Step by step
The question
Question: graph y = tan 2x.
The x values are in degrees.
1 Read A and B
A tangent graph can be written like this: y = A tan Bx.
Here A is 1: the number in front.
B is 2: 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 = 90°: straight up on the unit circle, where across is 0.
Here the angle is 2x.
2x is 90° when x is 45°.
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: 180°.
B squeezes it: the period is 180° over B.
Careful!
B squeezes the wave, so the period gets shorter.
We divide by B.
We do not multiply.
B is 2.
So the period is 90°: one full piece every 90°.
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 −22.5° to 22.5°, a quarter period at a time: 22.5°.
A quarter of 90° is 22.5°.
tan at these points: −1, 0, 1.
Like tan −45°, tan 0° and tan 45°.
Each y is A times that: the same numbers.
5 Draw the graph
Draw the grid.
The x-axis is in degrees: a mark every 22.5°.
The gaps: dashed lines at x = −45° and x = 45°.
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, 90°: the same piece again after each gap.
6 Check
Let's check one point from the table: x = 22.5°.
Put it in the equation.
The inside is 45°, and tan 45° is 1.
So y is 1, 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.