# Graph of Rational Functions

How to graph a rational function by using its asymptotes: basic graphs, examples, and their solutions.

## Graph: *y* = 1/*x*

This is the graph of *y* = 1/*x*.

The graph is at the right top and left bottom of the axes.

## Asymptotes

The asymptotes are the lines

that the graph follows.

The asymptotes of a rational function are found by

setting (denominator) = 0 and (left side) = 0.

For example,

the asymptotes of *y* = 1/*x* are*x* = 0 and *y* = 0.

The denominator cannot be 0.

So *x* = 0 is the asymptote.

The left side cannot be 0.

(The right side's numerator is not 0.)

So *y* = 0 is the asymptote.

## Graph: *y* = -1/*x*

This is the graph of *y* = -1/*x*.

The asymptotes are *x* = 0 and *y* = 0.

(the same as *y* = 1/*x*)

But the graph is

at the left top and right bottom of the asymptotes.

(different from *y* = 1/*x*)

This is because

the sign of the right side's numerator is (-): -1.

## Example 1

Make the rational function in standard form.

Move +2 to the left side.

Find the asymptotes.*x* - 1 = 0, *y* - 2 = 0

So *x* = 1, *y* = 2.

This is the graph of *y* - 2 = 4/*x*.

The right side's numerator (4) is (+).

So the graph is

at the right top and left bottom of the asymptotes.

## Example 2

Make the rational function in standard form.

Move -1 to the left side.

Change 1/(3 - *x*) to -1/(*x* - 3)

to make the coefficient of *x* to 1.

Find the asymptotes.*x* - 3 = 0, *y* + 1 = 0

So *x* = 3, *y* = -1.

This is the graph of *y* + 1 = -1/(*x* - 3).

The right side's numerator (-1) is (-).

So the graph is

at the left top and right bottom of the asymptotes.

## Example 3

Make the rational function in standard form.

To remove 3*x* in the right side's numerator,

write 3(*x* - 2),

and write +3⋅2 to undo the former change (-3⋅2).

Split the fraction into

the factored numerator part, 3(*x* - 2)/(*x* - 2),

and the remaining part, +5/(*x* - 2).

Cancel (*x* - 2) on both sides.

Move 3 to the left side.

Then the rational function is in standard form.

Find the asymptotes.*x* - 2 = 0, *y* - 3 = 0

So *x* = 2, *y* = 3.

This is the graph of *y* - 3 = 5/(*x* - 2).

The right side's numerator (5) is (+).

So the graph is

at the right top and left bottom of the asymptotes.

## Graph: *y* = 1/*x*^{2}

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

Its asymptotes are *x* = 0 and *y* = 0.

The graph is

at the right and left top of the asymptotes.

1/*x*^{2} is always (+).

So *y* is always (+).

So the graph is

at the top of the horizontal asymptote (*y* = 0).