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### 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.

## Want to join the conversation?

• 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?
• 2pi on a unit circle represents a full rotation around the unit circle, the circumference of a circle is 2pi*r, which is 2pi in a unit circle. Radians are a measure of an angle with a value of how many radii you move along the circumference of the circle. The 2pi representing a full cycle/revolution only applies to a unit circle. 2pi's value of 6.28 does not change and is still 6.28. If you "shrink" a sinusoidal function that means it takes less of a input value to reach the same amount of rotation than before, you can still travel 2pi around the circle but you probably wont end up with the same number of full rotations. That is the period, how many rotations you do traveling 2pi around the circumference. If the period is a bigger number you make many rotations per 2pi travel, if it is smaller you make less or no rotations per 2pi travelled. Yes, to that last question.

FINAL REMINDER:
2pi's value does not change it is always approximately 6.28, the thing that is changing the period, the amount of rotations per 2pi.
• I am super confused on this topic, any help on this?
(1 vote)
• This topic observes the periodicity of functions, or studying the "periods" of graphs. We mostly observe the maximums (peaks) and minimums (low points).

In the first example, we observe a sine wave. The question asks how long Divya completes a full revolution around the ferris wheel. We notice that when `t=0`, it is approximately equal to `h=2`. We also notice that it reaches to a maximum of around `h=34`. We can draw out that a full rotation starts at `h=2` and ends back on `h=2`. Now we have this information, we can see that on the `t` axis, a full rotation is around 60 seconds long. This is an explanation on what you should see on the practices.