# 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* = *nπ* + *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 1*x*

Period: *π* / |1|

*y* = tan 2*x*

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.