How do you graph #y=3cos2pix# and include two full periods?

When graphing any sinusoidal graph except for #tanx# and #cotx# the period is #(2pi)/w# where #w# is the value next to #x# in this case #2pi#.

So, our period is represented as:

Per. #\ T = (2pi)/w#

Per. #\ T = (2pi)/(2pi)#

Per. #\ T = 1#

Now let's find our amplitude, which is always the number to the left of the trigonometric function, in this case, #3#. This means that instead of having a vertical range of #[-1, 1]# our graph will range from #[-3, 3]#.

Once we have this information, the easiest way to graph for two periods is to go out four points on the graph's #x-"axis"# and mark our period: #1#.

So it would look like this: Origin, point, point, point, 1

(I know it's difficult to visualize but bear with me until the end)

Then take half of the period and put it at half of that point's distance:

Origin, point, #1/2#, point, #1# (since #1/2# is half of #1#)

Then do it again:

Origin, #1/4#, #1/2#, point, #1#

Now that we know each increment is by one fourth, we can find the missing point

Origin, #1/4#, #1/2#, #3/4#, #1#.

And since we know that positive #cos# graphs always start at #(0, "amplitude")#, we can now graph our equation, as seen below:

Hopefully, this helps you visualize it

For two full periods, just keep moving by #1/4# on the #x#-axis and #3# up or down on the #y#-axis following the up and down pattern, it's very consecutive and easy to follow.