# Rational Function: Graph

How to graph a rational function by finding its asymptotes: graph, 3 examples, and their solutions.

## Graph

### y = a/x (a > 0)

This is the graph of y = a/x

(a > 0).

If the numerator is plus (a > 0),

the graph is

on the right top and the left bottom

of the axes.

### y = a/x (a < 0)

This is the graph of y = a/x

(a < 0).

If the numerator is minus (a < 0),

the graph is

on the left top and the right bottom

of the axes.

## Asymptote

### Graph

The graph of y = a/x

has two asymptotes:

a vertical asymptote

and a horizontal asymptote.

(An asymptote is a line that the graph follows.)

### Formula

To find the asymptotes:

1) Set the denominator, x, 0.

x = 0.

2) Set the fraction part, a/x, 0.

Then y = 0.

## Example 1

### Example

### Solution

To graph the rational function,

first find the asymptotes.

The denominator is (x - 1).

So set

x - 1 = 0.

Move -1 to the right side.

Then x = 1.

Next, the fraction part is 4/(x - 1).

Set this part 0.

Then y = 0 + 2.

Then y = 2.

So the asymptotes are

x = 1 and y = 2.

Then graph y = 4/(x - 1) + 2

on a coordinate plane.

First draw the asymptotes

x = 1 and y = 2.

Next, see the numerator of 4/(x - 1).

The numerator 4 is plus.

So draw the graph

on the right top and the left bottom

of the asymptotes.

This is the graph of y = 4/(x - 1) + 2.

## Example 2

### Example

### Solution

The coefficient of x should be 1.

So change 1/(3 - x)

to -1/(x - 3).

(Multiply -1

to both of the numerator and the denominator.)

To graph the rational function,

first find the asymptotes.

The denominator is (x - 3).

So set

x - 3 = 0.

Move -3 to the right side.

Then x = 3.

Next, the fraction part is -1/(x - 3).

Set this part 0.

Then y = 0 - 1.

Then y = -1.

So the asymptotes are

x = 3 and y = -1.

Then graph y = -1/(x - 3) - 1

on a coordinate plane.

First draw the asymptotes

x = 3 and y = -1.

Next, see the numerator of -1/(x - 3).

The numerator -1 is minus.

So draw the graph

on the left top and the right bottom

of the asymptotes.

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

## Example 3

### Example

### Solution

The numerator is 3x - 5.

To find the asymptotes,

the numerator should be a constant.

(= no variable)

So change the numerator to a constant.

First write the fraction bar

and the denominator (x - 2).

The x term of the numerator is 3x.

So write 3(x.

The denominator is (x - 2).

So write -2).

3(-2) is added.

To undo this change,

write +3⋅2.

Write -5.

So [3x - 5]/(x - 2)

= [3(x - 2) + 3⋅2 - 5]/(x - 2).

+3⋅2 = +6

+6 - 5 = +1

Split this into two parts:

3(x - 2) and +1.

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

Simplify a Rational Expression

So the given function is

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

Now the numerator of the fraction part

is a constant.

So you can find the asymptotes.

Find the asymptotes.

The denominator is (x - 2).

So set

x - 2 = 0.

Move 2 to the right side.

Then x = 2.

Next, the fraction part is 1/(x - 2).

Set this part 0.

Then y = 0 + 3.

So y = 3.

So the asymptotes are

x = 2 and y = 3.

Then graph y = 1/(x - 2) + 3

on a coordinate plane.

First draw the asymptotes

x = 2 and y = 3.

Next, see the numerator of 1/(x - 2).

The numerator 1 is plus.

So draw the graph

on the right top and the left bottom

of the asymptotes.

This is the graph of y = 1/(x - 2) + 3.