In all three cases the function is of the form `y=af(b(x-h))+k`
a gives the amplitude for sinusoids which is the vertical stretch/compression of the parent function f(x). If a<0 the graph is reflected over the horizontal axis.
b gives the horizontal compression/stretch. It is associated with the period as `b=(2pi)/p` . If b<0 the graph is reflected over a vertical axis.
h gives the horizontal translation or phase shift.
k gives the vertical translation -- this is the midline for the sinusoids.
(a) Graph `y=3cos(2x)+1` for `0<x<360^@`
The parent function is y=cosx. The amplitude is 3, the period is `pi` or `180^@` , there is no phase shift or reflection and the midline is 1 (meaning the maximum is 4 and the minimum is -2.)
** The vertical lines are spaced at `90^@` .
(b) Graph `y=-sin(x/3)-2` for `-pi/2<=x<=pi/2` :
The amplitude is 1 but the graph is reflected across the horizontal axis. The period is `6pi` . The midline is y=-2 so the maximum is -1 and the minimum is -3.
Here it is as part of the complete period:
(c) Graph `y=-tan(x/2)+2`
The graph will be reflected over the horizontal axis. Since the period of the tangent is `pi` , the period for this function is `2pi` . There is a vertical shift of 2 units up.