# Hyperbola: Asymptotes

How to find the asymptotes of a hyperbola: formulas, examples, and their solutions.

## Formula: Asymptotes of *x*^{2}/*a*^{2} - *y*^{2}/*b*^{2} = 1

The asymptote is a line the graph follows.

For the hyperbola *x*^{2}/*a*^{2} - *y*^{2}/*b*^{2} = 1,

the equations of the asymptotes are*y* = ±(*b*/*a*)*x*.

## Example 1: Asymptotes of *x*^{2}/9 - *y*^{2}/16 = 1

Change the equation in standard form.

9 = 3^{2}

16 = 4^{2}

So *x*^{2}/3^{2} - *y*^{2}/4^{2} = 1.

*a* = 3*b* = 4

So the asymptotes are*y* = ±(4/3)*x*.

This is the graph of the hyperbola *x*^{2}/3^{2} - *y*^{2}/4^{2} = 1

and its asymptotes *y* = ±(4/3)*x*.

You can see that

as *x* goes to ∞ or -∞,

the hyperbola follows the asymptotes.

## Formula: Asymptotes of *y*^{2}/*b*^{2} - *x*^{2}/*a*^{2} = 1

For the hyperbola *y*^{2}/*b*^{2} - *x*^{2}/*a*^{2} = 1,

the equations of the asymptotes are also*y* = ±(*b*/*a*)*x*.

## Example 2: Asymptotes of *x*^{2} - *y*^{2} = -4

Change the equation in standard form.

Divide both sides by -4.

Switch the order of the terms

in the left side.

4 = 2^{2}

So *y*^{2}/2^{2} - *x*^{2}/2^{2} = 1.

*b* = 2*a* = 2

So the asymptotes are*y* = ±(2/2)*x*.

2/2 = 1

So the asymptotes are *y* = ±*x*.

This is the graph of the hyperbola *y*^{2}/2^{2} - *x*^{2}/2^{2} = 1

and its asymptotes *y* = ±*x*.

You can see that

as *x* goes to ∞ or -∞,

the hyperbola follows the asymptotes.