How do you graph #y=3cos(4x)#?

This is what the graph of y=3 cos(4x) looks like:
graph{y=3 cos (4x) [-10, 10, -5, 5]}

This is how we get this graph:

1) Start with the graph of cos(x):
graph{cos x [-10, 10, -5, 5]}

2) Now, the first thing to notice is that our equation has been modified in two ways y = 3 cos( 4 x)

3) First, let's deal with the 4 inside the parenthesis. Numbers inside the parenthesis affect the period of a graph, or in other words, how much width does it take for the graph to repeat.

The period is given by the following formula where B is the constant inside of the parenthesis:
#omega=(2pi)/B=(2pi)/(4)=pi/2#

This means that the period of the graph is 4 times more compressed than the normal graph, which looks like this:
graph{y=cos(4x) [-10, 10, -5, 5]}

4) Now lets take a look at 3 in the equation: y = 3 cos( 4 x).
Numbers in the front of the equation affect the amplitude of a graph, or how tall or short the graph is. The relationship here is direct, the larger the constant the larger the graph.

In this case, it means the the graph would be 3 times as tall as a normal cos graph. So, if we apply that to our previous graph we get the anser:
graph{y=3 cos (4x) [-10, 10, -5, 5]}