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(π2)=0,
cos(π)=−1,
cos(3π2)=0,
cos(2π)=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=π2⇒y(π8)=−1
4x=π⇒y(π4)=2
4x=3π2⇒y(3π8)=−1
4x=2π⇒y(π2)=−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.
