Question

How do you graph `f(x)=2sin(3x)`?

Answer 1

See below:

Explanation:

Let's start with the graph for `f(x)=sinx`:

graph{sinx [-6.25, 6.25, -3, 3]}

I've set the graph to be `-2pi<=x<=2pi`. The graph intersects the x-axis at `x=pi+npi` where n is an integer. The max is `y=1` and occurs at `x=pi/2+n2pi` and min at `y=-1` at `x=(3pi)/2+n2pi`.

So how does the graph change with `y=2sin(3x)`?

The first thing I'll look at is the 2. That will change the expansion along the y-axis by a factor of 2 - or in other words, each max and min will be 2 and `-2` respectively.

The other thing that happens is that the `3x` part of this will compress the graph along the x-axis by a factor of 3 - so what had been happening at `pi` now happens at `pi/3`.

When we put these two changes together, we get this:

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

The graph is still set at `-2pi<=x<=2pi`. The graph intersects the x-axis at `x=pi/3+npi/3` where n is an integer. The max is `y=2` and occurs at `x=pi/6+n(2/3)pi` and min at `y=-2` at `x=(pi)/2+n(2/3)pi`.