbols and language?
     Have I discussed why I chose the methods and processes I used?
     Have I discussed other ways in which I could have approached the problem and why I chose not to use those
         ways?
     Have I suggested improvements to my method?
     Have I discussed how I know that my model is correct?
     Have I discussed the accuracy of my model in the context of the problem?
     Have I discussed the significance of my findings?
     Have I given a detailed explanation of my findings in connection to real life?
Part 3 – Using Trigonometric Models to Examine Global Warming – Due Tuesday, January 26th.
You will be assessed on MYP Criteria A – Knowledge and Understanding
                                       C – Communication
                                       D – Reflection and Evaluation

Your task is to do research on temperature trends over time and determine whether global warming is happening.
You will need to find appropriate data that you can examine. You will then need to come up with models that you
think fit this data. You should use technology as well as your mathematical skills to produce these models. You will
then need to analyze your different models and make a judgment on the seriousness of global warning.

This is an open-ended problem with a number of different approaches, so it gives you a lot of opportunity for
reflection and evaluation.

To get you started thinking:
The table below gives the 24-hour Average Temperature at Dar es Salaam Airport over an entire century.

      Jan     Feb    Mar     Apr     May     Jun     Jul     Aug    Sep     Oct    Nov     Dec     Year
°C     27.5   27.6    27.2    26.2    25.1    23.9    23.3   23.5    24.0   25.1    26.2    27.3    25.6
°F     81.5   81.7    81.0    79.2    77.2    75.0    73.9   74.3    75.2   77.2    79.2    81.1    78.1

Source: DAR ES SALAAM AIRPORT data derived from GHCN 2 Beta. 607 months between 1895 and 1991
(http://www.worldclimate.com/, accessed Nov 6, 2007)

What are your variables? What is an appropriate domain and range for this data? How well does this data fit a
trigonometric model? Can you come up with a trigonometric model by hand? Can you use technology to produce a
better model? (Hint: How do you find a regression line of best fit on the calculator?)

Would this example give a good idea of global warming? Do you think Dar es Salaam is a good indicator? Why or
why not? Is there other data which might be better? What additional data might you need to make a judgment on
the effects of global warming?

Some questions to consider:
    Have I collected appropriate data?
    Have I shown that I thoroughly understand the mathematics involved?
    Have I been able to make appropriate conclusions in this situation based on my mathematical knowledge?
    Have I used different suitable ways of representing my ideas (tables, graphs, figures, formulas, etc.)?
    How have I used technology effectively to solve the problem?
    Have I used technology effectively to present my information?
    Have I presented all my information clearly and logically in report format?
    Have I clearly explained my reasoning and my solution to the problem?
    Have I used a wide range of mathematical symbols and language?
    Have I discussed why I chose the methods and processes I used?
    Have I discussed other ways in which I could have approached the problem and why I chose not to use those
      ways?
    Have I suggested improvements to my method when needed?
    Have I discussed how I know that my models are correct?
    Have I explained whether my results make sense in the problem situation?
    Have I discussed the accuracy of my models in the context of the problem?
    Have I discussed the significance of my findings in relation to trends in global warming?
                                          Grade 10 Mathematics Extended
                                      Global Warming – Is it Really Happening?
                                      Name ____________________________

Criterion A: Knowledge and Understanding
Level Descriptors                                                                       Part Part Part
                                                                                         1    2    3
  0     I have not met any of the descriptors
 1-2    I attempt to make conclusions when solving simple problems in familiar
         situations.
 3-4    I sometimes make appropriate conclusions when solving simple and more
         complex problems in familiar situations.
 5-6    I generally make appropriate conclusions when solving challenging problems
         in a variety of familiar situations.
 7-8    I consistently make appropriate conclusions when solving challenging
         problems in a variety of situations including unfamiliar ones.
       Comments:




Criterion B: Investigating Patterns
Level Descriptors                                                                       Part Part Part
                                                                                         1    2    3
  0     I have not met any of the descriptors
 1-2    With some help, I apply mathematical problem-solving techniques to
         recognize simple patterns.
 3-4    I choose and then apply mathematical problem-solving techniques to recognize
         patterns
        I suggest relationships or general rules.
 5-6    I choose and then apply mathematical problem-solving techniques to recognize
         patterns
        I describe the patterns as relationships or general rules
        I make conclusions that fit the results.
 7-8    I choose and then apply mathematical problem-solving techniques to recognize
         patterns
        I describe the patterns as relationships or general rules
        I make conclusions that fit the results.
        I justify or prove my mathematical relationships or general rules.
       Comments:
Criterion C: Communication In Mathematics
Level Descriptors                                                                            Part Part Part
                                                                                              1    2    3
  0     I have not met any of the descriptors
 1-2    I use basic mathematical language (notation, symbols, vocabulary) and/or ways
         of representing mathematical ideas (formulae, diagrams, tables, charts, graphs,
         models).
        My reasoning is difficult to follow.
 3-4    I use sufficient mathematical language (notation, symbols, vocabulary) and
         ways of representing mathematical ideas (formulae, diagrams, tables, charts,
         graphs, models).
        My reasoning is clear but not always logical or complete.
        I move between different forms of representation with some success.
 5-6    I use good mathematical language (notation, symbols, vocabulary) and ways of
         representing mathematical ideas (formulae, diagrams, tables, charts, graphs,
         models).
        My reasoning is to the point, logical and complete.
        I move effectively between different forms of representation.
       Comments:




Criterion D: Reflection in Mathematics
Level Descriptors                                                                            Part Part Part
                                                                                              1    2    3
          I have not met any of the descriptors
          I try to explain whether my results make sense in the problem situation.
          I try to describe the importance of my findings in connection to real life.
          I correctly but briefly explain whether my results make sense in the problem
           situation.
          I describe the importance of my findings in connection to real life.
          I try to justify the degree of accuracy in my results, where appropriate.
          I critically explain whether my results make sense in the problem situation.
          I give a detailed explanation of the importance of my findings in connection to
           real life.
          I justify the degree of accuracy in my results, where appropriate.
          I suggest improvements to my method when needed.
       Comments:
                                 Grade 10 Extended Mathematics
                               Investigating Trigonometric Graphs

Here is some terminology that might be helpful in your investigation.
Periodic function: a function that repeats its values in regular intervals or periods.
Period: how many degrees it takes to complete one cycle.
Frequency: how many cycles there would be in 360°.
Amplitude: how far up and how far down the graph goes from the center line.
Translation: a transformation that "slides" an object in a fixed distance and in a given direction.
   The original object and its translation have the same shape and size, and they face in the same direction.
Stretch: Just what it sounds like!



              2
                                                       center line

                   amplitude




                                          amplitude

              -2



Task 1 – Investigating y = sin x and y = cos x

Set your WINDOW as follows:           Y min = -2              Sketch the graphs of y = sin x and y = cos x
                                      Y max = 2               on the set of axes below.
                                      Y scl = 1
                                      X min = -360
                                      X max = 360
                                      X scl = 90
                                                  y




                                                                                               x
                                                                                                   Degrees




Find the period and amplitude of these graphs:
y = sin x                            y = cos x

Period ________________               Period ___________________

Amplitude ____________                Amplitude ________________

Explain how the graphs of y = sin x and y = cos x are related to each other.
Task 2 – Investigating y = -sin x

Graph y = sin x and y = - sin x on the same set of axes
                                            y




                                                                                     x

                                                                                         Degrees




      What effect does the negative have on the graph?




      Can you give a justification or proof as to why you think this is the case?




Task 3 – Investigating graphs in the form y = A sin x
                                                 1
Compare the graphs of y = sin x, y = 2 sin x, y = sin x on the axes below.
                                                 2
                                            y




                                                                                     x
                                                                                         Degrees




      Explore y = A sin x for other values for ‘A’. Make sure you include a good range of values.
      Explain what happens to y = sin x as A changes. Describe any patterns you see as relationships or
       general rules.




      Can you give a justification or proof for your relationship or rule?
Task 4 – Investigating graphs of y = sin x + D

Compare the graphs of y = sin (x), y = sin (x) + 1 and y = sin (x) – 2 on the axes below.
                                              y




                                                                                            x
                                                                                                Degrees




      Explore y = sin x + D for other values of ‘D’. Make sure you include a good range of values.
      Explain what happens to y = sin x + D as ‘D’ changes. Describe any patterns you see as
       relationships or general rules.




      Can you give a justification or proof for your relationship or rule?




Task 5 – Investigating y = sin Bx
                                                           1
Compare the graphs of y = sin x, y = sin 2x, and y = sin     x on the axes below.
                                                           2
                                              y




                                                                                            x

                                                                                                Degrees




      Explore y = sin Bx for other values of ‘B’. Make sure you include a good range of values.
      Explain what happens to y = sin Bx as ‘B’ changes. Describe any patterns you see as relationships
       or general rules.




      Can you give a justification or proof for your relationship or rule?
Task 6 – Investigating the graphs of y = sin (x – C)

Compare the graphs of y = sin x, y = sin (x -90º), y = sin (x - 45º) on the axes below.
                                               y




                                                                                          x
                                                                                              Degrees




      Explore y = sin (x – C) for other values of ‘C’. Make sure you include a good range of values.
      Explain what happens to y = sin (x – C) as ‘C’ changes. Describe any patterns you see as
       relationships or general rules.




      Can you give a justification or proof for your relationship or rule?




Summary
What would happen if you combined values? Describe the effects of A, B, C, and D in the graph of:
               y = A sin [B(x – C)] + D
Make sure you include positive and negative values of A, B, C and D in your summary.
Also, be sure to include justification as to why you think your generalizations work.




If Time: Investigate y = A cos [B(x – C)] + D and describe the effects of A, B, C, and D on the graph.
What are the similarities and differences with the sine function?
Drawing Conclusions – NO CALCULATORS ALLOWED!

1. Use your results from above to find the period and amplitude of these graphs.
   Then explain how to transform y = sin x to form these graphs.
   Finally, sketch the graph.

a) y = 2 sin x – 1
   period ____ amplitude ____                 Transformations: _________________________________________

                                 3   y


                                 2


                                 1

                                                             x

 - 360   - 270   - 180   - 90            90      180   270   360

                                -1


                                -2


                                -3


                                -4


                                -5




b) y = - 1/2sin (x - 45º)
   period ____ amplitude ____                 Transformations: _________________________________________
                                     y




                                                                   x




c) y = 0.5 sin 2x
  period ____ amplitude ____                  Transformations: _________________________________________
                                     y




                                                                   x




d) y = sin (x/2) + 2
   period ____ amplitude ____                 Transformations: _________________________________________

                                 3   y


                                 2


                                 1

                                                             x

 - 360   - 270   - 180   - 90            90      180   270   360

                                -1


                                -2


                                -3


                                -4


                                -5
e)       y = 3sin (x + 90º) – 2
          period ____ amplitude ____                 Transformations: _________________________________________

                                        3   y


                                        2


                                        1

                                                                      x

 - 360      - 270       - 180   - 90            90      180   270     360

                                       -1


                                       -2


                                       -3


                                       -4


                                       -5




f)       y = -sin (x - 90º)
          period ____ amplitude ____                 Transformations: _________________________________________

                                        3   y


                                        2


                                        1

                                                                      x

 - 360      - 270       - 180   - 90            90      180   270     360

                                       -1


                                       -2


                                       -3


                                       -4


                                       -5




2. For each of these graphs determine the: a) period and amplitude and
                                           b) the sine or cosine equation of the graph
1.                                                      3.
                    y                                                                y




                                                              x                                                           x




Period ____             Amplitude ____                                      Period ____   Amplitude ____

Equation of the graph _____________________                                 Equation of the graph _____________________




2.                                                                          4.
                    y                                                                y


                                                                  x




                                                                                                                          x




Period ____             Amplitude ____                                      Period ____   Amplitude ____
Equation of the graph _____________________                                 Equation of the graph _____________________
                               10   y




                                5




                                                         x

- 360   - 270   - 180   - 90            90   180   270   360




                                2   y

                                1
                                                         x

- 360   - 270   - 180   - 90            90   180   270   360

                               -1

                               -2

                               -3

                               -4

                               -5

                               -6

                               -7

                               -8

								
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