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.