This is a one-day unit combining the use of the TI-82 graphing calculator, its overhead projector calculator, the overhead projector itself, and everyday classroom tools including graph paper especially suited for graphing sinusoids. A second day which uses the computer--Sunburst's "Green Globs", a computer program which has the student type various functions into the computer and observe the computer graph them--could follow the first day. "Green Globs" has games for the student to play; these games have the student graph various functions, including sinusoids, to intersect various points on the Cartesian plane. The more points a student can intersect with one function, the more points he/she can score. The major emphasis here will be the lesson using the TI-82 graphing calculator. The student can certainly graph (and should) some of the basic sinusoids by hand, using a table for x- and y- coordinates, but through the use of the graphing calculator will be able to observe what happens when the values of a, b, c, and d are changed in the sinusoid function "y=a sin b (x-c) + d"
In this guide, I have included the worksheets for the students to use as they graph the functions on the graphing calculator. There are more than enough problems for students to try, and they could make up some of their own. They should write their conclusions on the provided paper to turn in at the end of the period.
LEVEL; ninth through twelfth grade--any student who is enrolled in trigonometry
MATERIALS NEEDED: TI-82 graphing calculators, the overhead projector and the graphing calculator used to demonstrate, graph paper
Students will work as a whole class, as an individual, and in small assigned groups
LINKS TO THE NCTM STANDARDS:
Standard 2: Mathematics As Communication "formulate mathematical definitions and express generalizations discovered through investigations" "express mathematical ideas orally and in writing"
Standard 5: Algebra "use tables and graphs as tools to interpret expressions, equations, and inequalities"
Standard 6: Functions "represent and analyze relationships using tables, verbal rules, equations, and graphs" "translate among tabular, symbolic, and graphical representations of functions" "analyze the effects of parameter changes on the graphs of functions"
Standard 9: Trigonometry "apply general graphing techniques to trigonometric functions"
This lesson should be an effective means to teach students about sinusoidal variation. They should, by the end of the lesson, understand how to graph y=a sin b (x-c) + d using the concepts of period, amplitude, phase shift, and vertical shift. This will also extend their knowledge of graphing calculators and Sunburst's "Green Globs". It will give them an opportunity to work both on their own independently and in groups. All students will have had some previous classroom experience with graphing calculators. Depending upon the makeup of the class--whether composed mostly of average students or advanced students--this lesson with the graphing calculator could take one day or two days.
NAME _________________________________ DATE __________________
CLASS _________________________
Our goal is to learn more about graphing "sinusoids"--the sine and cosine functions and their variations. We will be using the TI-82 graphing calculator today to help us observe patterns and make conclusions. Primarily, work will be done in your assigned group of three. Take notes (if you need to) from the graphing calculator examples I will be demonstrating on the overhead. Then work with your group--and try some on your own--as you work through the problems. Although your group will discuss your conclusions together, each person should complete and turn in their worksheet for a grade. We will discuss your conclusions tomorrow and you will have a chance to try some graphs on the computer using "Green Globs". What you learn about graphing sinusoids will be included on the next quiz.
1. Graph each of the following on your calculator.
y=sin x
y=2 sin x
y=5 sin x
y=10 sin x
Make a conclusion. _____________________________________________________
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What do you think “amplitude” means? ___________________________________
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Now do #1 again, but replace “sin” with “cos”. Does your conclusion change, or do you think it will apply here also? __________________________
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Now graph the following:
y=sin x
y=(1/2)sin x
y=(1/4)sin x
y=-sin x
y=-2sin x
Make a conclusion. _____________________________________________________
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Try the same four graphs again, only replace "sin" with "cos". Does your previous conclusion change? Explain.
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2. Graph each of the following on your calculator. Then repeat the process using "cos" rather than "sin".
y=sin x
y=sin x + 2
y=sin x - 3
y=sin x + 4
Make a conclusion. _____________________________________________________
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What do you think "Vertical Shift" means?
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3. Graph each of the following on your calculator.
y=sin x
y=sin (2x)
y=sin (5x)
y=sin (1/2)x
Now do these four graphs again, replacing "sin" with "cos". Make a conclusion.
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Remembering that for y=sin x and for y=cos x, the amount necessary for one complete motion is 360 degrees or 2 pi radians, try to make another conclusion. Make up more examples and try them if you need to.
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4. Graph each of the following on your calculator.
y=sin x
y=sin (x-3)
y=sin (x+2)
y=sin (x+1)
y=sin (x-4)
Now, do these five graphs again, replacing "sin" with "cos". Then try them again, but use increments of pi, such as pi/2, pi, 3pi/2. Make a conclusion.
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5. Let's put it all together. Graph each of the following on your calculator.
y=sin x + 1
y= 3 sin x + 1
y= 3 sin (x-pi) + 1
y= 3 sin 2(x - pi) + 1
Try it again using "cos" instead of "sin".
Now, try one more, and then repeat it using "cos".
y= sin x - 1
y=-2 sin x - 1
y= -2 sin (x + (pi/2)) - 1
y = -2 sin 3(x + (pi/2)) - 1
From what you have observed, define the following:
Amplitude:
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Vertical Shift:
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Phase Shift:
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Period:
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