Step by step
The question
Question: graph y = csc 2x.
1 Write it as 1 over
csc x means 1 over sin x.
csc goes with sin: each one is sin turned upside down.
So y = 1 over sin 2x.
Careful!
csc x is 1 over sin x.
It is not the inverse sine, sin to the minus 1.
That is a different thing.
2 Draw the helper graph
Draw the grid.
A mark every π/4: a quarter of the period.
Our helper is y = sin 2x.
Its amplitude is 1, and its period is 2π over 2: π.
We draw the helper wave lightly, dashed.
csc will be built from it.
3 Find the gaps
Where the helper is 0, csc is 1/0.
We can never divide by 0: there is a gap.
We draw a dashed line there: an asymptote.
The helper crosses the x-axis at 0, π/2 and π, and so on.
4 Find the turning points
Where the helper is at its top or bottom, sin 2x is 1 or −1.
1/1 is 1, and 1 over −1 is −1: the value stays the same.
csc touches the helper there.
5 Draw the U shapes
Between two gaps, the graph is a U.
It starts at the turning point.
As the helper gets closer to 0, 1 over it gets bigger and bigger: up to the gap.
Where the helper is below the axis, the U opens down.
6 Check
Let's check one more point: x = π/12.
sin π/6 is one half.
1 over one half is 2.
So y is 2: the point is on the U.
Right!
Let's look back.
First: write it as 1 over.
Then: draw the helper graph.
Then: find the gaps.
Then: find the turning points.
Then: draw the U shapes.
Last: check.