First off, rearanging terms gives, #y=-3cos4x-1#.
This means you will have a negative cosine curve, and because of the #-1# the curves center will be moved down by 1.
Also notice how for all #x#, #cos(x)# is between #-1,1#, and because #cosx# is multiplied by #3# in #y# it will now oscillate between #-3,3#.
We now know the center and its max/min value. Now for the shape.
Start with plugging in values into #y# that will give us known values for cosine which is:
#cos(0)=1#,
#cos(\pi/2)=0#,
#cos(\pi)=-1#,
#cos({3\pi}/2)=0#,
#cos(2\pi)=1#.
(These values can easily be seen in the unitcircle where cosine is the x value.)
We do only need these values to know how the curve will look for all #x# as cosine is periodic and will repeat forever. This will also apply to the negative side as cosine is an even function (#cos(-x)=cos(x)#).
So we want to insert som #x# into #y# such that we get these known values. Lets start off with getting #cos(0)=1# in #y#.
#4x=0=>y(0)=-4#
#4x=\pi/2=>y(\pi/8)=-1#
#4x=\pi=>y(\pi/4)=2#
#4x={3\pi}/2=>y({3\pi}/8)=-1#
#4x=2\pi=>y(\pi/2)=-4#
Now insert these #(x,y)# pairs into a cordinate system and connect the dots! Note, as its the negative cosine we want to plot, the function will start by increasing from #x=0# instead of decreasing.
