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## Algebra 2

### Course: Algebra 2>Unit 12

Lesson 2: Interpreting features of functions

# Periodicity of algebraic models

Sal analyzes the periodicity of graphs that model real world situations.

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• It has occurred to me that the sine could be defined in terms of periodic motion of a spring, where F = -kx is proportional to the negative of the displacement, and results in a periodic displacement function. I suspect that it has been proven at some point in the past that this is the same function as the sine (the y coordinate) in the unit circle, since this is how we calculate the displacement of an oscillating spring + weight. However, how do we know they are the same? What is the proof of this?
Intuitively it seems that there is a simple relationship between these two kinds of motion, circular and oscillating.

With gravity, acceleration is constant, velocity is a linear function v = at + s, and distance is quadratic.
With an oscillating spring, acceleration is a linear function of and in the opposite direction of displacement. But the displacement itself oscillates, so the acceleration also oscillates. Then so does the velocity. I imagine all 3 of these are sine functions.
Is there an exponential function (with a common factor of displacement, or something) somehow involved in this? • I used to think of 2Pi as a constant approximately 6.28 but I'm beginning to realize is that 2Pi of a Sin or Cos function can be greater or less that 6.28 of whatever unit be used. For example if we squeeze a sinusoidal function down we can to less than 2Pi as a hard constant of 6.28. Well then it really is 2Pi because it represents a full revolution or a full cycle, but not necessarily 6.28 on the x axis. So how should Pi and 2Pi be regarded? Is it just a measurement of rotation?
(1 vote) • Just out of curiosity, if this was a very rough COS function would it look like -16.35 (PI/30X)+17.85 ? Going from a min height of (0, 1.7) to a max height of (30, 34). plus or minus  