Question

How do you graph `r=2sin3theta`?

Answer 1

Start with `y=sinx` then adjust.

Explanation:

First it's good to remember the basic graph of `y=sinx`:

graph{sinx [-6.25, 6.25, -2.2, 2.2]}

The graph spans `-2pi<=theta<=2pi`

Now let's make the needed adjustments.

First there's the 2. It is going to take the result of the sin function and increase it by a factor of 2. So the maximum and minimum on the y-axis will go from 1 to 2. That will look like this:

graph{2sinx [-6.25, 6.25, -2.2, 2.2]}

And now the `3theta`. This is going to make the result of any point along the x-axis compress in. For instance, at `theta=pi`, the function is going to return what we'd expect at `3pi`. This adjustment, without the first one we did for the 2, looks like this:

graph{sin(3x) [-6.25, 6.25, -2.2, 2.2]}

Put it all together and we get:

graph{2sin(3x) [-6.25, 6.25, -2.2, 2.2]}