Graph y = tan x Easy

The question

Question: graph y = tan x.

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 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: the same numbers.

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 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.

▶ Watch it solved

Graph y = tan 2x Medium

The question

Question: graph y = tan 2x.

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 = π/2: straight up on the unit circle, where across is 0.

Here the angle is 2x.

2x is π/2 when x is π/4.

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 2.

So the period is π/2: one full piece every π/2.

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/8 to π/8, a quarter period at a time: π/8.

A quarter of π/2 is π/8.

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 radians: a mark every π/8.

The gaps: dashed lines at x = −π/4 and x = π/4.

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/2: the same piece again after each gap.

6 Check

Let's check one point from the table: x = π/8.

Put it in the equation.

The inside is π/4, and tan π/4 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.

▶ Watch it solved

Graph y = 2 tan 2x Harder

The question

Question: graph y = 2 tan 2x.

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 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 = π/2: straight up on the unit circle, where across is 0.

Here the angle is 2x.

2x is π/2 when x is π/4.

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 2.

So the period is π/2: one full piece every π/2.

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/8 to π/8, a quarter period at a time: π/8.

A quarter of π/2 is π/8.

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/8.

The gaps: dashed lines at x = −π/4 and x = π/4.

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/2: the same piece again after each gap.

6 Check

Let's check one point from the table: x = π/8.

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.

▶ Watch it solved