Lesson 3 of 3
Use extremes, timing, and a starting observation to write and check an equation.
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Big ideas · 1 of 8
Big ideas
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A sinusoidal model with maximum M and minimum m has midline and amplitude . Output units carry through to both quantities.
The PeriodThe shortest positive input shift that repeats the whole pattern.Exactly: A fundamental period satisfies wherever defined, with no smaller positive period.Mix-up alert: For a nonconstant transformed sine or cosine, the period is ; for tangent it is .See this in Key words P is the input distance between consecutive matching cycle positions, such as one maximum and the next. The angular coefficient magnitude is ; maximum to minimum is only half a cycle.
A cosine model with positive A and B starts at a maximum when ; negative A starts at a minimum. A positive sine with rises through its midline at .
A useful check tests the starting point, one later point, and the range. Different formulas can describe the same curve. State input/output units and the observation interval rather than assuming the pattern lasts forever.
Step-through
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A buoy height is modeled between 2 m and 8 m, repeats every 12 s, and starts at its minimum.
Whole picture, 4 steps
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Worked example
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Worked example
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Mix-up alert
Do this: Explain the wrong idea and the repair.
Wrong idea: the time from one maximum to the next minimum is the full period. Repair: those positions are half a cycle apart. Double that time before finding .
Tip
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Explain why observing just a maximum value is not enough to determine a unique sinusoidal model.
Worked example
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A model has minimum 1, maximum 7, and period 8, starting at its minimum.
Watch
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Flashcards
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Card 1 of 8
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The midline is halfway between the extremes.
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How do you find the midline from maximum and minimum?
Average them: .
How do you find amplitude from maximum and minimum?
Take half their difference: .
What is the time between consecutive maxima?
One full period.
What is the time from a maximum to the next minimum?
Half a period.
Which unshifted positive-A function starts at its maximum?
Cosine.
Which unshifted positive-A, positive-B function starts at the midline rising?
Sine.
How can cosine start at a minimum?
Use a negative outside coefficient or an equivalent phase shift.
Why check a model at more than one input?
To test phase and timing as well as the starting height.
True or false
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Self-test
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6 questions. One at a time.
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Self-test
Do this: Work on paper, choose an answer, then read the feedback.
6 questions. One at a time.
Practice, not a grade. No timer. Your best score stays in this browser.
Write it
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Write it
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Match
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Connect the ideas
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