Step by step
The question
Question: graph y = 2 tan 3x.
1 Read A and B
A tangent graph can be written like this: y = A tan Bx.
Here A is 2: the number in front.
B is 3: the number in front of x.
2 Why tan has gaps
tan x is sin x over cos x.
When cos x is 0, we would divide by 0.
That is never allowed.
So tan has no value there: a gap.
cos x is 0 at x = π/2: straight up on the unit circle, where across is 0.
Here the angle is 3x.
3x is π/2 when x is π/6.
At each gap we draw a dashed line, called an asymptote.
The graph gets closer and closer to it, but never touches it.
3 Find the period
The tangent graph repeats every half turn: π.
B squeezes it: the period is π over B.
Careful!
B squeezes the wave, so the period gets shorter.
We divide by B.
We do not multiply.
B is 3.
So the period is π/3: one full piece every π/3.
4 Make the key points
We use a table.
tan is 0 in the middle, at x = 0, and the piece reaches up and down from there.
x goes from −π/12 to π/12, a quarter period at a time: π/12.
A quarter of π/3 is π/12.
tan at these points: −1, 0, 1.
Like tan −45°, tan 0° and tan 45°.
Each y is A times that: times 2.
5 Draw the graph
Draw the grid.
The x-axis is in radians: a mark every π/12.
The gaps: dashed lines at x = −π/6 and x = π/6.
Plot the key points from the table.
Join them with a smooth curve.
Near the dashed lines it shoots up and down.
The graph repeats every period, π/3: the same piece again after each gap.
6 Check
Let's check one point from the table: x = π/12.
Put it in the equation.
The inside is π/4, and tan π/4 is 1.
So y is 2, like in the table.
So the graph is right!
Let's look back.
First: read A and B.
Then: why tan has gaps.
Then: find the period.
Then: make the key points.
Then: draw the graph.
Last: check.