# Graphing Tangent Functions

How to graph tangent functions: graph, period, example, and its solution.

## Graph: One Cycle

This is one cycle of y = tan x.

As x changes from -π/2 to π/2,
tan x moves -∞ → 0 → ∞.

It passes through the central point: (0, 0).

x = ±π/2 are the vertical asymptotes.

This table shows the coordinates of the blue points.

## Period, Asymptotes

This is the graph of y = tan x.

For every π, the blue cycle repeats.

So the period of y = tan x is π.

And the vertical asymptotes are x = + π/2.

For every π, the blue cycle repeats.

So if y = tan x is under the translation
(x, y) → (x + π, y),

its image, y = tan (x - π),
will exactly cover y = tan x.

Translation of a function

So tan x = tan (x - π).

This property is true for any periodic functions:
f(x) = f(x - [period]).

## Formula: y = tan bx

y = tan [1/2]x

Period: π / |1/2|

y = tan 1x

Period: π / |1|

y = tan 2x

Period: π / |2|

So, for y = tan bx,Period: π / |b|

## Example

y = tan (x - π/2) + 1 is under the translation
(x, y) → (x + π/2, y + 1).

Translation of a function

Roughly draw one cycle of the function
with its details.

The 'central point' is (π/2, 1).

The period is π / |1| = π.

So the half-period is π/2.

So the left asymptote is x = π/2 - π/2 = 0

And the right asymptote is x = π/2 + π/2 = π.

Use these values to sketch one cycle.

Mark the central point: (π/2, 1).

Draw the asymptotes:
x = 0 (y-axis, already existing),
x = π.

Draw one cycle between the asymptotes
that passes the central point.