# Graphing Rational Functions

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

## Graphs: *y* = *a*/*x*

This is the graph of the rational function*y* = *a*/*x* (*a* > 0).

The numerator of the right side (*a*) is (+).

Then the graph is

at the [right top] and the [left bottom]

of the axes (= asymptotes).

(Asymptotes: the line that the graph follows.

You'll learn about the asymptotes below.)

This is the graph of the rational function*y* = *a*/*x* (*a* < 0).

The numerator of the right side (*a*) is (-).

Then the graph is

at the [left top] and the [right bottom]

of the axes (= asymptotes).

## Asymptotes of *y* = *a*/*x*

The asymptotes of a rational function are found by

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

So the asymptotes of *y* = *a*/*x* are

[*x* = 0] and [*y* = 0].

The denominator cannot be 0.

So [*x* = 0] is the asymptote.*a* is not 0.

So the left side cannot be 0.

So [*y* = 0] is the other asymptote.

By finding the asymptotes,

you can draw the graph of a rational function.

## Example 1: Graph *y* = 4/(*x* - 1) + 2

Move +2 to the left side.

Then *y* - 2 = 4/(*x* - 1).

To find the first asymptote of*y* - 2 = 4/(*x* - 1),

set (denominator) = 0.

Then *x* - 1 = 0.

So the asymptote is [*x* = 1].

To find the other asymptote of*y* - 2 = 4/(*x* - 1),

set (left side) = 0.

Then *y* - 2 = 0.

So the other asymptote is [*y* = 2].

Graph *y* - 2 = 4/(*x* - 1)

on the coordinate plane.

First, lightly draw the asymptotes

[*x* = 1] and [*y* = 2].

The numerator of 4/(*x* - 1) is (+).

So draw two curves

at the [right top] and the [left bottom]

of the asymptotes.

## Example 2: Graph *y* = 1/(3 - *x*) - 1

Change the coefficient of *x* to 1.

So change 1/(3 - *x*) to -1/(*x* - 3).

Move -1 to the left side.

Then *y* + 1 = -1/(*x* - 3).

To find the first asymptote of*y* + 1 = -1/(*x* - 3),

set (denominator) = 0.

Then *x* - 3 = 0.

So the asymptote is [*x* = 3].

To find the other asymptote of*y* + 1 = -1/(*x* - 3),

set (left side) = 0.

Then *y* + 1 = 0.

So the other asymptote is [*y* = -1].

Graph *y* + 1 = -1/(*x* - 3)

on the coordinate plane.

First, lightly draw the asymptotes

[*x* = 3] and [*y* = -1].

The numerator of -1/(*x* - 3) is (-).

So draw two curves

at the [left top] and the [right bottom]

of the asymptotes.

## Example 3: Graph *y* = (3*x* - 1)/(*x* - 2)

The numerator of the right side

should only have a constant.

So, to cancel the 3*x* term,

change 3*x* to 3(*x* - 2).

(*x* - 2) is the denominator.

Then, to undo -2,

write +3⋅2.

Split the right side fraction.

Then *y* = 3(*x* - 2)/(*x* - 2) + (3⋅2 - 1)/(*x* - 2).

3(*x* - 2)/(*x* - 2) = 3

3⋅2 = 6

Move 3 to the left side.

6 - 1 = 5

Then *y* - 3 = 5/(*x* - 2).

To find the first asymptote of*y* - 3 = 5/(*x* - 2),

set (denominator) = 0.

Then *x* - 2 = 0.

So the asymptote is [*x* = 2].

To find the other asymptote of*y* - 3 = 5/(*x* - 2),

set (left side) = 0.

Then *y* - 3 = 0.

So the other asymptote is [*y* = 3].

Graph *y* - 3 = 5/(*x* - 2)

on the coordinate plane.

First, lightly draw the asymptotes

[*x* = 2] and [*y* = 3].

The numerator of 5/(*x* - 2) is (+).

So draw two curves

at the [right top] and the [left bottom]

of the asymptotes.