I need some help understanding how to find the amplitude, period, vertical translation, and phase shift of a function.
The standard equation is given as y=Asin(Bx-C)+D. This equation applies to sine and cosine and it’s respective reciprocals.
A represents the amplitude of you equation. A is always positive due to absolute value |A|
B is the period which represents how long is one cycle of the wave function.
C is the phase shift which just represents where the graph starts (wave graphs are infinite but this indicates where to start for your equation js.)
D is self explanatory, vertical translation, up or down how many units.
Now referencing the example you gave: y=3cos2(x-pi/4)
A - is 3 as given by the standard equation.
B - the concept of the unit circle is that one cycle is 2pi. Now to find how long one cycle is, you would have to divide 2pi by the value of B (2pi/B). The period will be 2pi, as 2pi/1 = 2pi
(*Graphing wave functions requires knowing increments. To find increments, divide the period by 4. For this graph the increments are pi/2, as 2pi/4 = pi/2. It helps to imagine pi as 1 if it confuses you.)
C - To find C you simply divide the value of B (C/B) (Not the period or increments, ONLY B)(Notice how in the standard equation is (Bx-C). So if you ever see an equation where (Bx+C) occurs, C is negative). So in your equation B is 1 so POSITIVE pi/4 divided by 1 is pi/4
D - Phase shift is obvious, a positive value is up the number of units and negative is down the number of units.