Graph of the Harmonic Oscillation `y=Asin(omega x+alpha)`

Trigonometric functions are used to describe oscillatory processes (for example, oscillation of pendulum). One of the most important formulas that describes such processes is `y=Asin(omega x+alpha)`, which is called formula of harmonic (or sinusoidal) oscillations. `A` is called amplitude of oscillation. `omega` is called frequency of oscillation. The bigger `omega` the more oscillations per unit of time. `alpha` is called starting phase of oscillation.

Example. Draw graph of the function `y=2sin(x/3-pi/6)`.

Let's first rewrite function as `y=2sin(1/3(x-pi/2))`.

Now we can obtain required graph from the graph of the function `y=sin(x)` in following four steps:

  1. Draw graph of the function `y=sin(x)`.graph of harmonic oscillations
  2. Move obtained graph `pi/2` units to the right.
  3. Compress obtained graph to y-axis with coeffcient `1/3` (actually this means to stretch graph from y-axis with coefficient 3).
  4. Stretch obtained graph from x-axis with coefficient 2.

Obtained graph is graph of the function `y=2sin(x/3-pi/6)` (see figure).