Step by step
The question
Question: graph y = cot 2x.
1 Why cot has gaps
cot x is cos x over sin x: tan turned upside down.
When sin x is 0, we would divide by 0.
That is never allowed: a gap.
sin x is 0 at x = 0, at π, and every π after that: on the x-axis of the unit circle.
Here the angle is 2x.
2x is π when x is π/2.
2 Find the period
Like tan, cot repeats every half turn: π over B.
B is 2: the period is π/2.
Careful!
Each piece of cot goes down from left to right.
tan goes up.
They are upside down of each other.
3 Make the key points
Between the gaps at 0 and π/2, three key points.
x goes a quarter period at a time: π/8.
cot at these points: 1, 0, −1.
Like cot 45°, cot 90° and cot 135°.
y is the same.
4 Draw the graph
Draw the grid: a mark every π/8.
The gaps: dashed lines at x = 0 and every π/2.
Plot the key points.
Join them with a smooth curve, going down.
Near the dashed lines it shoots up and down.
The same piece again after each gap.
5 Check
Let's check one point from the table: x = π/8.
The inside is π/4, and cot of it is 1.
So y is 1, like in the table.
Right!
Let's look back.
First: why cot has gaps.
Then: find the period.
Then: make the key points.
Then: draw the graph.
Last: check.