Step by step
The question
Question: graph y = −3 tan x.
1 Read A and B
A tangent graph can be written like this: y = A tan Bx.
Here A is −3: the number in front.
B is 1: 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.
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.
B is 1.
So the period is π: one full piece every π.
Careful!
The tangent graph goes up and down forever.
So it has no amplitude.
A only stretches it: at a quarter period, y is A.
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/4 to π/4, a quarter period at a time: π/4.
A quarter of π is π/4.
tan at these points: −1, 0, 1.
Like tan −45°, tan 0° and tan 45°.
Each y is A times that: times −3.
5 Draw the graph
Draw the grid.
The x-axis is in radians: a mark every π/4.
The gaps: dashed lines at x = −π/2 and x = π/2.
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: the same piece again after each gap.
6 Check
Let's check one point from the table: x = π/4.
Put it in the equation.
The inside is π/4, and tan π/4 is 1.
So y is −3, 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.