Question

How do you make a sin graph with a period of 4pi?

Answer 1

I would manipulate the numerical coefficient inside the argument of the sine:

Explanation:

If you have the normal sine function (period`=2pi`):
`y=sin(x)`
the coefficient conected with the period is the `1` multiplying the argument as in: `y=sin(x)=sin(1*x)`;
this coefficient (call it `c`) helps you to "see" the period of your function that can be evaluated as:
`period=(2pi)/c=(2pi)/1=2pi`

Now if you want a period of `4pi` you need that `c=1/2` so that you get:
`period=(2pi)/c=(2pi)/(1/2)=2pi*2=4pi~~12.6rad`

Finally the complete function will be:

`y=sin(c*x)=sin(1/2x)`

and graphically:
graph{sin(1/2x) [-18.02, 18.03, -9.01, 9.01]}