How to graph y= -2cos(4x) ?

This is a sinusoid, and you may graph it in the normal way by substituting values of x in the function. The general equation suggests a simple way to graph the function.

A sinusoid is curve similar to the sine function but possibly shifted in phase, period, amplitude, or any combination thereof. The general sinusoid of amplitude a, angular frequency omega (and period 2pi/omega), and phase c is given by

f(x)=asin(omegax+c).
That implies (in the mathematical sense) that you may take a graph of Y = sin (x) and shift and scale the coordinate system to fit your equation.

The line representing the y axis is positioned where y is a minimum (-2) and x is zero. The maximum and minimum values are 2 and -2 on this line may be used to scale the y values. Whereas the full variation of x values for one complete harmonic range the sin function takes place over 2 pi radians, your function occurs in 1/4 of that and is represented in pi/2 radians. Mark it off and scale to the crossings of the x axis and the max and min values.

It's much easier to show than to explain without a picture in words. I wish we had a chalk board. Perhaps I should make a YouTube video.

I also did it in Excel, with the formula, =-2*(COS(4*A1)) copied down to lower cells to get values for x and y. I inserted a graph to check it. Excel takes angular values in radians, so account for that.

As I noted above, the shape of the graph is similar to Y=cos(x) except that it goes between -2 and plus 2 and begins at -2 instead of 1. The value, 4x means that the oscillation is four times faster than if it were simply x.

Anyway, for 3.1 radians or two full oscillations, here is a table of X and Y with X in radians.
0 -2.000000000
0.1 -1.842121988
0.2 -1.393413419
0.3 -0.724715509
0.4 0.058399045
0.5 0.832293673
0.6 1.474787431
0.7 1.884444681
0.8 1.996589552
0.9 1.793516833
1 1.307287242
1.1 0.61466574
1.2 -0.174997967
1.3 -0.937033343
1.4 -1.551131757
1.5 -1.920340573
1.6 -1.986369838
1.7 -1.738794981
1.8 -1.216702629
1.9 -0.502519685
2 0.291000068
2.1 1.038577308
2.2 1.622186028
2.3 1.949687243
2.4 1.969375712
2.5 1.678143058
2.6 1.121968515
2.7 0.388659813
2.8 -0.406009728
2.9 -1.13657926
3 -1.687707917
3.1 -1.972384605

You may enter these values by hand or copy and paste this table into Excel as numbers instead of using the formula. The table worked fine when I checked in the edit window, but copy and paste changed the formatting when I "submitted" it. You need to adjust some entries for alignment before creating and inserting the graph.

I took a nice mathematics course about 1952 called Analytic Geometry, I don't know it it is offered today. It was great for visualizing a graph from the formula of the function.

http://en.wikipedia.org/wiki/Analytic_geometry

Good luck and have fun with the graphing,

google it: https://support.google.com/websearch/answer/3284611?#plotting